{"id":16989,"date":"2022-03-05T16:34:05","date_gmt":"2022-03-05T11:04:05","guid":{"rendered":"https:\/\/envisionmathanswerkey.com\/?p=16989"},"modified":"2022-03-14T17:43:50","modified_gmt":"2022-03-14T12:13:50","slug":"envision-math-common-core-grade-6-answer-key-topic-4","status":"publish","type":"post","link":"https:\/\/envisionmathanswerkey.com\/envision-math-common-core-grade-6-answer-key-topic-4\/","title":{"rendered":"Envision Math Common Core Grade 6 Answer Key Topic 4 Represent And Solve Equations And Inequalities"},"content":{"rendered":"

Go through the\u00a0enVision Math Common Core Grade 6 Answer Key<\/a>\u00a0Topic 4 Represent And Solve Equations And Inequalities<\/strong>\u00a0regularly and improve your accuracy in solving questions.<\/p>\n

Envision Math Common Core 6th Grade Answers Key Topic 4 Represent And Solve Equations And Inequalities<\/h2>\n

?Topic Essential Question <\/strong>What procedures can be used to write and solve equations and inequalities?
\nAnswer:
\nThe three methods most commonly used to solve systems of equation are substitution, and elimination. Substitution and elimination are simple methods that can effectively solve most systems of two equations in a few straightforward steps
\nThe steps to solve an inequality are:
\na. Add the same number to both sides.
\nb. Subtract the same number from both sides.
\nc. Multiply both sides by the same positive number.
\nd. Divide both sides by the same positive number.
\ne. Multiply both sides by the same negative number and reverse the sign<\/p>\n

3-ACT MATH<\/strong><\/p>\n

\"Envision<\/p>\n

Checking a Bag
\nA large plane flying across the ocean can weigh almost 1 million pounds! The heavier an airplane is, the more fuel it needs for a flight. The cost of fuel has led many airlines to add a weight restriction on luggage.
\nIf you were to fly somewhere, what would you bring? What would you leave at home to minimize the weight of your luggage? Packing light is important, not only to avoid a fee but also to do your part to conserve fuel. Think about this during the 3-Act Mathematical Modeling lesson.
\n\"Envision<\/p>\n

Topic 4 EnVision STEM Project<\/h3>\n

Did You Know?
\nThe design of a bridge depends on factors such as the distance the bridge will cover, the expected number of vehicles that will cross the bridge daily, and the geographic conditions.
\n\"Envision<\/p>\n

Your Task: Design a Bridge
\nNow that you have defined the problem, identified the criteria and constraints, and performed some data collection, it is time to focus on the solution. You and your classmates will continue to be engineers as you brainstorm solutions and develop prototypes for your bridge.
\n\"Envision<\/p>\n

Topic 4 Get Ready!<\/h3>\n

Review What You Know!<\/strong><\/p>\n

Vocabulary<\/strong>
\nChoose the best term from the box to complete each definition.<\/p>\n

    \n
  • algebraic expression<\/li>\n
  • coefficient equation<\/li>\n
  • evaluate<\/li>\n
  • variable<\/li>\n<\/ul>\n

    Question 1.
    \nIn 6x, x is a(n) _________ .
    \nAnswer:
    \nWe know that,
    \nIn 6x,
    \n“6” is called as “Coefficient”
    \n“x” is called as “Variable”
    \nHence, from the above,
    \nWe can conclude that
    \nThe best term that is suitable for the given statement is: Variable<\/p>\n

    Question 2.
    \nx + 5 is an example of a(n) __________
    \nAnswer:
    \nWe know that,
    \nThe expressions are of 2 types. They are:
    \na. Numeric Expressions – The expression that contains only numbers and arithmetic symbols
    \nEx: 6 + 2 – 5
    \nb. Algebraic expressions – The expression that contains variables, numbers,and arithmetic symbols
    \nEx: x + 2 – 5 + y
    \nSo,
    \nx + 5 is an example of “Algebraic expression”
    \nHence, from the above,
    \nWe can conclude that
    \nThe best term that is suitable for the given statement is: Algebraic expression<\/p>\n

    Question 3.
    \n__________ an expression to find its value.
    \nAnswer:
    \nWe know that,
    \n“Evaluate” an expression to find its value
    \nHence, from the above,
    \nWe can conclude that
    \nThe best term that is suitable for the given statement is: Evaluate<\/p>\n

    Question 4.
    \nThe expressions on each side of the equal sign in a(n) __________ are equal.
    \nAnswer:
    \nWe know that,
    \nThe expressions on each side of the equal sign in an “Equation” are equal
    \nHence, from the above,
    \nWe can conclude that
    \nThe best term that is suitable for the given statement is: Equation<\/p>\n

    Equality<\/strong><\/p>\n

    Tell whether the equation is true or false.<\/strong>
    \nQuestion 5.
    \n6 + 2 = 2 + 6
    \nAnswer:
    \nThe given equation is:
    \n6 + 2 = 2 + 6
    \nNow,
    \nWe know that,
    \nAccording to the Commutative Property of Addition,
    \na + b = b + a
    \nHence, from the above,
    \nWe can conclude that
    \nThe given equation is: True<\/p>\n

    Question 6.
    \n2.5 \u2013 1 = 1 – 2.5
    \nAnswer:
    \nThe given equation is:
    \n2.5 – 1 = 1 – 2.5
    \nNow,
    \nWe know that,
    \nAccording to the Commutative Property,
    \na – b \u2260 b – a
    \nHence, from the above,
    \nWe can conclude that
    \nThe given equation is: False<\/p>\n

    Question 7.
    \n\\(\\frac{1}{2}\\) \u00d7 3= 3 \u00d7 \\(\\frac{1}{2}\\)
    \nAnswer:
    \nThe given equation is:
    \n\\(\\frac{1}{2}\\) \u00d7 3 = 3 \u00d7 \\(\\frac{1}{2}\\)
    \nNow,
    \nWe know that,
    \nAccording to the Commutative Property of Multiplication,
    \na \u00d7 b = b \u00d7 a
    \nHence, from the above,
    \nWe can conclude that
    \nThe given equation is: True<\/p>\n

    Question 8.
    \n\\(\\frac{3}{4} \\div 5=\\frac{3}{4} \\times \\frac{1}{5}\\)
    \nAnswer:
    \nThe given equation is:
    \n\\(\\frac{3}{4}\\) \u00f7 5 = \\(\\frac{3}{4}\\) \u00d7 \\(\\frac{1}{5}\\)
    \nNow,
    \nWe know that,
    \nWhen any number is present after the “Division” symbol, then it will turn into its reciprocal when it is converted into “Multiplication” and vice-versa
    \nSo,
    \n\u00f7 5 will convert into \u00d7 \\(\\frac{1}{5}\\)
    \nHence, from the above,
    \nWe can conclude that
    \nThe given equation is: True<\/p>\n

    Question 9.
    \n\\(5 \\div \\frac{1}{3}=\\frac{5}{3}\\)
    \nAnswer:
    \nThe given equation is:
    \n5 \u00f7 \\(\\frac{1}{3}\\) = \\(\\frac{5}{3}\\)
    \nNow,
    \nWe know that,
    \nWhen any number is present after the “Division” symbol, then it will turn into its reciprocal when it is converted into “Multiplication” and vice-versa
    \nSo,
    \n\u00f7 \\(\\frac{1}{3}\\) will be converted into \u00d7 3
    \nHence, from the above,
    \nWe can conclude that
    \nThe given equation is: False<\/p>\n

    Question 10.
    \n\\(\\frac{2}{3} \\times 5=\\frac{10}{15}\\)
    \nAnswer:
    \nThe given equation is:
    \n\\(\\frac{2}{3}\\) \u00d7 5 = \\(\\frac{10}{15}\\)
    \nNow,
    \n\\(\\frac{2}{3}\\) \u00d7 5
    \n= \\(\\frac{2 \u00d7 5}{3}\\)
    \n= \\(\\frac{10}{3}\\)
    \nHence, from the above,
    \nWe can conclude that
    \nThe given equation is: False<\/p>\n

    Expressions<\/strong><\/p>\n

    Evaluate each expression.<\/strong>
    \nQuestion 11.
    \nx – 2 for x = 8
    \nAnswer:
    \nThe given expression is: x – 2
    \nNow,
    \nFor x = 8,
    \nx – 2 = 8 – 2
    \n= 6
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of the given expression is: 6<\/p>\n

    Question 12.
    \n2b for b = 9
    \nAnswer:
    \nThe given expression is: 2b
    \nNow,
    \nFor b = 9,
    \n2b = 2 \u00d7 b
    \n= 2 \u00d7 9
    \n= 18
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of the given expression is: 18<\/p>\n

    Question 13.
    \n3\\(\\frac{3}{4}\\) + y for y = \\(\\frac{5}{6}\\)
    \nAnswer:
    \nThe given expression is: 3\\(\\frac{3}{4}\\) + y
    \nNow,
    \nFor y = \\(\\frac{5}{6}\\),
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of the given expression is: 4\\(\\frac{7}{12}\\)<\/p>\n

    Question 14.
    \n\\(\\frac{15}{x}\\) for x = 3
    \nAnswer:
    \nThe given expression is: \\(\\frac{15}{x}\\)
    \nNow,
    \nFor x = 3,
    \n\\(\\frac{15}{x}\\) = \\(\\frac{15}{3}\\)
    \n= 5
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of the given expression is: 5<\/p>\n

    Question 15.
    \n5.6t for t = 0.7
    \nAnswer:
    \nThe given expression is: 5.6t
    \nNow,
    \nFor t = 0.7,
    \n5.6t = 5.6 \u00d7 t
    \n= 5.6 \u00d7 0.7
    \n= 3.92
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of the given expression is: 3.92<\/p>\n

    Question 16.
    \n4x for x = \\(\\frac{1}{2}\\)
    \nAnswer:
    \nThe given expression is: 4x
    \nNow,
    \nFor x = \\(\\frac{1}{2}\\),
    \n4x = 4 \u00d7 \\(\\frac{1}{2}\\)
    \n= \\(\\frac{4}{2}\\)
    \n= 2
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of the given expression is: 2<\/p>\n

    Order of Operations<\/strong><\/p>\n

    Question 17.
    \nExplain the order in which you should compute the operations in the expression below. Then evaluate the expression.
    \n[(33 \u00f7 3) + 1] – 22<\/sup>
    \nAnswer:
    \nThe given expression is: [(33 \u00f7 3) + 1] – 2\u00b2
    \nNow,
    \nWe know that,
    \nWe will follow the order of operations by uisng the BODMAS rule
    \nWhere,
    \nB – Brackets
    \nO – Of
    \nD – Division
    \nM – Multiplication
    \nA – Addition
    \nS – Subtraction
    \nSo,
    \nIn the given expression,
    \nThe order of expressions we have to compute from the first to the last is:
    \na. 33 \u00f7 3
    \nb. [(33 \u00f7 3) + 1]
    \nc. 2\u00b2
    \nd. [(33 \u00f7 3) + 1] – 2\u00b2
    \nSo,
    \n[(33 \u00f7 3) + 1] – 2\u00b2
    \n= [11 + 1] – 2\u00b2
    \n= 12 – 4
    \n= 8
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of the given expression is: 8<\/p>\n

    Graphing in the Coordinate Plane<\/strong><\/p>\n

    Question 18.
    \nDescribe how to plot point A(-6, 2) on a coordinate plane.
    \nAnswer:
    \nThe given point is: A (-6, 2)
    \nNow,
    \nWhen we observe the given point,
    \nThe x-coordinate of point A is negative and the y-coordinate of point A is positive
    \nSo,
    \nFor point A,
    \nThe x-coordinate is at the left side of the coordinate plane and the y-coordinate is at the top side of the coordinate plane
    \nHence,
    \nThe representation of A (-6, 2) in the coordinate plane is:
    \n\"\"<\/p>\n

    Language Development<\/strong>
    \nUse the graphic organizer to help you understand new vocabulary terms.
    \n\"Envision
    \nAnswer:
    \n\"\"<\/p>\n

    Pick A Project<\/strong><\/p>\n

    PROJECT 4A
    \nIf you were going to try a new exercise, what would it be? Why?
    \nPROJECT: ANALYZE AN EXERCISE ROUTINE
    \n\"Envision<\/p>\n

    PROJECT 4B
    \nWhat is the most interesting book you have read?
    \nPROJECT: WRITE AND ILLUSTRATE A CHILDREN’S BOOK
    \n\"Envision<\/p>\n

    PROJECT 4C
    \nIf you were a carpenter what sorts of things would you build?
    \nPROJECT: MAKE A MODEL OF A STAIRCASE
    \n\"Envision<\/p>\n

    PROJECT 4D
    \nWhat skills would you need if you wanted to move as slowly as a snail?
    \nPROJECT: PLAN A RACE
    \n\"Envision<\/p>\n

    Lesson 4.1 Understand Equations and Solutions<\/h3>\n

    Solve & Discuss It!<\/strong>
    \nUnit cubes are placed on a pan balance. There are 3 cubes on one pan and 9 cubes on the other pan. What can you do to make the pans balance?<\/p>\n

    Model with Math<\/strong>
    \nA pan balance can be used to represent the relationship between two quantities. You can write an equation with a variable to show this relationship.
    \nI can… determine if a value for a variable makes an equation true.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nUnit cubes are placed on a pan balance. There are 3 cubes on one pan and 9 cubes on the other pan.
    \nNow,
    \nLet the number of cubes on one pan that has to be added as to make the pan balance be: x
    \nLet the number of cubes on other pan that has to be added as to make the pan balance be: y
    \nSo,
    \nThe number of cubes present on one pan = x + 3
    \nThe number of cubes present on other pan = y + 9
    \nSo,
    \nTo make the given pans balanced,
    \nx + 3 = y + 9
    \nHence, from the above,
    \nWe can conclude that
    \nThe equation that makes the pans balances is:
    \nx + 3 = y + 9<\/p>\n

    Focus on math practices<\/strong>
    \nUse Structure<\/strong> Suppose that you added 10 cubes to the pan with 3 cubes and then added 4 cubes to the pan with 9 cubes. Would the pans balance? Write an equation to show this relationship.
    \nAnswer:
    \nIt is given that
    \nyou added 10 cubes to the pan with 3 cubes and then added 4 cubes to the pan with 9 cubes
    \nSo,
    \nThe number of cubes present on one pan = 10 + 3 = 13 cubes
    \nThe number of cubes present on other pan = 9 + 4 = 13 cubes
    \nSo,
    \nThe number of cubes present on both pans are the same
    \nHence, from the above,
    \nWe can conclude that
    \nThe pans are balanced<\/p>\n

    Essential Question<\/strong>
    \nHow can you determine whether a given number makes an equation true?
    \nAnswer:
    \nWe can check if the given solution to an expression is actually the solution the expression by replacing the variables in the equation with the given solution and solving the equation. If LHS = RHS with the given solution, the solution is correct. If not, it’s incorrect.<\/p>\n

    Try It!<\/strong><\/p>\n

    Tracy received a $21.00 gift card for phone apps. She has used $9.00 of the value and wants to buy one more app from the list above to use up the balance. Complete the bar diagram and use the equation $21.00 = x + $9.00 to determine which app she should buy.
    \n\"Envision
    \nThe solution is _________, so Tracy should buy the _________ app.
    \nAnswer:
    \nIt is given that
    \nTracy received a $21.00 gift card for phone apps. She has used $9.00 of the value and wants to buy one more app from the list above to use up the balance
    \nNow,
    \nThe given figure is:
    \n\"\"
    \nNow,
    \nLet x be the cost of the another app that Tracy wanted to buy
    \nSo,
    \nAccording to the given information,
    \nThe equation that represents the used up balance is:
    \n$9 + $x = $21
    \nNow,
    \nThe representation of the above equation in the form of a bar diagram is:
    \n\"\"
    \nSo,
    \n$x = $21 – $9
    \n$x = $12
    \nSo,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nTracy should buy the “Remote Desktop” app with her balance of $12<\/p>\n

    Convince Me!<\/strong> What do you notice about the expression on the left side of an equation compared to the expression on the right side when a value is substituted for the variable? How do you know which value is a solution?
    \nAnswer:
    \nWhen we observe the expression that is present on the left side of an equation and the expression that is present on the right side of an equation when a value is substituted for the variable,
    \nWe can say that
    \nThe left hand side of the equation = The right hand side of the equation
    \nNow,
    \nThe solution for the equation is the value we substituted for a “Variable”<\/p>\n

    Try It!<\/strong><\/p>\n

    Anthony has a total of y marbles and 4 boxes. He puts 13 marbles in each box and has none left over. Which of his friends, if any, correctly guessed how many marbles Anthony has in all? Use the equation y \u00f7 4 = 13.
    \nSubstitute each guess for y and evaluate.
    \n\"Envision
    \nTry y = 48: _______ \u00f7 4 = _______
    \nTry y = 60: _______ \u00f7 4 = _______
    \nTry y = 120: _______ \u00f7 4 = _______<\/p>\n

    Of Anthony’s three friends, _______ correctly guessed the number of marbles he has in all.
    \nNo solution is given in the set of values.
    \nAnthony has _______ marbles in all.
    \nAnswer:
    \nIt is given that
    \nAnthony has a total of y marbles and 4 boxes. He puts 13 marbles in each box and has none left over.
    \nNow,
    \nThe representation of the given situation in the form of a bar diagram is:
    \n\"\"
    \nNow,
    \nThe given table is:
    \n\"\"
    \nNow,
    \nTo find whether any friend of Anthony guesses correctly or not,
    \nSubstitute the values of marbles below
    \n\"\"
    \nSo,
    \n\"\"
    \nNow,
    \nThe given equation that represents the number of marbles Anthony has is:
    \ny \u00f7 4 = 13
    \ny = 13 \u00d7 4
    \ny = 52 marbles
    \nSo,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe number of marbles Anthony has in all is: 52 marbles<\/p>\n

    KEY CONCEPT<\/strong>
    \nA solution of an equation is a value for the variable that makes the equation true. Substitute values from a given set for the variable and evaluate. x – 4 = 12 x = 9, 16
    \n\"Envision<\/p>\n

    Do You Understand?<\/strong>
    \nQuestion 1.
    \nEssential Question<\/strong> How can you determine whether a given number makes an equation true?
    \nAnswer:
    \nWe can check if the given solution to an expression is actually the solution the expression by replacing the variables in the equation with the given solution and solving the equation. If LHS = RHS with the given solution, the solution is correct. If not, it’s incorrect.<\/p>\n

    Question 2.
    \nWhen is an equation true?
    \nAnswer:
    \nTo make a true equation,\u00a0check your math to make sure that the values on each side of the equals sign are the same. Ensure that the numerical values on both sides of the “=” sign are the same to make a true equation<\/p>\n

    Question 3.
    \nReasoning<\/strong> Ben says that n = 5 is the solution of the equation 7n = 45. How can you check whether Ben is correct?
    \nAnswer:
    \nIt is given that
    \nBen says that n = 5 is the solution of the equation 7n = 45
    \nNow,
    \nThe given equation is:
    \n7n = 45
    \nNow,
    \nsubstitute the given value of n in the given equation
    \nSo,
    \nFor the given equation,
    \nIf\u00a0 LHS = RHS, then
    \nBen is correct
    \nIf LHS \u2260 RHS, then
    \nBen is not correct<\/p>\n

    Question 4.
    \nA pan balance has 3 cubes on one pan and 11 cubes on the other pan. Lucy thinks she should add 7, 8, 9, or 10 cubes to make the pans balance. How can you use the equation 3 + C = 11 to find the number of cubes Lucy should add?
    \nAnswer:
    \nIt is given that
    \nA pan balance has 3 cubes on one pan and 11 cubes on the other pan. Lucy thinks she should add 7, 8, 9, or 10 cubes to make the pans balance.
    \nNow,
    \nAccording to the given information,
    \nThe number of cubes on one pan = 3 cubes
    \nThe number of cubes on other pan = 11 cubes
    \nNow,
    \nIt is given that
    \nLucy wants to add the extra cubes with 3 cubes
    \nSo,
    \nThe number of cubes in the second pan is constant
    \nNow,
    \nLet the number of extra cubes on one pan to be added so that the pan should be balanced be: C
    \nSo,
    \n3 + C = 11
    \nSo,
    \nC = 11 – 3
    \nC = 9 cubes
    \nSo,
    \nLucy should add 9 cubes to make the pans balance
    \nHence, from the above,
    \nWe can conclude that
    \nLucy should add 9 cubes on one pan to make both the pans balance<\/p>\n

    Do You Know How?<\/strong>
    \nIn 5-8, substitute each given value of the variable to find which, if any, is a solution of the equation.<\/strong>
    \nQuestion 5.
    \nd + 9 = 35; d = 16, 22, 26, 36
    \nAnswer:
    \nThe given equation is:
    \nd + 9 = 35
    \nNow,
    \nFor d = 16,
    \n16 + 9 = 35
    \n25 \u2260 35
    \nFor d = 22,
    \n22 + 9 = 35
    \n31 \u2260 35
    \nFor d = 26,
    \n26 + 9 = 35
    \n35 = 35
    \nFor d = 36,
    \n36 + 9 = 35
    \n45 \u2260 35
    \nHence, from the above,
    \nWe can conclude that
    \nd = 26 is a solution to the given equation<\/p>\n

    Question 6.
    \n14n = 35; n = 2, 3, 3.5, 4
    \nAnswer:
    \nThe given equation is:
    \n14n = 35
    \nNow,
    \nFor n = 2,
    \n14 (2) = 35
    \n28 \u2260 35
    \nFor n = 3,
    \n14 (3) = 35
    \n42 \u2260 35
    \nFor n = 3.5,
    \n14 (3.5) = 35
    \n49 \u2260 35
    \nFor n = 4,
    \n14 (4) = 35
    \n56 \u2260 35
    \nHence, from the above,
    \nWe can conclude that
    \nThe given values of n are not any solutions for the given equation<\/p>\n

    Question 7.
    \n13.4 – g = 8.1; g = 4.3, 5.3, 5.5, 6.5
    \nAnswer:
    \nThe given equation is:
    \n13.4 – g = 8.1
    \nNow,
    \nFor g = 4.3,
    \n13.4 – 4.3 = 8.1
    \n9.1 \u2260 8.1
    \nFor g = 5.3,
    \n13.4 – 5.3 = 8.1
    \n8.1 = 8.1
    \nFor g = 5.5,
    \n13.4 – 5.5 = 8.1
    \n7.9 \u2260 8.1
    \nFor g = 6.5,
    \n13.4 – 6.5 = 8.1
    \n6.9 \u2260 8.1
    \nHence, from the above,
    \nWe can conclude that
    \ng = 5.3 is a solution to the given equation<\/p>\n

    Question 8.
    \n4 = 36 \u00f7 m; m = 4, 6, 8, 9
    \nAnswer:
    \nThe given equation is:
    \n4 = 36 \u00f7 m
    \nNow,
    \nFor m = 4,
    \n4 = 36 \u00f7 4
    \n4 \u2260 9
    \nFor m = 6,
    \n4 = 36 \u00f7 6
    \n4 \u2260 6
    \nFor m = 8,
    \n4 = 36 \u00f7 8
    \n4 \u2260 4.5
    \nFor m = 9,
    \n4 = 36 \u00f7 9
    \n4 = 4
    \nHence, from the above,
    \nWe can conclude that
    \nm = 9 is a solution to the given equation<\/p>\n

    In 9-12, tell whether each equation is true or false for n = 8.<\/strong>
    \nQuestion 9.
    \nn = 54 – 36
    \nAnswer:
    \nThe given equation is:
    \nn = 54 – 36
    \nNow,
    \nn = 54 – 36
    \n= 18
    \nSo,
    \n8 \u2260 18
    \nHence, from the above,
    \nWe can conclude that
    \nThe given equation is False for n = 8<\/p>\n

    Question 10.
    \n5n = 40
    \nAnswer:
    \nThe given equation is:
    \n5n = 40
    \nNow,
    \n5n = 40
    \nn = \\(\\frac{40}{5}\\)
    \n= 8
    \nSo,
    \n8 = 8
    \nHence, from the above,
    \nWe can conclude that
    \nThe given equation is True for n = 8<\/p>\n

    Question 11.
    \n152 \u00f7 n = 21
    \nAnswer:
    \nThe given equation is:
    \n152 \u00f7 n = 21
    \nNow,
    \n152 \u00f7 n = 21
    \nn = \\(\\frac{152}{21}\\)
    \n= 0.7
    \nSo,
    \n0.7 \u2260 8
    \nHence, from the above,
    \nWe can conclude that
    \nThe given equation is False for n = 8<\/p>\n

    Question 12.
    \nn + 46 = 54
    \nAnswer:
    \nThe given equation is:
    \nn + 46 = 54
    \nNow,
    \nn + 46 = 54
    \nn = 54 – 46
    \n= 8
    \nSo,
    \n8 = 8
    \nHence, from the above,
    \nWe can conclude that
    \nThe given equation is True for n = 8<\/p>\n

    Practice & Problem Solving<\/strong><\/p>\n

    In 13-16, tell which given value, if any, is a solution of the equation.<\/strong>
    \nQuestion 13.
    \nt – 2.1 = 0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0t = 2.1, 2.4, 2.6, 2.8
    \nAnswer:
    \nThe given equation is:
    \nt – 2.1 = 0
    \nNow,
    \nt – 2.1 = 0
    \nNow,
    \nAdd with 2.1 on both sides
    \nSo,
    \nt – 2.1 + 2.1 = 0 + 2.1
    \nt = 2.1
    \nHence, from the above,
    \nWe can conclude that
    \nt = 2.1 is the solution for the given equation<\/p>\n

    Question 14.
    \n49 = 7r\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 r = 3, 6, 7, 9
    \nAnswer:
    \nThe given equation is:
    \n49 = 7r
    \nNow,
    \n49 = 7r
    \nDivide by 7 on both sides
    \nSo,
    \n\\(\\frac{49}{7}\\) = \\(\\frac{7r}{7}\\)
    \n7 = r
    \nHence, from the above,
    \nWe can conclude that
    \nr = 7 is the solution for the given equation<\/p>\n

    Question 15.
    \n$4.10 = $6.25 – y\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0y = $2.15, $2.95, $3.05, $3.15
    \nAnswer:
    \nThe given equation is:
    \n$4.10 = $6.25 – y
    \nNow,
    \n$4.10 = $6.25 – y
    \ny = $6.25 – $4.10
    \ny = $2.15
    \nHence, from the above,
    \nWe can conclude that
    \ny = $2.15 is the solution for the given equation<\/p>\n

    Question 16.
    \n24 \u00f7 h = 6\u00a0 \u00a0 \u00a0 \u00a0 \u00a0h = 1, 3, 6, 8
    \nAnswer:
    \nThe given equation is:
    \n24 \u00f7 h = 6
    \nNow,
    \n24 \u00f7 h = 6
    \nh = \\(\\frac{24}{6}\\)
    \nh = 4
    \nHence, from the above,
    \nWe can conclude that
    \nThere is not any solution from the given values for the given equation<\/p>\n

    Question 17.
    \nIn the past, Marcie’s father rode his bike 108 miles in 7.5 hours. Her mother rode the same distance in 8 hours. Marcie plans to ride her bike 108 miles at a steady rate of 18 mph for y hours. Will she match her father’s or mother’s time? Use the equation 108 \u00f7 y = 18 to justify your answer.
    \nAnswer:
    \nIt is given that
    \nIn the past, Marcie’s father rode his bike 108 miles in 7.5 hours. Her mother rode the same distance in 8 hours. Marcie plans to ride her bike 108 miles at a steady rate of 18 mph for y hours.
    \nNow,
    \nWe know that,
    \nSpeed = \\(\\frac{Distance}{Time}\\)
    \nTime = \\(\\frac{Distance}{Speed}\\)
    \nSo,
    \nAccording to the given information,
    \nThe time took for Marcie to ride her bike at a steady rate (t) = (The distance rode by Marcie) \u00f7 (The speed of Marcie’s bike)
    \n= 108 \u00f7 18
    \n= 6 hours
    \nNow,
    \nFrom the given information,
    \nThe time took for Marcie’s father to ride his bike is: 7.5 hours
    \nThe time took for Marcie’s mother to ride her bike is: 8 hours
    \nSo,
    \n6 \u2260 7.5 \u2260 8
    \nHence, from the above,
    \nWe can conclude that
    \nMarcie will not match her father’s or mother’s time<\/p>\n

    Question 18.
    \nWrite if b = 6 is a solution or is not a solution of each equation.
    \na. 8b = 48
    \nb. 11 – b = 6
    \nc. b + 3 = 9
    \nd. 54 \u00f7 b = 9
    \nAnswer:
    \nThe given expression sare:
    \na. 8b = 48
    \nb. 11 – b = 6
    \nc. b + 3 = 9
    \nd. 54 \u00f7 b = 9
    \nNow,
    \na.
    \nFor b = 6,
    \n8 (6) = 48
    \n48 = 48
    \nb.
    \nFor b = 6,
    \n11 – 6 = 6
    \n5 \u2260 6
    \nc.
    \nFor b = 6,
    \n6 + 3 = 9
    \n9 = 9
    \nd.
    \nFor b = 6,
    \n54 \u00f7 6 = 9
    \n9 = 9
    \nHence, from the above,
    \nWe can conclude that
    \nThe equations that have a solution b = 6 are:
    \n\"\"<\/p>\n

    Question 19.
    \nA group of 4 friends is planning a fun day trip. The equations in the table represent the number of people n who can participate in each activity for $29.
    \n\"Envision
    \nWhich activity should the friends choose if they want to spend exactly $29?
    \nAnswer:
    \nIt is given that
    \nA group of 4 friends is planning a fun day trip. The equations in the table represent the number of people n who can participate in each activity for $29.
    \nNow,
    \nThe given table is:
    \n\"\"
    \nNow,
    \nFor the friends to choose an activity so as they spend exactly $29,
    \nThe solutions of the variables should satisfy their own equations
    \nNow,
    \na. For the Raft Trip activity:
    \nThe given equation is:
    \n6n + 5 = 29
    \n6n = 29 – 5
    \n6n = 24
    \nn = \\(\\frac{24}{6}\\)
    \nn = 4
    \nVerification:
    \n6 (4) + 5 = 29
    \n24 + 5 = 29
    \n29 = 29
    \nb. For the Amusement park Trip activity:
    \nThe given equation is:
    \n14n = 29
    \nn = \\(\\frac{29}{14}\\)
    \nn = 2.07
    \nn \u2245 2
    \nVerification:
    \n14 (2) = 29
    \n28 \u2260 29
    \nc. For the Balloon Ride activity:
    \n30n – 40 = 29
    \n30n = 20 + 49
    \n30n = 69
    \nn = \\(\\frac{69}{30}\\)
    \nn = \\(\\frac{23}{10}\\)
    \nn = 2.3
    \nn \u2245 2
    \nVerification:
    \n30 (2) – 40 = 29
    \n60 – 40 = 29
    \n20 \u2260 29
    \nHence, from the above,
    \nWe can conclude that
    \nThe activity the friends should choose if they want to spend exactly $29 is: Raft Trip<\/p>\n

    Question 20.
    \nThere are 27 pennies on one pan of a pan balance and 18 pennies on the other. To make the pans balance, Hillary thinks 5 pennies should be added to the higher pan. Sean thinks 8 pennies should be added, and Rachel thinks 9 pennies should be added. Use the equation 27 = 18 + p to determine who is correct.
    \nAnswer:
    \nIt is given that
    \nThere are 27 pennies on one pan of a pan balance and 18 pennies on the other. To make the pans balance, Hillary thinks 5 pennies should be added to the higher pan. Sean thinks 8 pennies should be added, and Rachel thinks 9 pennies should be added
    \nNow,
    \nLet the nuber of pennies added be: p
    \nNow,
    \nThe given equation in the problem that represents the situation is:
    \n27 = 18 + p
    \nSo,
    \nOn the basis of the given information,
    \nAccording to Hillary, p = 5
    \nAccording to Sean, p = 8
    \nAccording to Rachel, p = 9
    \nNow,
    \nFor p = 5,
    \n27 = 18 + p
    \n27 = 18 + 5
    \n27 \u2260 23
    \nNow,
    \nFor p = 8,
    \n27 = 18 + p
    \n27 = 18 + 8
    \n27 \u2260 26
    \nNow,
    \nFor p = 9,
    \n27 = 18 + p
    \n27 = 18 + 9
    \n27 = 27
    \nHence, from the above,
    \nWe can conclude that
    \nRachel is correct<\/p>\n

    Question 21.
    \nConstruct Arguments<\/strong> Gerard spent $5.12 for a drink and a sandwich. His drink cost $1.30. Did he have a ham sandwich for $3.54, a tuna sandwich for $3.82, or a turkey sandwich for $3.92? Use the equation s + 1.30 = 5.12 to justify your answer.
    \nAnswer:
    \nIt is given that
    \nGerard spent $5.12 for a drink and a sandwich. His drink cost $1.30
    \nNow,
    \nLet the cost of sandwich be: $s
    \nSo,
    \nAccording to the given information,
    \n$5.12 = s + $1.30
    \nSubtract $1.30 on both sides
    \nSo,
    \n$5.12 – $1.30 = s + $1.30 – $1.30
    \ns = $3.82
    \nNow,
    \nAccording to the given information,
    \nThe cost of tuna sandwich is: $3.82
    \nHence, from the above,
    \nWe can conclude that
    \nGerard will have a tuna sandwich<\/p>\n

    Question 22.
    \nHigher Order Thinking<\/strong> Write an equation that has a solution of 12. Show how you know that 12 is the solution.
    \nAnswer:
    \nLet the equation that has a solution of 12 be:
    \n6x = 72
    \nNow,
    \nWe know that,
    \nTo find whether the solution is correct or not, substitute the solution in the place of the variable present in that equation and if LHS = RHS, then that solution is true for the equation
    \nSo,
    \nThe equation is: 6x = 72
    \nNow,
    \nSubstitute 12 in place of x
    \nSo,
    \n6 (12) = 72
    \nHence,
    \n72 = 72<\/p>\n

    Question 23.
    \nGina’s family is driving 255 miles to visit Tallahassee. After driving for a while, they pass a sign that reads “Tallahassee: 124 miles.” Substitute the values m = 111, 121, 131, and 141 in the equation 255 – m = 124 to find the number of miles the family has already driven.
    \nAnswer:
    \nIt is given that
    \nGina’s family is driving 255 miles to visit Tallahassee. After driving for a while, they pass a sign that reads “Tallahassee: 124 miles.”
    \nNow,
    \nThe given equation to represent the number of miles the Gina’s family has already driven is:
    \n255 – m = 124 with m = 111, 121, 131, and 141
    \nNow,
    \nFor m = 111,
    \n255 – m = 124
    \n255 – 111 = 124
    \n144\u00a0 \u2260 124
    \nNow,,
    \nFor m = 121,
    \n255 – m = 124
    \n255 – 121 = 124
    \n134\u00a0 \u2260 124
    \nNow,
    \nFor m = 131,
    \n255 – m = 124
    \n255 – 131 = 124
    \n124\u00a0 = 124
    \nNow,
    \nFor m = 141,
    \n255 – m = 124
    \n255 – 141 = 124
    \n114\u00a0 \u2260 124
    \nHence, from the above,
    \nWe can conclude that
    \nThe number of miles the Gina’s family has already driven is: 131 miles<\/p>\n

    Question 24.
    \nLisa is making a quilt that uses a pattern of triangles like the one shown. Write an equation that represents the missing side length if the perimeter is 19 centimeters
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nLisa is making a quilt that uses a pattern of triangles like the one shown
    \nNow,
    \nThe given figure is:
    \n\"Envision
    \nNow,
    \nWe know that,
    \nThe “Perimeter” of a figure is the sum of all the side lengths of a figure
    \nSo,
    \nFor the given figure,
    \nPerimeter = 5 + (3 + 3) + m + 3
    \n19 = 5 + 6 + m + 3
    \n19 = 14 + m
    \nm = 19 – 14
    \nm = 5 cm
    \nHence, from the above,
    \nWe can conclude that
    \nThe missing side length is: 5 cm<\/p>\n

    Question 25.
    \nAlisa’s family planted 7 palm trees in their yard. The park down the street has 147 palm trees. Alisa guessed that the park has either 11 or 31 times as many palm trees as her yard has. Is either of Alisa’s guesses correct? Use the equation 7n = 147 to justify your answer.
    \nAnswer:
    \nIt is given that
    \nAlisa’s family planted 7 palm trees in their yard. The park down the street has 147 palm trees. Alisa guessed that the park has either 11 or 31 times as many palm trees as her yard has.
    \nNow,
    \nThe equation that represents the given information that is given in the problem is:
    \n7n = 147
    \nSo,
    \nn = \\(\\frac{147}{7}\\)
    \nn = 21 palm trees
    \nBut,
    \nAccording to the given information,
    \nAlisa had guessed that the park has either 11 or 31 times as many palm trees as her yard has
    \nHence, from the above,
    \nWe can conclude that
    \nAlisa’s guesses are not correct<\/p>\n

    Assessment Practice<\/strong><\/p>\n

    Question 26.
    \nTrish has $26.00 to spend at a craft store. She buys fabric that costs $18.62. She also wants to buy knitting needles for $7.32, silk flowers for $7.38, or oil paints for $8.48.
    \nUse the equation $18.62 + c = $26.00, where c is the item cost, to find the most expensive item Trish can buy Explain how you found your answer.
    \nAnswer:
    \nIt is given that
    \nTrish has $26.00 to spend at a craft store. She buys fabric that costs $18.62. She also wants to buy knitting needles for $7.32, silk flowers for $7.38, or oil paints for $8.48.
    \nNow,
    \nThe equation that represents the given information and is mentioned in the problem is:
    \n$18.62 + c = $26.00
    \nSo,
    \nc = $26.00 – $18.62
    \nc = $7.38
    \nSo,
    \nTrish can buy silk flowers with her balance
    \nHence, from the above,
    \nWe can conclude that
    \nThe most expensive item Trish can buy with the remaining amount of money is: Silk flowers<\/p>\n

    Lesson 4.2 Apply Properties of Equality<\/h3>\n

    Solve & Discuss It!<\/strong>
    \nStart with the equation 4 + 8 = 12 and complete each computation listed below. Do each computation individually. Which of the computations keeps the equation true? Explain.<\/p>\n

    I can… use the properties of equality to write equivalent equations.
    \n\"Envision
    \nAnswer:
    \nThe given equation is:
    \n4 + 8 = 12
    \nNow,
    \nWe know that,
    \nThe equation will be true only when the equations’s
    \nLHS = RHS
    \nNow,
    \nThe completed table with the individual computations and the result as “True” and “False” is:
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe computations that keep the equation true are:
    \na. Add 5 to both sides of the equation
    \nb. Subtract 4 from both sides of the equation<\/p>\n

    Reasoning<\/strong>
    \nHow can you determine whether an equation is true?
    \nAnswer:
    \nFrom above,
    \nThe completed table is:
    \n\"\"
    \nNow,
    \nWe know that,
    \nAccording to the Properties of Equality,
    \nIn an equation,
    \nThe operation we performed with the given number on the left side of the equation will be as the same as the operation we performed with the same number on the right side of the equation
    \nSo,
    \nFrom the completed table,
    \nOnly 2 computations follows the “Properties of equality”
    \nHence, from the above,
    \nWe can conclude that
    \nWe can determine the equation by using the “Properties of equality”<\/p>\n

    Focus on math practices<\/strong>
    \nUse Structure<\/strong> Complete the equation 7 + \u2610 = 10 – \u2610 by filling in the missing numbers. Describe at least two other operations with numbers that you can do to each side of the completed equation to keep it true.
    \nAnswer:
    \nLet the missing number on the left side and right side of the given equation be: x
    \nNow,
    \nWe know that,
    \nAccording to the Properties of Equality,
    \nIn an equation,
    \nThe operation we performed with the given number on the left side of the equation will be as the same as the operation we performed with the same number on the right side of the equation
    \nSo,
    \n7 + x = 10 – x
    \nx + x = 10 – 7
    \n2x = 3
    \nx = \\(\\frac{3}{2}\\)
    \nx = 1.5
    \nHence, from the above,
    \nWe can conclude that
    \nThe completed equation is:
    \n\"\"<\/p>\n

    Essential Question<\/strong>
    \nHow can you use the properties of equality to write equivalent equations?
    \nAnswer:
    \nJust as you can add or subtract the same exact quantity on both sides of an equation, you can also\u00a0multiply both sides of an equation by the same quantity\u00a0to write an equivalent equation.<\/p>\n

    Try It!<\/strong><\/p>\n

    If 5y = 25, which property of equality was used to keep the equation 5y – 7 = 25 – 7 equal?
    \nAnswer:
    \nThe given equation is:
    \n5y = 25
    \nNow,
    \nWe know that,
    \nThe “Subtraction Property of equality” states that when you subtract the same amount from both sides of an equation, the two sides of the equation stay equal
    \nSo,
    \nAccording to the above property,
    \n5y = 25 changed into 5y – 7 = 25 – 7
    \nHence, from the above,
    \nWe can conclude that
    \nThe “Subtraction Property of Equality” was used to keep the equation 5y – 7 = 25 – 7 equal<\/p>\n

    Convince Me!<\/strong> What other properties of equality could you apply to keep the equation 5y = 25 equal? Give an example of each.
    \nThe given equation is:
    \n5y = 25
    \nNow,
    \nWe know that,
    \nWe can use any “Property of Equality” like “Addition Property of equality”, “Subtraction Property of equality” , “Multiplication Property of equality”, and “Division Property of equality” to keep the given equation true
    \nHence, from the above,
    \nWe can conclude that
    \nThe other properties of equality you could apply to keep the equation 5y = 25 equal are:
    \na. Addition Property of equality
    \nb. Multiplication Property of equality
    \nc. Division Property of equality<\/p>\n

    Try It!<\/strong><\/p>\n

    A. A scale balances with four blue x-blocks on one side and 36 green blocks on the other side. Complete the equation to balance the scale with only one blue x-block.
    \n4 \u2022 x = 36
    \n(4 \u2022 x) \u00f7 \u2610 = 36 \u00f7 \u2610
    \nx = 9
    \nAnswer:
    \nIt is given that
    \nA scale balances with four blue x-blocks on one side and 36 green blocks on the other side.
    \nNow,
    \nAccording to the given information,
    \nTo balance the equation with only one blue x-block,
    \nDivide the both sides of the given equation by 4 so that only one blue x rock will be present
    \nSo,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe number of blue x-rocks are: 9 blue x-rocks<\/p>\n

    B. If 25 + d = 36, does 25 + d – 25 = 36 \u2013 20? Explain.
    \nAnswer:
    \nThe given equations are: 25 + d = 36 and 25 + d – 25 = 36 – 20
    \nNow,
    \nWe know that,
    \nThe “Subtraction Property of equality” states that when you subtract the same amount from both sides of an equation, the two sides of the equation stay equal
    \nSo,
    \nFor 25+ d = 36,
    \nAccording to the given second equation,
    \n25 + d – 25 = 36 – 25
    \nHence, from the above,
    \nWe can conclude that
    \nIf 25 +d = 36, then 25 + d – 25 = 36 – 25<\/p>\n

    KEY CONCEPT<\/strong>
    \nYou can use the properties of equality to write equivalent equations.<\/p>\n

    Addition Property of Equality
    \n7 + 3 = 10
    \n(7 + 3) + a = 10 + a
    \nAdd the same amount to each side to keep the equation balanced.<\/p>\n

    Subtraction Property of Equality
    \n7 + 3 = 10
    \n(7 + 3) – a = 10 – a
    \nSubtract the same amount from each side to keep the equation balanced.<\/p>\n

    Multiplication Property of Equality
    \n7 + 3 = 10
    \n(7 + 3) \u00d7 a = 10 \u00d7 a
    \nMultiply each side of the equation by the same amount to keep the equation balanced.<\/p>\n

    Division Property of Equality
    \n7+ 3 = 10
    \n(7 + 3) \u00f7 a = 10 \u00f7 a
    \nDivide each side of the equation by the same non-zero amount to keep the equation balanced.<\/p>\n

    Do You Understand?<\/strong>
    \nQuestion 1.
    \nEssential Question<\/strong> How can you use the properties of equality to write equivalent equations?
    \nAnswer:
    \nJust as you can add or subtract the same exact quantity on both sides of an equation, you can also\u00a0multiply both sides of an equation by the same quantity\u00a0to write an equivalent equation.<\/p>\n

    Question 2.
    \nA pan balance shows 7 + 5 = 12. If 4 units are removed from one side, what needs to be done to the other side to keep the pans balanced?
    \nAnswer:
    \nIt is given that
    \nA pan balance shows 7 + 5 = 12. and 4 units are removed from one side
    \nNow,
    \nWe know that,
    \nAccording to the “Subtraction Property of Equality”,
    \nSubtract the same amount from each side to keep the equation balanced.
    \nSo,
    \nTo keeps the pans balanced, we have to remove 4 from the other side also
    \nHence, from the above,
    \nWe can conclude that
    \nWe have to remove 4 from other side also to keep the pans balanced<\/p>\n

    Question 3.
    \nIf one side of the equation 23 + 43 = 66 is multiplied by 3, what needs to be done to the other side of the equation to keep the sides equal?
    \nAnswer:
    \nIt is given that
    \nOne side of the equation 23 + 43 = 66 is multiplied by 3
    \nNow,
    \nWe know that,
    \nAccording to the “Multiplication Property of Equality”,
    \nMultiply the same amount from each side to keep the equation balanced.
    \nSo,
    \nTo keeps the pans balanced, we have to multiply 3 to the other side also
    \nHence, from the above,
    \nWe can conclude that
    \nWe have to multiply 3 to the other side also to keep the pans balanced<\/p>\n

    Question 4.
    \nReasoning<\/strong> If one side of the equation x + 5 = 8 has 9 added to it and the other side has (4 + 5) added to it, will the equation stay equal?
    \nAnswer:
    \nIt is given that
    \nOne side of the equation x + 5 = 8 has 9 added to it and the other side has (4 + 5) added to it
    \nNow,
    \nWe know that,
    \nAccording to the “Addition Property of Equality”,
    \nAdd the same amount from each side to keep the equation balanced.
    \nNow,
    \nWe know that,
    \n4 + 5 = 9
    \nSo,
    \nThe equation stays equal
    \nHence, from the above,
    \nWe can conclude that
    \nThe equation will stay equal even we added (4 + 5) to the other side of the equation<\/p>\n

    Do You Know How?<\/strong>
    \nIn 5 and 6, answer yes or no and explain why or why not.<\/strong>
    \nQuestion 5.
    \nIf 23 + 37 = 60, does 23 + 37 + 9 = 60 + 9?
    \nAnswer:
    \nThe given equations are:
    \n23 + 37 = 60 and 23 + 37 + 9 = 60 + 9
    \nNow,
    \nWe know that,
    \nAccording to the “Addition Property of Equality”,
    \nAdd the same amount from each side to keep the equation balanced.
    \nSo,
    \nFrom the above equation,
    \nWe can observe that
    \n9 is added to both sides
    \nHence, from the above,
    \nWe can conclude that
    \nThe given equation is true<\/p>\n

    Question 6.
    \nIf 16 + 1 = 17, does (16 + 1) – 1 = 17 – 2?
    \nAnswer:
    \nThe given equations are:
    \n16 + 1 = 17 and (16 + 1) – 1 = 17 – 2
    \nNow,
    \nWe know that,
    \nAccording to the “Subtraction Property of Equality”,
    \nSubtract the same amount from each side to keep the equation balanced.
    \nSo,
    \nFrom the above equation,
    \nWe can observe that
    \n1 is subtracted from one side and 2 is subtracted from the other side
    \nHence, from the above,
    \nWe can conclude that
    \nThe given equation is false<\/p>\n

    Question 7.
    \nApply the Multiplication Property of Equality to write an equation equivalent to 7n = 28.
    \nAnswer:
    \nThe given equation is:
    \n7n = 28
    \nNow,
    \nWe know that,
    \nAccording to the “Multiplication Property of Equality”,
    \nMultiply the same amount from each side to keep the equation balanced.
    \nNow,
    \nLet the number be multiplied on both sides of the equation be 2
    \nSo,
    \nAccording the Multiplication Property of Equality,
    \n7n \u00d7 2 = 28 \u00d7 2
    \nHence, from the above,
    \nWe can conclude that
    \nAccording to the Multiplication Property of equation,
    \nThe equation that is equivalent to the given equation is:
    \n7n \u00d7 2 = 28 \u00d7 2<\/p>\n

    Question 8.
    \nCritique Reasoning<\/strong> Tomas says that if one side of the equation 6m = 9 is divided by 2 and the other side is divided by 3, the equation will stay equal because the result will be 3m = 3. Is Tomas correct? Explain.
    \nAnswer:
    \nIt is given that
    \nTomas says that one side of the equation 6m = 9 is divided by 2 and the other side is divided by 3
    \nNow,
    \nWe know that,
    \nAccording to the “Division Property of Equality”,
    \nDivide the same amount from each side to keep the equation balanced.
    \nSo,
    \nWe have to divide the both sides of the equation by either only 2 or only 2 but not with 2 on one side and with 3 on other side
    \nHence,f rom the above,
    \nWe can conclude that
    \nTomas is not correct<\/p>\n

    Practice & Problem Solving<\/strong><\/p>\n

    In 9-12, tell which property of equality was used.<\/strong>
    \nQuestion 9.
    \n5m + 4 = 19
    \n5m + 4 – 3 = 19 – 3
    \nAnswer:
    \nThe given equations are:
    \n5m + 4 = 19
    \n5m + 4 – 3 = 19 – 3
    \nNow,
    \nWe know that,
    \nAccording to the “Subtraction Property of Equality”,
    \nSubtract the same amount from each side to keep the equation balanced.
    \nSo,
    \nFrom the given equation,
    \nWe can observe that
    \nWe subtracted 3 on both sides
    \nHence, from the above,
    \nWe can conclude that
    \n“Subtraction Property of Equality” was used for the given equation<\/p>\n

    Question 10.
    \n3t = 20
    \n3t \u00f7 2 = 20 \u00f7 2
    \nAnswer:
    \nThe given equations are:
    \n3t = 20
    \n3t \u00f7 2 = 20 \u00f7 2
    \nNow,
    \nWe know that,
    \nAccording to the “Division Property of Equality”,
    \nDivide the same amount at each side to keep the equation balanced.
    \nSo,
    \nFrom the given equation,
    \nWe can observe that
    \nWe divided by 2 on both sides
    \nHence, from the above,
    \nWe can conclude that
    \n“Division Property of Equality” was used for the given equation<\/p>\n

    Question 11.
    \n\\(\\frac{n}{6}\\) = 9
    \n\\(\\left(\\frac{n}{6}\\right)\\) \u00d7 5 = 9 \u00d7 5
    \nAnswer:
    \nThe given equations are:
    \n\\(\\frac{n}{6}\\) = 9
    \n\\(\\frac{n}{6}\\) \u00d7 5 = 9 \u00d7 5
    \nNow,
    \nWe know that,
    \nAccording to the “Multiplication Property of Equality”,
    \nMultiply the same amount at each side to keep the equation balanced.
    \nSo,
    \nFrom the given equation,
    \nWe can observe that
    \nWe multiplied 5 on both sides
    \nHence, from the above,
    \nWe can conclude that
    \n“Multiplication Property of Equality” was used for the given equation<\/p>\n

    Question 12.
    \n5b – 6 = 14
    \n(5b – 6) + 2 = 14 + 2
    \nAnswer:
    \nThe given equations are:
    \n5b – 6 = 14
    \n(5b – 6) + 2 = 14 + 2
    \nNow,
    \nWe know that,
    \nAccording to the “Addition Property of Equality”,
    \nAdd the same amount at each side to keep the equation balanced.
    \nSo,
    \nFrom the given equation,
    \nWe can observe that
    \nWe added 2 on both sides
    \nHence, from the above,
    \nWe can conclude that
    \n“Addition Property of Equality” was used for the given equation<\/p>\n

    Question 13.
    \nIf r + 9 = 42, does r + 9 – 9 = 42 + 9? Why or why not?
    \nAnswer:
    \nThe given equations are:
    \nr + 9 = 42
    \nr + 9 – 9 = 42 + 9
    \nNow,
    \nWe know that,
    \nAccording to the “Subtraction Property of Equality”,
    \nSubtract the same amount from each side to keep the equation balanced.
    \nSo,
    \nFrom the given equation,
    \nWe can observe that
    \nWe subtracted 9 from one side and added 9 at the other side
    \nHence, from the above,
    \nWe can conclude that
    \nr + 9 = 42 is not the equivalent of r + 9 – 9 = 42 + 9<\/p>\n

    Question 14.
    \nIf 6s = 24, does 6s \u00f7 6 = 24 \u00f7 6? Why or why not?
    \nAnswer:
    \nThe given equations are:
    \n6s = 24
    \n6s \u00f7 6 = 24 \u00f7 6
    \nNow,
    \nWe know that,
    \nAccording to the “Division Property of Equality”,
    \nDivide the same amount at each side to keep the equation balanced.
    \nSo,
    \nFrom the given equation,
    \nWe can observe that
    \nWe divided with 6 on both sides
    \nHence, from the above,
    \nWe can conclude that
    \n6s = 24 is the equivalent of 6s \u00f7 6 = 24 \u00f7 6<\/p>\n

    Question 15.
    \nThis scale was balanced. Find the number to add that makes the scale become balanced again. Then complete the equation to make it true.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nThis scale was balanced
    \nNow,
    \nThe given figure is:
    \n\"\"
    \nNow,
    \nFrom the given figure,
    \nWe can observe that
    \nTo make the scale balanced again,
    \nWe have to add some quantity to 12 on the left side of the pan
    \nNow,
    \nLet the some quantity that is added on the left side of the pan be x
    \nSo,
    \nTo make the scale balanced,
    \n12 + x = 2 + 7+ 3 + 16
    \nSo,
    \n12 + x = 28
    \nSubtract with 12 on both sides
    \n12 + x – 12 = 28 – 12
    \nx = 28 – 12
    \nx = 16
    \nHence, from the above,
    \nWe can conclude that
    \nThe completed equation so that the equation is true is:
    \n\"\"<\/p>\n

    Question 16.
    \nThis scale balanced with 3 green blocks on one side and 1 blue x-block on the other side. Find the number to multiply by that makes the scale balance. Then complete the equation to make it true.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nThis scale balanced with 3 green blocks on one side and 1 blue x-block on the other side
    \nNow,
    \nThe given figure is:
    \n\"\"
    \nNow,
    \nFrom the given information,
    \nThe equation that represents the given situation is:
    \n1x = 3
    \nSo,
    \nx = 3
    \nNow,
    \nMultiply both sides with 5
    \nSo,
    \n5 \u00d7 x = 3 \u00d7 5
    \n5x = 15
    \n15 = 5x
    \nHence, from the above,
    \nWe can conclude that
    \nThe completed equation so that the equation will be true is:
    \n\"\"<\/p>\n

    Question 17.
    \nYou start with the equation 8x = 24. Your friend changes the equation as follows.
    \n8x = 24 \u00f7 4
    \nHow can you make your friend’s equation equivalent to the original equation?
    \nAnswer:
    \nThe given equations are:
    \n8x = 24
    \n8x = 24 \u00f7 4
    \nNow,
    \nFrom the above equation,
    \nWe can observe that
    \nYour friend wants to change the equation by dividing the equation by 4
    \nNow,
    \nWe know that,
    \nAccording to the “Division Property of Equality”,
    \nDivide the same amount at each side to keep the equation balanced.
    \nSo,
    \nFor the given equation,
    \nWe have to divide by 4 on both sides instead of only one side like your friend
    \nSo,
    \n8x \u00f7 4 = 24 \u00f7 4
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of your friend’s equation equivalent to the original given equation is:
    \n8x \u00f7 4 = 24 \u00f7 4<\/p>\n

    Question 18.
    \nA scale balanced with 1 blue x-block and 20 green blocks on the left side and 40 green blocks on the right side. A student bumped into the scale and knocked some blocks off so that only 1 blue x-block and 3 green blocks remained on the left side. How many blocks do you need to remove from the right side to make the scale balance?
    \nAnswer:
    \nIt is given that
    \nA scale balanced with 1 blue x-block and 20 green blocks on the left side and 40 green blocks on the right side. A student bumped into the scale and knocked some blocks off so that only 1 blue x-block and 3 green blocks remained on the left side
    \nSo,
    \nAccording to the given information,
    \nNow,
    \nBefore a student bumped into the scale,
    \nThe representation of the equation is:
    \n1x blue rocks + 20 green rocks = 40 green rocks
    \nx blue rocks + 20 green rocks = 40 green rocks
    \nNow,
    \nAfter a student bumped into the scale,
    \nx blue rocks + 20 green rocks – 0 blue rocks – 17 green rocks = 40 green rocks – x
    \nWhere,
    \nx is the variable to make the equation balanced
    \nNow,
    \nWe know that,
    \nAccording to the “Subtraction Property of Equality”,
    \nSubtract the same amount from each side to keep the equation balanced.
    \nSo,
    \nx blue rocks + 20 green rocks – 0 blue rocks – 17 green rocks = 40 green rocks – 0 blue rocks – 17 green rocks
    \nHence, from the above,
    \nWe can conclude that
    \nThe number of blocks you needed to remove from the right side to make the scale balance is: 17 green rocks<\/p>\n

    Question 19.
    \nBobbie wrote y + 6 = 15. Then she wrote (y + 6) \u00f7 3 = 15. Explain why the second equation is not equivalent to the first. What can Bobbie do to make the two equations equivalent?
    \nAnswer:
    \nIt is given that
    \nBobbie wrote y + 6 = 15. Then she wrote (y + 6) \u00f7 3 = 15
    \nNow,
    \nThe given equations are:
    \ny + 6 = 15
    \n(y + 6) \u00f7 3 = 15
    \nNow,
    \nWe know that,
    \nAccording to the “Division Property of Equality”,
    \nDivide the same amount at each side to keep the equation balanced.
    \nSo,
    \nFrom the given equation,
    \nWe can observe that
    \nWe have to divide by 3 on both sides
    \nBut,
    \nAccording to the second equation,
    \nIt has been divided only on the left side
    \nSo,
    \n(y + 6) \u00f7 3 \u2260 15
    \nHence, from the above,
    \nWe can conclude that
    \nDue to the “Division Property of Equality”,
    \n(y + 6) \u00f7 3 \u2260 15<\/p>\n

    Question 20.
    \nConstruct Arguments<\/strong> John wrote that 5 + 5 = 10. Then he wrote that 5 + 5 + n = 10 + n. Are the equations John wrote equivalent? Explain.
    \nAnswer:
    \nIt is given that
    \nJohn wrote that 5 + 5 = 10. Then he wrote that 5 + 5 + n = 10 + n
    \nNow,
    \nThe given equations are:
    \n5 + 5 = 10
    \n5 + 5 + n = 10 + n
    \nNow,
    \nWe know that,
    \nAccording to the “Addition Property of Equality”,
    \nAdd the same amount at each side to keep the equation balanced.
    \nSo,
    \nFrom the given equation,
    \nWe can observe that
    \nJohn added n at both sides
    \nSo,
    \n5 + 5 = 10 is equivalent to 5 + 5 + n = 10 + n
    \nHence, from the above,
    \nWe can conclude that
    \nDue to the Addition Property of Equality,
    \n5 + 5 = 10 is equivalent to 5 + 5 + n = 10 + n<\/p>\n

    Question 21.
    \nReasoning<\/strong> Scientists often use a pan balance to measure mass when doing experiments. The equation 4 + 3 – 1 = 7 – 1 represents a scientist taking away one unit of mass from each side of a pan balance. Construct an argument to explain how the scientist knows that the pans are still in balance.
    \nAnswer:
    \nIt is given that
    \nScientists often use a pan balance to measure mass when doing experiments. The equation 4 + 3 – 1 = 7 – 1 represents a scientist taking away one unit of mass from each side of a pan balance.
    \nNow,
    \nThe given expression is:
    \n4 + 3 – 1 = 7 – 1
    \nNow,
    \nWe know that,
    \nAccording to the “Subtraction Property of Equality”,
    \nSubtract the same amount from each side to keep the equation balanced.
    \nSo,
    \nFrom the given equation,
    \nWe can observe that
    \n1 is subtracted from both sides of the equation
    \nSo,
    \nThe pans are still in balance according to “Subtraction Property of Equality”
    \nHence, from the above,
    \nWe can conclude that
    \nThe pans are still in balance according to “Subtraction Property of Equality”<\/p>\n

    Question 22.
    \nBryce wrote the equation n – 3 = 4. Lexi used a property of equality to write an equivalent equation. Write an equation Lexi could have written. Explain how you know the equations are equivalent.
    \nAnswer:
    \nIt is given that
    \nBryce wrote the equation n – 3 = 4. Lexi used a property of equality to write an equivalent equation
    \nNow,
    \nThe given equation is:
    \nn – 3 = 4
    \nNow,
    \nWe know that,
    \nAccording to the “Addition Property of Equality”,
    \nAdd the same amount at each side to keep the equation balanced.
    \nSo,
    \nn – 3 + 3 = 4 + 3
    \nn = 7
    \nSo,
    \nThe equivalent equation of the given equation is: n = 7
    \nHence, from the above,
    \nWe can conclude that
    \nThe equation Lexi could have written is: n = 7
    \nThe equation swritten by Bryce and Lexi are equivalent by using the “Additie Property of Equality”<\/p>\n

    Question 23.
    \nHigher Order Thinking<\/strong> Emil has $1 and a quarter. Jade has 5 quarters. If Emil gives Jade $1 and Jade gives Emil 4 quarters, will they each still have the same amount of money? Explain.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nEmil has $1 and a quarter. Jade has 5 quarters and Emil gives Jade $1 and Jade gives Emil 4 quarters
    \nNow,
    \nWe know that,
    \n1 Quarter = $0.25
    \nNow,
    \nThe amount of money Emil contains = $1 + $0.25
    \n= $1.25
    \nThe amount of money Jade contains = 5 \u00d7 $0.25
    \n= $1.25
    \nNow,
    \nIf Emil gives Jade $1, then
    \nThe amount of money Jade contains = $1.25 + $1
    \n= $2.25
    \nThe remaining amount of money Emil contains = $1.25 – $1
    \n= $0.25
    \nIf Jade gives 4 Quarters to Emil, then
    \nThe amount of money Emil contains = $0.25 + 4 \u00d7 $0.25
    \n= $0.25 + $1
    \n= $1.25
    \nHence, from the above,
    \nWe can conclude that
    \nEmil and Jade each still have the same amount of money<\/p>\n

    Question 24.
    \nVocabulary<\/strong> If 7w = 49, which property of equality was used to find the equivalent equation 7w \u00f7 7 = 49 \u00f7 7?
    \nAnswer:
    \nThe given equations are:
    \n7w = 49
    \n7w \u00f7 7 = 49 \u00f7 7
    \nNow,
    \nWe know that,
    \nAccording to the “Division Property of Equality”,
    \nDivide the same amount at each side to keep the equation balanced.
    \nSo,
    \nFrom the given equation,
    \nWe can observe that
    \n7 is divided on both sides
    \nHence, from the above,
    \nWe can conclude that
    \n“Division Property of Equality” was used to find the equivalent equation\u00a0 of the given equation 7w \u00f7 7 = 49 \u00f7 7<\/p>\n

    Question 25.
    \nYou start with the equation 12b = 24. What step should you take to find the quantity that equals 4b?
    \nAnswer:
    \nThe given equation is:
    \n12b = 24
    \nNow,
    \nIt is given that
    \nAfter using the “Equality Property”, we have to make 12b as 4b
    \nNow,
    \nWe know that,
    \nAccording to the “Division Property of Equality”,
    \nDivide the same amount at each side to keep the equation balanced.
    \nSo,
    \nDivide the given equation by 3 on both sides
    \nSo,
    \n\\(\\frac{12b}{3}\\) = \\(\\frac{24}{3}\\)
    \n4b = 8
    \nHence, from the above,
    \nWe can conclude that
    \nTo find the quantity “4b”,
    \nWe have to use “Division Property of Equality” for the given equation<\/p>\n

    Assessment Practice<\/strong><\/p>\n

    Question 26.
    \nWhich equation is equivalent to n + 4 = 11?.
    \nA. (n + 4) \u00d7 2 = 11
    \nB. (n + 4) \u00d7 2 = 11 \u00f7 2
    \nC. (n + 4) \u00d7 2 = 11 \u00d7 4
    \nD (n + 4) \u00d7 2 = 11 \u00d7 2
    \nAnswer:
    \nThe given equation is:
    \nn + 4 = 11
    \nNow.
    \nWe know that,
    \nAccording to the “Properties of Equality”,
    \nThe operation we are doing at the left side will be the same as the operation we are doing at the right side
    \nHence, from the above,
    \nWe can conclude that
    \nThe equation that is equivalent to n + 4 = 11 is:
    \n\"\"<\/p>\n

    Question 27.
    \nWhich of the equations is NOT equivalent to 8p = 12? Select all that apply.
    \n\u2610 8p \u00f7 8 = 12 \u00f7 8
    \n\u2610 8p \u00f7 8 = 12 \u00f7 12
    \n\u2610 8p + 4 = 12 + 4
    \n\u2610 8p – 2 = 12 – 2
    \n\u2610 8p \u00d7 8 = 12 \u00d7 12
    \nAnswer:
    \nThe given equation is:
    \n8p = 12
    \nNow.
    \nWe know that,
    \nAccording to the “Properties of Equality”,
    \nThe operation we are doing at the left side will be the same as the operation we are doing at the right side
    \nHence, from the above,
    \nWe can conclude that
    \nThe equations that are not equivalent to 8p = 12 are:
    \n\"\"<\/p>\n

    Lesson 4.3 Write and Solve Addition and Subtraction Equations<\/h3>\n

    Solve & Discuss It!<\/strong>
    \nA group of students were on a school bus. How many students were on the bus before the last stop?
    \nI can… write and solve an addition or subtraction equation.
    \n\"Envision<\/p>\n

    Use Appropriate Tools<\/strong>
    \nYou use a pan balance to help solve for the unknown.
    \nAnswer:
    \nIt is given that
    \nA group of students were on a school bus
    \nNow,
    \nThe given figure is:
    \n\"Envision
    \nNow,
    \nFrom the given figure,
    \nWe can observe that
    \nThe total number of students arrived at school = (The total number of students present on the bus before the last stop) + (The total number of students present on the bus at the last stop)
    \nNow,
    \nLet the total number of students present on the bus before the last stop be x
    \nSo,
    \n25 = x + 16
    \nx = 25 – 16
    \nx = 9 students
    \nHence, from the above,
    \nWe can conclude that
    \nThe total number of students present on the bus before the last stop is: 9 students<\/p>\n

    Focus on math practices<\/strong>
    \nReasoning<\/strong> How does using cubes on the pan balance demonstrate the Addition and Subtraction Properties of Equality?
    \nAnswer:
    \nIn the pan balance,
    \nThe cubes represent the variables
    \nRemember that the pans must be in a balanced position and it is represented by “=”
    \nEx:
    \nIn a pan balance,
    \nAt one side, 3x cubes are present
    \nAt other side, 2x cubes are present
    \nSo,
    \nThe balanced pan will be:
    \n3x = 2x
    \nEx:
    \nIn a pan balance,
    \nAt one side, 10x cubes are added
    \nAt other side, 25x cubes are added and later, 6x quantities are removed
    \nSo,
    \nThe balance pan will be:
    \n10x = 25x – 6x<\/p>\n

    Essential Question<\/strong>
    \nHow can you write and solve an addition or subtraction equation?
    \nAnswer:
    \na.
    \nTo write a proper addition equation, you have an equals sign. One side shows you the total. The other side shows you what things are being added together. You can have a variable on either side of the equation
    \nb.
    \nWhen we write a subtraction equation, we use two symbols: – and =. The minus sign (-) means one thing is being subtracted from another<\/p>\n

    Try It!<\/strong><\/p>\n

    Cabrini had some markers. After she bought 12 more markers, she had 16. How many markers did Cabrini have at the start?
    \nLet n represent the number of markers Cabrini had at the start.
    \nSolve the addition equation.
    \nn + 12 = 16
    \nn + 12 ___ = 16 ____
    \nn = _____
    \nCabrini had ______ markers at the start.
    \nAnswer:
    \nIt is given that
    \nCabrini had some markers. After she bought 12 more markers, she had 16
    \nNow,
    \nLet the number of markers Cabrini had before she bought more markers be: x
    \nSo,
    \nAccording to the given information,
    \nThe equation will be:
    \nx + 12 = 16
    \nNow,
    \nWe know that,
    \nAccording to the “Subtraction Property of Equality”,
    \nSubtract the same amount from each side to keep the equation balanced.
    \nSo,
    \nSubtract 12 from both sides in the given equation
    \nSo,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nCabrini had 4 markers at the start<\/p>\n

    Convince Me!<\/strong> Which property of equality is used to solve the equation n + 12 = 16? Could one of the other properties of equality have also been used? Explain.
    \nThe given equation is:
    \nn + 12 = 16
    \nNow,
    \nTo solve the above equation,
    \nThe property of equality we used is “Subtraction Property of Equality”
    \nNow,
    \nWe know that,
    \nAccording to the “Subtraction Property of Equality”,
    \nSubtract the same amount from each side to keep the equation balanced.
    \nNow,
    \nWe know that,
    \nIf we observe the above equation,
    \nWe can say that,
    \nWe can use other “Properties of Equality” like “Addition Property of Equality”, “Multiplication Property of Equality”, and “Division Property of Equality”
    \nHence, from the above,
    \nWe can conclude that
    \nWe used “Subtraction Property of Equality” to solve the given equation n + 12 = 16
    \nThe other properties of equality can also be used to solve the given equation n + 12 = 16<\/p>\n

    Try It!<\/strong><\/p>\n

    Vivian read 14 fewer pages than she was assigned to read. She read 60 pages. Write and solve an equation to find how many pages, p, Vivian was assigned to read.
    \nAnswer:
    \nIt is given that
    \nVivian read 14 fewer pages than she was assigned to read. She read 60 pages.
    \nNow,
    \nLet the number of pages Vivian was assigned to read be: x pages
    \nNow,
    \nAccording to the given information,
    \nThe equation that represents the given situation is:
    \nx – 14 = 60
    \nNow,
    \nWe know that,
    \nAccording to the “Addition Property of Equality”,
    \nAdd the same amount at each side to keep the equation balanced.
    \nSo,
    \nAdd 14 on both sides to the given equation
    \nSo,
    \nx – 14 + 14 = 60 + 14
    \nx = 74 pages
    \nHence, from the above,
    \nWe can conclude that
    \nThe number of pages Vivian was assigned to read is: 74 pages<\/p>\n

    KEY CONCEPT<\/strong>
    \nYou can use inverse relationships and the properties of equality to solve equations.
    \n\"Envision<\/p>\n

    Do You Understand?<\/strong>
    \nQuestion 1.
    \nEssential Question<\/strong> How can you write and solve an addition or subtraction equation?
    \nAnswer:
    \na.
    \nTo write a proper addition equation, you have an equals sign. One side shows you the total. The other side shows you what things are being added together. You can have a variable on either side of the equation
    \nb.
    \nWhen we write a subtraction equation, we use two symbols: – and =. The minus sign (-) means one thing is being subtracted from another<\/p>\n

    Question 2.
    \nExplain how you can use the inverse relationship of addition and subtraction to solve the equation n + 7 = 25.
    \nAnswer:
    \nThe given equation is:
    \nn + 7 = 25
    \nNow,
    \nWe know that,
    \nAddition and subtraction are inverse operations because\u00a0one operation can “undo” the other operation
    \nNow,
    \nWe know that,
    \nAccording to the “Subtraction Property of Equality”,
    \nSubtract the same amount from each side to keep the equation balanced.
    \nNow,
    \nsubtract 7 from both sides
    \nSo,
    \nn + 7 – 7 = 25 – 7
    \nn = 18
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of n for the given equation is: 18<\/p>\n

    Question 3.
    \nModel with Math<\/strong> Clare had t seashells. After she bought 8 more seashells, she had 24 seashells. Write and solve an equation to find the number of seashells Clare started with.
    \nAnswer:
    \nIt is given that
    \nClare had t seashells. After she bought 8 more seashells, she had 24 seashells.
    \nSo,
    \nThe number of seashells = (The number of seashells Clare had at the start) + (The number of seashells she bought after the start)
    \n24 = t + 8
    \nNow,
    \nWe know that,
    \nAccording to the “Subtraction Property of Equality”,
    \nSubtract the same amount from each side to keep the equation balanced.
    \nNow,
    \nsubtract 8 from both sides
    \nSo,
    \n24 – 8 = t + 8 – 8
    \nt = 16 seashells
    \nHence, from the above,
    \nWe can conclude that
    \nThe number of seashells Clare had at the start is: 16 seashells<\/p>\n

    Question 4.
    \nModel with Math<\/strong> The outside temperature dropped 20\u00b0F from the time Arianna ate breakfast until the time she ate dinner. When she ate dinner the temperature was 35\u00b0F. Write and solve an equation to find the outside temperature t when Arianna ate breakfast.
    \nAnswer:
    \nIt is given that
    \nThe outside temperature dropped 20\u00b0F from the time Arianna ate breakfast until the time she ate dinner. When she ate dinner the temperature was 35\u00b0F
    \nSo,
    \nThe outside temperature from the time Arianna ate breakfast until the time she ate dinner = (The outside temperature when Arianna ate breakfast) + (The outside temperature when Arianna ate dinner)
    \n20\u00b0 F = x\u00b0 F + 35\u00b0 F
    \nNow,
    \nWe know that,
    \nAccording to the “Subtraction Property of Equality”,
    \nSubtract the same amount from each side to keep the equation balanced.
    \nNow,
    \nsubtract 35\u00b0 F from both sides
    \nSo,
    \n20\u00b0 F – 35\u00b0 F = x\u00b0 F + 35\u00b0 F – 35\u00b0 F
    \nx\u00b0 F = – 15\u00b0 F
    \nHence, from above,
    \nWe can conclude that
    \nThe outside temperature when Arianna ate breakfast is: -15\u00b0 F<\/p>\n

    Do You Know How?<\/strong>
    \nIn 5-10, solve each equation.<\/strong>
    \nQuestion 5.
    \n24 + m = 49
    \nAnswer:
    \nThe given equation is:
    \n24 + m = 49
    \nNow,
    \nSubtract with 24 on both sides
    \nSo,
    \n24 + m – 24 = 49 – 24
    \nm = 25
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of m is: 25<\/p>\n

    Question 6.
    \n12 = y – 11
    \nAnswer:
    \nThe given equation is:
    \n12 = y – 11
    \nNow,
    \nAdd 11 on both sides
    \nSo,
    \n12 + 11 = y – 11 + 11
    \ny = 23
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of y is: 23<\/p>\n

    Question 7.
    \n22 = 13 + a
    \nAnswer:
    \nThe givene quation is:
    \n22 = 13 + a
    \nNow,
    \nSubtract 13 on both sides
    \nSo,
    \n22 – 13 = 13 + a – 13
    \na = 9
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of a is: 9<\/p>\n

    Question 8.
    \nt – 40 = 3
    \nAnswer:
    \nThe given equation is:
    \nt – 40 = 3
    \nNow,
    \nAdd 40 on both sides
    \nSo,
    \nt – 40 + 40 = 3 + 40
    \nt = 43
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of t is: 43<\/p>\n

    Question 9.
    \nd + 11 = 15
    \nAnswer:
    \nThe given equation is:
    \nd + 11 = 15
    \nNow,
    \nSubtract 11 on both sides
    \nSo,
    \nd + 11 – 11 = 15 – 11
    \nd = 4
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of d is: 4<\/p>\n

    Question 10.
    \n32 = s – 19
    \nAnswer:
    \nThe given equation is:
    \n32 = s – 19
    \nNow,
    \nAdd 19 on both sides
    \nSo,
    \n32 + 19 = s – 19 + 19
    \ns = 51
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of s is: 51<\/p>\n

    Practice & Problem Solving<\/strong><\/p>\n

    Leveled Practice
    \nIn 11-16, solve each equation.<\/strong>
    \nQuestion 11.
    \ny – 12 = 89
    \ny – 12 + _____ = 89 + 12
    \ny = ______
    \nAnswer:
    \nThe given equation is:
    \ny – 12 = 89
    \nNow,
    \nAdd 12 on both sides
    \nSo,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of y is: 101<\/p>\n

    Question 12.
    \n80 + r = 160
    \n80 + r – ______ = 160 – ______
    \nr = ______
    \nAnswer:
    \nThe given equation is:
    \n80 + r = 160
    \nNow,
    \nSubtract 80 on both sides
    \nSo,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of r is: 80<\/p>\n

    Question 13.
    \n60 = x \u2013 16
    \n60 + ______ = x – 16 + ______
    \n______ = x
    \nAnswer:
    \nThe given equation is:
    \n60 = x – 16
    \nNow,
    \nAdd 16 on both sides
    \nSo,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of x is: 76<\/p>\n

    Question 14.
    \n20 = y + 12
    \nAnswer:
    \nThe givene quation is:
    \n20 = y + 12
    \nNow,
    \nSubtract 12 on both sides
    \nSo,
    \n20 – 12 = y + 12 – 12
    \ny = 8
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of y is: 8<\/p>\n

    Question 15.
    \nx + 2 = 19
    \nAnswer:
    \nThe given equation is:
    \nx + 2 = 19
    \nNow,
    \nSubtract 2 on both sides
    \nSo,
    \nx + 2 – 2 = 19 – 2
    \nx = 17
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of x is: 17<\/p>\n

    Question 16.
    \nz – 313 = 176
    \nAnswer:
    \nThe given equation is:
    \nz – 313 = 176
    \nNow,
    \nAdd 313 on both sides
    \nSo,
    \nz – 313 + 313 = 176 + 313
    \nz = 489
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of z is: 489<\/p>\n

    Question 17.
    \nYou have some baseball trading cards. You give 21 baseball cards to a friend and have 9 left for yourself. How many baseball cards were in your original deck? Write and solve an equation to find t, the number of baseball cards in your original deck.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nYou have some baseball trading cards. You give 21 baseball cards to a friend and have 9 left for yourself
    \nNow,
    \nThe representation of the given information in the form of a bar diagram is:
    \n\"Envision
    \nNow,
    \nFrom the given bar diagram,
    \nWe can observe that
    \n(The total number of baseball trading cards) – (The number of baseball cards you have given to your friend) = (The number of baseball cards you have left)
    \nSo,
    \nt – 21 = 9
    \nNow,
    \nAdd 21 on both sides
    \nSo,
    \nt – 21 + 21 = 9 + 21
    \nt = 30
    \nHence, from the above,
    \nWe can conclude that
    \nThe total number of baseball trading cards in your original deck is: 30 cards<\/p>\n

    Question 18.
    \nModel with Math<\/strong> Joy added 26 new contacts to her phone list. She now has a total of 100 contacts. Let c represent how many contacts Joy had on her phone list before she updated it. Write an equation and solve for c.
    \nAnswer:
    \nIt is given that
    \nJoy added 26 new contacts to her phone list. She now has a total of 100 contacts. Let c represent how many contacts Joy had on her phone list before she updated it
    \nNow,
    \nAccording to the given information,
    \nThe representation of the equation for the given information is:
    \nc + 26 = 100
    \nNow,
    \nSubtract 26 on both sides
    \nSo,
    \nc + 26 – 26 = 100 – 26
    \nc = 84 contacts
    \nHence, from the above,
    \nWe can conclude that
    \nThe number of contacts Joy had on her phone list before she updated is: 84 contacts<\/p>\n

    Question 19.
    \nReasoning<\/strong> Jeremy bought a sandwich and a drink that cost him $7. His drink cost $1.75. Solve the equation 7 = s + 1.75 to find s, the cost of Jeremy’s sandwich.
    \nAnswer:
    \nIt is given that
    \nJeremy bought a sandwich and a drink that cost him $7. His drink cost $1.75
    \nNow,
    \nLet s be the cost of Jeremy’s sandwich
    \nSo,
    \nAccording to the given information,
    \nThe total cost pf a sandwich and a drink that Jeremy bought = (The cost of Jeremy’s sandwich) + (The cost of Jeremy’s drink)
    \n$7 = s + $1.75
    \nNow,
    \nsubtract $1.75 on both sides
    \nSo,
    \n$7 – $1.75 = s + $1.25 – $1.25
    \ns = $5.25
    \nHence, from the baove,
    \nWe can conclude that
    \nThe cost of Jeremy’s sandwich is: $5.25<\/p>\n

    Question 20.
    \nA triathlon is about 51 Kilometers. One participant completed two of the three legs of the race and traveled 42 Kilometers. Solve the equation 42 + d = 51 for the distance, d, of the third leg of the race.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nA triathlon is about 51 Kilometers. One participant completed two of the three legs of the race and traveled 42 Kilometers
    \nNow,
    \nThe representation of the given information in the form of a bar diagram is:
    \n\"Envision<\/p>\n

    Now,
    \nThe representation of the equation that represents the given information ad as given in the problem is:
    \n42 + d = 51
    \nWhere,
    \nd is the Distance of the third leg
    \nNow,
    \nSubtract 41 from both sides
    \nSo,
    \n42 + d – 42 = 51 – 42
    \nd = 9 Kilometers
    \nHence, from the above,
    \nWe can conclude that
    \nThe distance of the third leg of the race is: 9 Kilometers<\/p>\n

    Question 21.
    \nWhat operation should be used to solve the equation 153 = 9 + 45? Solve the equation.
    \nAnswer:
    \nThe given equation is:
    \n153 = 9 + 45
    \nNow,
    \nWhen we observe the equation,
    \nThere is “Addition” operation taking place in the given equation
    \nNow,
    \nWe have to subtract the required number
    \nHence, from the above,
    \nWe can conclude that
    \nThe operation we should be used to solve the given equation 153 = 9 + 45 is: Subtraction<\/p>\n

    Question 22.
    \nHigher Order Thinking<\/strong> In the equation 6 + 3y = 4y + 2 the variable y represents the same value. Is y = 2, 3, 4, or 5 the solution of this equation? Explain.
    \nAnswer:
    \nIt is given that
    \nIn the equation 6 + 3y = 4y + 2 the variable y represents the same value
    \nNow,
    \nThe given equation is:
    \n6 + 3y = 4y + 2
    \nNow,
    \nFor y = 2:
    \n6 + 3y = 4y + 2
    \n6 + 3 (2) = 4 (2) + 2
    \n6 + 6 = 8 + 2
    \n12 \u2260 10
    \nFor y = 3:
    \n6 + 3y = 4y + 2
    \n6 + 3 (3) = 4 (3) + 2
    \n6 + 9 = 12 + 2
    \n15 \u2260 14
    \nFor y = 4;
    \n6 + 3y = 4y + 2
    \n6 + 3 (4) + 4 (4) + 2
    \n6 + 12 = 16 + 2
    \n18 = 18
    \nFor y = 5:
    \n6 + 3y = 4y + 2
    \n6 + 3 (5) = 4 (5) + 2
    \n6 + 15 = 20 + 2
    \n21 \u2260 22
    \nHence, from the above,
    \nWe can conclude that
    \ny = 4 is the solution for the given equation<\/p>\n

    Question 23.
    \nA traffic helicopter descends to hover 477 meters above the ground. Let h be the original height of the helicopter. What is a subtraction equation that represents the problem? What was the original height of the helicopter?
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nA traffic helicopter descends to hover 477 meters above the ground. Let h be the original height of the helicopter
    \nNow,
    \nThe given figure is:
    \n\"Envision
    \nNow,
    \nFrom the given figure,
    \nWe can observe that
    \n(The original height of helicopter) + 127 m = 477 m
    \nh – 127 = 477
    \nNow,
    \nAdd 127 on both sides
    \nSo,
    \nh + 127 – 127 = 477 + 127
    \nh = 604 meters
    \nHence, from the above,
    \nWe can conclude that
    \nThe subtraction equation that represents the given problem is:
    \nh\u00a0 – 127 = 477
    \nThe original height of the helicopter is: 604 meters<\/p>\n

    Question 24.
    \nThe drama club sold all the tickets for its annual production in three days. The club sold 143 tickets the first day and 295 tickets the second day. If the drama club sold 826 tickets, how many tickets were sold on the third day of sales? Solve the equation 438 + t = 826 for the number of tickets, t, sold on the third day of ticket sales.
    \nAnswer:
    \nIt is given that
    \nThe drama club sold all the tickets for its annual production in three days. The club sold 143 tickets the first day and 295 tickets the second day. If the drama club sold 826 tickets
    \nSo,
    \nThe total number of tickets sold by the drama club in two days = (The number of tickets sold on the first day) + (The number of tickets sold on the second day)
    \n= 143 + 295
    \n= 438 tickets
    \nNow,
    \nLet the number of tickets sold on third day be: t
    \nSo,
    \nAccording to the given information,
    \nThe equation that represents the given information is:
    \n438 + t = 826
    \nNow,
    \nSubtract 438 on both sides
    \nSo,
    \n438 + t – 438 = 826 – 438
    \nt = 388 tickets
    \nHence, from the above,
    \nWe can conclude that
    \nThe number of tickets sold on the third day of ticket sales is: 388 tickets<\/p>\n

    Question 25.
    \nIn a bag of mixed nuts, there are 35 almonds, 34 pecans, 32 walnuts, and p pistachios. The bag has a total of 134 nuts. Find the total number of almonds, pecans, and walnuts. Then write and solve an equation to find the number of pistachios in the bag.
    \nAnswer:
    \nIt is given that
    \nIn a bag of mixed nuts, there are 35 almonds, 34 pecans, 32 walnuts, and p pistachios. The bag has a total of 134 nuts
    \nSo,
    \nThe total number of almonds, pecans, and walnuts = 35 + 34 + 32
    \n= 101
    \nNow,
    \nThe total number of almonds, peacons, walnuts, and pistachios = (The total number of almond, pecans, and walnuts) + (The number of pistachios)
    \nSo,
    \n134 = 101 + p
    \nNow,
    \nSubtract 101 on both sides
    \nSo,
    \n134 – 101 = 101 + p – 101
    \np = 33
    \nHence, from the above,
    \nWe can conclude that
    \nThe number of pistachios in the bag is: 33 pistachios<\/p>\n

    Assessment Practice<\/strong><\/p>\n

    Question 26.
    \nWhich equation has g = 6 as the solution?
    \nA. g+ 2 = 10
    \nB. 9- 1 = 10
    \nC. 58 +9 = 60
    \nD. 44 – g = 38
    \nAnswer:
    \nThe given equations are:
    \nA. g+ 2 = 10
    \nB. 9- 1 = 10
    \nC. 58 +g = 60
    \nD. 44 – g = 38
    \nSo,
    \nA.
    \ng + 2 = 10
    \ng = 10 – 2
    \ng = 8
    \nB.
    \ng – 1 = 10
    \ng = 10 + 1
    \ng = 11
    \nC.
    \n58 + g = 60
    \ng = 60 – 58
    \ng = 2
    \nD.
    \n44 – g = 38
    \ng = 44- 38
    \ng = 6
    \nHence, from the above,
    \nWe can conclude that
    \nThe equation that has g = 6 as the solution is:
    \n\"\"<\/p>\n

    Question 27.
    \nSelect all the equations that have the same solution as 36 = x + 32.
    \n\u2610 42 = 38 + x
    \n\u2610 x + 15 = 19
    \n\u2610 18 = x – 2
    \n\u2610 36 = x – 32
    \n\u2610 52 – x = 46
    \nAnswer:
    \nThe given equation is:
    \n36 = x + 32
    \nSo,
    \nx = 36 – 32
    \nx = 4
    \nNow,
    \nThe given equations are:
    \nA.
    \n42 = 38 + x
    \nx = 42 – 38
    \nx = 4
    \nB.
    \nx + 15 = 19
    \nx = 19 – 15
    \nx = 4
    \nC.
    \n18 = x – 2
    \nx = 18 + 2
    \nx = 20
    \nD.
    \n36 = x – 32
    \nx = 36 + 32
    \nx = 68
    \nE.
    \n52 – x = 46
    \nx = 52 – 46
    \nx = 6
    \nHence, from the above,
    \nWe can conclude that
    \nAll the equations that have the same solution as 36 = x + 32 are:
    \n\"\"<\/p>\n

    Lesson 4.4 Write and Solve Multiplication and Division Equations<\/h3>\n

    Solve & Discuss It!<\/strong>
    \nA school group is planning a trip to New York City. There are 29 people going on the trip. They agreed to share the total cost of the trip equally. Let s equal each person’s share of the cost. What is each person’s share of the cost?<\/p>\n

    I can… write and solve a multiplication or division equation.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nA school group is planning a trip to New York City. There are 29 people going on the trip. They agreed to share the total cost of the trip equally. Let s equal each person’s share of the cost
    \nNow,
    \nThe given table is:
    \n\"\"
    \nNow,
    \nFrom the given table,
    \nWe can observe that
    \nThe total cost is: $19,111
    \nSo,
    \nAccording to the given information,
    \n(The total number of people going to a trip) \u00d7 (The share of the cost of each person) = $19,111
    \n29 \u00d7 s = $19,111
    \ns = \\(\\frac{$19,111}{29}\\)
    \ns = $659
    \nHence, from the above,
    \nWe can conclude that
    \nEach person’s share of the cost is: $659<\/p>\n

    Generalize<\/strong>
    \nHow can you use what you know about dividing lesser numbers to write equations and solve problems involving greater numbers?
    \nAnswer:
    \nLet a smaller faction be: \\(\\frac{a}{b}\\)
    \nWhere,
    \na = 1, 2, ……… n
    \nb = 1, 2, ……… n
    \nNow,
    \nWhen we divide lesser numbers say the lesser numbers only have 2 digits, then the quotient we will also get will be a small number
    \nNow,
    \nWhen we divide larger numbers say the larger numbers have 4 digits, then the quotient we will also get will be a large number with 3 (or) 4 digits<\/p>\n

    Focus on math practices<\/strong>
    \nConstruct Arguments<\/strong> Can you use the same strategy that you used above to find each person’s share of the hotel bill? Explain.
    \nAnswer:
    \nYes,
    \nWe can use the same strategy that you used above to find each person’s share of the hotel bill<\/p>\n

    Essential Question<\/strong>
    \nHow can you write and solve a multiplication or division equation?
    \nAnswer:
    \nThe steps to solve a multiplication or division problem is:
    \na. Always perform the same operation to both sides of the equation.
    \nb. When you multiply or divide, you have to multiply and divide by the entire side of the equation.
    \nc. Try to perform addition and subtraction first to get some multiple of x by itself on one side.<\/p>\n

    Try It!<\/strong><\/p>\n

    Theresia picked the same number of tomatoes each day. In 4 days she picked 52 tomatoes. How many tomatoes did Theresia pick each day? Let n represent the number of tomatoes Theresia picked each day.
    \n4n = 52
    \n4n \u00f7 4 = 52 ______
    \nn = _______
    \nTheresia picked _______ tomatoes each day.
    \nAnswer:
    \nIt is given that
    \nTheresia picked the same number of tomatoes each day. In 4 days she picked 52 tomatoes
    \nNow,
    \nLet n represent the number of tomatoes Theresia picked each day.
    \nSo,
    \n4 \u00d7 (The number of tomatoes Theresia picked each day) = (The number of tomatoes Theresia picked in 4 days)
    \n4 \u00d7 n = 52
    \nNow,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nTheresia picked 13 tomatoes each day<\/p>\n

    Convince Me!<\/strong> Which property or equality can you use to solve Theresia’s equation? Explain.
    \nAnswer:
    \nFrom the above problem,
    \nThe equation is:
    \n4n = 52
    \nNow,
    \nTo find the value of n (or) the number of tomatoes Theresia picked each day,
    \nWe have to use “Division Property of Equality”
    \nNow,
    \nWe know that,
    \nAccording to the “Division Property of Equality”,
    \nDivide the same amount to the equation to keep the equation balanced
    \nHence, from the above,
    \nWe can conclude that
    \nWe used the “Division Property of equality” to solve Theresia’s equation<\/p>\n

    Try It!<\/strong><\/p>\n

    Meghann is reading a 630-page book. She reads 18 pages each day. Write and solve a division equation to find the number of days, d, it will take Meghann to finish her book.
    \nAnswer:
    \nIt is given that
    \nMeghann is reading a 630-page book. She reads 18 pages each day
    \nNow,
    \nLet d be the number of days it will take Meghann to finish her book
    \nNow,
    \n(The number of pages Meghann reads each day) \u00d7 (The number of days it will take Meghann to finish her book) = (The total number of pages in a book)
    \n18 \u00d7 d = 630
    \nd = \\(\\frac{630}{18}\\)
    \nd = 35 days
    \nHence, from the above,
    \nWe can conclude that
    \nIt will take 35 days for Meghann to finish her 630-page book<\/p>\n

    KEY CONCEPT<\/strong>
    \nYou can multiply or divide both sides of an equation by the same number and it will remain balanced.
    \n\"Envision<\/p>\n

    Do You Understand?<\/strong>
    \nQuestion 1.
    \nEssential Question<\/strong> How can you write and solve a multiplication or division equation?
    \nAnswer:
    \nThe steps to solve a multiplication or division problem is:
    \na. Always perform the same operation to both sides of the equation.
    \nb. When you multiply or divide, you have to multiply and divide by the entire side of the equation.
    \nc. Try to perform addition and subtraction first to get some multiple of x by itself on one side.<\/p>\n

    Question 2.
    \nWhich property of equality would you use to solve the equation 8n = 16?
    \nAnswer:
    \nThe given equation is:
    \n8n = 16
    \nNow,
    \nDivide by 8 to find the value of n
    \nSo,
    \nWe have to use “Division Property of Equality”
    \nNow,
    \nWe know that,
    \nAccording to the “Division Property of Equality”,
    \nDivide the same amount at each side to keep the equation balanced.
    \nSo,
    \n\\(\\frac{8n}{8}\\) = \\(\\frac{16}{8}\\)
    \nn = 2
    \nHence, from the above,
    \nWe can conclude that
    \nYou would use “Division Property of Equality” to solve the given equation 8n = 16<\/p>\n

    Question 3.
    \nWhich property of equality would you use to solve the equation a \u00f7 9 = 2?
    \nAnswer:
    \nThe given equation is:
    \na \u00f7 9 = 2
    \nNow,
    \nMultiply with 9 to find the value of ‘a’
    \nSo,
    \nWe have to use “Multiplication Property of Equality”
    \nNow,
    \nWe know that,
    \nAccording to the “Multiplication Property of Equality”,
    \nMultiply the same amount at each side to keep the equation balanced.
    \nSo,
    \n(a \u00f7 9) \u00d7 9 = 2 \u00d7 9
    \na = 18
    \nHence, from the above,
    \nWe can conclude that
    \nYou would use “Multiplication Property of Equality” to solve the given equation a \u00f7 9 = 2<\/p>\n

    Question 4.
    \nThere are 30 students in the drama club. They are carpooling in 5 vans to perform a play. They want each van to carry an equal number of students. Let s be the number of students in each van. Write and solve a multiplication equation to find the number of students in each van.
    \nAnswer:
    \nIt is given that
    \nThere are 30 students in the drama club. They are carpooling in 5 vans to perform a play. They want each van to carry an equal number of students.
    \nNow,
    \nLet s be the number of students in each van
    \nSo,
    \n(The number of vans carpooling to perform a play) \u00d7 (The number of students in each van) = (The total number of students in the drama club)
    \n5 \u00d7 s = 30
    \n5s = 80
    \nNow,
    \nDivide by 5 on both sides
    \nSo,
    \n\\(\\frac{5s}{5}\\) = \\(\\frac{80}{5}\\)
    \ns = 16 students
    \nHence, from the above,
    \nWe can conclude that
    \nThe number of students in each van is: 16 students<\/p>\n

    Do You Know How?<\/strong>
    \nIn 5-8, explain how to solve each equation.<\/strong>
    \nQuestion 5.
    \n18m = 36
    \nAnswer:
    \nThe given equation is:
    \n18m = 36
    \nNow,
    \nDivide the equation by 18 on both sides
    \nSo,
    \n\\(\\frac{18m}{18}\\) = \\(\\frac{36}{18}\\)
    \nm = 2
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of m in the given equation is: 2<\/p>\n

    Question 6.
    \nt \u00f7 3 = 10
    \nAnswer:
    \nThe given equation is:
    \nt \u00f7 3 = 10
    \nNow,
    \nMultiply the equation with 3 on both sides
    \nSo,
    \n(t \u00f7 3) \u00d7 3 = 10 \u00d7 3
    \nt = 30
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of t in the given equation is: 30<\/p>\n

    Question 7.
    \n12 = 2y
    \nAnswer:
    \nThe given equation is:
    \n12 = 2y
    \nNow,
    \nDivide the equation by 2 on both sides
    \nSo,
    \n\\(\\frac{12}{2}\\) = \\(\\frac{2y}{2}\\)
    \n6 = y
    \ny = 6
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of y in the given equation is: 6<\/p>\n

    Question 8.
    \n22 = a \u00f7 5
    \nAnswer:
    \nThe given equation is:
    \n22 = a \u00f7 5
    \nNow,
    \nMultiply the equation with 5 on both sides
    \nSo,
    \n22 \u00d7 5 = (a \u00f7 5) \u00d7 5
    \na = 110
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of a in the given equation is: 110<\/p>\n

    In 9-12, solve each equation.<\/strong>
    \nQuestion 9.
    \n23d = 2,392
    \nAnswer:
    \nThe given equation is:
    \n23d = 2,392
    \nNow,
    \nDivide the given equation by 23 on both sides
    \nSo,
    \n\\(\\frac{23d}{23}\\) = \\(\\frac{2,392}{23}\\)
    \nd = 104
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of d in the given equation is: 104<\/p>\n

    Question 10.
    \n74f = 6,179
    \nAnswer:
    \nThe given equation is:
    \n74f = 6,179
    \nNow,
    \nDivide the given equation by 74 on both sides
    \nSo,
    \n\\(\\frac{74f}{74}\\) = \\(\\frac{6,179}{74}\\)
    \nf = 83.5
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of f in the given equation is: 83.5<\/p>\n

    Question 11.
    \ny \u00f7 11 = 987
    \nAnswer:
    \nThe given equation is:
    \ny \u00f7 11 = 987
    \nNow,
    \nMultiply the given equation with 11 on both sides
    \nSo,
    \n(y \u00f7 11) \u00d7 11 = 987 \u00d7 11
    \ny = 10,857
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of y in the given equation is: 10,857<\/p>\n

    Question 12.
    \nr \u00f7 187 = 9
    \nAnswer:
    \nThe given equation is:
    \nr \u00f7 187 = 9
    \nNow,
    \nMultiply the given equation with 187 on both sides
    \nSo,
    \n(r \u00f7 187) \u00d7 187 = 9 \u00d7 187
    \ny = 1,683
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of r in the given equation is: 1,683<\/p>\n

    Practice & Problem Solving<\/strong><\/p>\n

    In 13-16, explain how to get the variable alone in each equation.<\/strong>
    \nQuestion 13.
    \n8y = 56
    \nAnswer:
    \nThe given equation is:
    \n8y = 56
    \nNow,
    \nDivide the given equation with 8 on both sides
    \nso,
    \n\\(\\frac{8y}{8}\\) = \\(\\frac{56}{8}\\)
    \ny = 7
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of y for the given equation is: 7<\/p>\n

    Question 14.
    \nt \u00f7 15 = 3
    \nAnswer:
    \nThe given equation is:
    \nt \u00f7 15 = 3
    \nNow,
    \nMultiply the given equation with 15 on both sides
    \nSo,
    \n(t \u00f7 15) \u00d7 15 = 3 \u00d7 15
    \nt = 45
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of t for the given equation is: 45<\/p>\n

    Question 15.
    \nu \u00f7 8 = 12
    \nAnswer:
    \nThe given equation is:
    \nu \u00f7 8 = 12
    \nNow,
    \nMultiply the given equation with 8 on both sides
    \nSo,
    \n(u \u00f7 8) \u00d7 8 = 12 \u00d7 8
    \nu = 96
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of u for the given equation is: 96<\/p>\n

    Question 16.
    \n31y = 310
    \nAnswer:
    \nThe given equation is:
    \n31y = 310
    \nNow,
    \nDivide the given equation with 31 on both sides
    \nso,
    \n\\(\\frac{31y}{31}\\) = \\(\\frac{310}{31}\\)
    \ny = 10
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of y for the given equation is: 10<\/p>\n

    In 17-20, solve each equation.<\/strong>
    \nQuestion 17.
    \nd \u00f7 2 = 108
    \nAnswer:
    \nThe given equation is:
    \nd \u00f7 2 = 108
    \nNow,
    \nMultiply the given equation with 2 on both sides
    \nSo,
    \n(d \u00f7 2) \u00d7 2 = 108 \u00d7 2
    \nd = 216
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of d for the given equation is: 216<\/p>\n

    Question 18.
    \n7,200 = 800s
    \nAnswer:
    \nThe given equation is:
    \n7,200 = 800s
    \nNow,
    \nDivide the given equation with 800 on both sides
    \nso,
    \n\\(\\frac{7,200}{800}\\) = \\(\\frac{800s}{800}\\)
    \n9 = s
    \ns = 9
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of s for the given equation is: 9<\/p>\n

    Question 19.
    \nx \u00f7 3 = 294
    \nAnswer:
    \nThe given equation is:
    \nx \u00f7 3 = 294
    \nNow,
    \nMultiply the given equation with 3 on both sides
    \nSo,
    \n(x \u00f7 3) \u00d7 3 = 294 \u00d7 3
    \nd = 882
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of x for the given equation is: 882<\/p>\n

    Question 20.
    \n99 = 3x
    \nAnswer:
    \nThe given equation is:
    \n99 = 3x
    \nNow,
    \nDivide the given equation with 3 on both sides
    \nso,
    \n\\(\\frac{99}{3}\\) = \\(\\frac{3x}{3}\\)
    \n33 = x
    \nx = 33
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of x for the given equation is: 33<\/p>\n

    In 21 and 22, write a division equation and a multiplication equation to represent each problem.<\/strong>
    \nQuestion 21.
    \nLolo typed 1,125 words in 15 minutes. Let w represent the number of words typed each minute. If Lolo typed the same number of words each minute, how many words did she type in 1 minute?
    \nAnswer:
    \nIt is given that
    \nLolo typed 1,125 words in 15 minutes. Let w represent the number of words typed each minute. and Lolo typed the same number of words each minute
    \nSo,
    \n(The number of minutes needed to type 1,125 words) \u00d7 (The number of words Lolo typed each minute) = 1,125
    \n15 \u00d7w = 1,125
    \n15w = 1,125
    \nNow,
    \nDivide by 15 on both sides
    \nSo,
    \n\\(\\frac{15w}{15}\\) = \\(\\frac{1,125}{15}\\)
    \nw = \\(\\frac{1,125}{15}\\)
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the multiplication equation and the division equation to represent the given situation is:
    \na. Multiplication equation: 15w = 1,125
    \nb. Division equation: w = \\(\\frac{1,125}{15}\\)<\/p>\n

    Question 22.
    \nIn 12 weeks Felipe earns $4,500 doing yard work. He earns the same amount each week. Let m stand for the amount earned each week. How much does Felipe make in 1 week?
    \nAnswer:
    \nIt is given that
    \nIn 12 weeks Felipe earns $4,500 doing yard work. He earns the same amount each week. Let m stand for the amount earned each week
    \nSo,
    \n(The number of weeks needed to earn $4,500) \u00d7 (The amount earned by Felipe each week) = $4,500
    \n12 \u00d7 m = $4,500
    \n12m = $4,500
    \nNow,
    \nDivide by 12 on both sides
    \nSo,
    \n\\(\\frac{12m}{12}\\) = \\(\\frac{$4,500}{12}\\)
    \nm = \\(\\frac{$4,500}{12}\\)
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the multiplication equation and the division equation to represent the given situation is:
    \na. Multiplication equation: 12m = $4,500
    \nb. Division equation: m = \\(\\frac{$4,500}{12}\\)<\/p>\n

    Question 23.
    \nModel with Math<\/strong> Abel has 3,330 toothpicks. He wants to use them all to make a floor mat with 18 equal rows. Use the bar diagram to write a division equation. Then solve the equation to find how many toothpicks Abel should use in each row.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nAbel has 3,330 toothpicks. He wants to use them all to make a floor mat with 18 equal rows
    \nNow,
    \nThe representation of the gven situation in the form of a bar diagram is:
    \n\"Envision
    \nNow,
    \nLet the number of toothpicks Abel shoud use in each row be: t
    \nNow,
    \nFrom the given bar diagram,
    \nWe can observe that
    \n(The total number of toothpicks) \u00f7 (The number of tooth picks Abel should use in each row) = (The total number of rows)
    \n\\(\\frac{3,330}{t}\\) = 18
    \nt = \\(\\frac{3,330}{18}\\)
    \nt = 185 toothpicks
    \nHence, from the above,
    \nWe can conclude that
    \nThe number of toothpicks Abel should use in each row is: 185 toothpicks<\/p>\n

    Question 24.
    \nModel with Math<\/strong> Emily took an airplane trip. Her plane flew an equal number of miles each hour. Let m stand for the miles flown each hour. Write an equation to represent one way you can find how many miles Emily’s plane flew each hour.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nEmily took an airplane trip. Her plane flew an equal number of miles each hour. Let m stand for the miles flown each hour.
    \nNow,
    \nThe given figure is:
    \n\"Envision
    \nNow,
    \nFrom the figure,
    \nWe can observe that
    \nThe total distance airplane travels is: 2,184 miles in 12 hours
    \nSo,
    \n(The time took for airplane to reach Emily’s destination) \u00d7 (The number of miles Emily’s plane flew each hour) = (The total distance an airplane travels in 12 hours)
    \n12 \u00d7 m = 2,184
    \n12m = 2,184
    \nNow,
    \nDivide by 12 on both sides
    \nSo,
    \nm = \\(\\frac{2,184}{12}\\)
    \nm = 182 miles
    \nHence, from the above,
    \nWe can conclude that
    \nThe number of miles Emily’s plane flew each hour is: 182 miles<\/p>\n

    In 25 and 26, use the triangle.<\/strong>
    \n\"Envision
    \nQuestion 25.
    \nThe area of the isosceles triangle is 44 square centimeters. Use the equation \\(\\frac{1}{2}\\)(8h) = 44 to find the height of the triangle.
    \nAnswer:
    \nIt is given that
    \nThe area of the isosceles triangle is 44 square centimeters
    \nNow,
    \nThe given figure is:
    \n\"Envision
    \nNow,
    \nWe know that,
    \nThe area of an Isosceles triangle (A) = \\(\\frac{1}{2}\\) \u00d7 Base \u00d7 Height
    \nSo,
    \nAccoring to the given information,
    \n\\(\\frac{1}{2}\\) \u00d7 8 \u00d7 h = 44
    \n\\(\\frac{8h}{2}\\) = 44
    \n8h = 44 \u00d7 2
    \n8h = 88
    \nh = \\(\\frac{88}{8}\\)
    \nh = 11 cm
    \nHence, from the above,
    \nWe can conclude that
    \nThe height of the given triangle is: 11 cm<\/p>\n

    Question 26.
    \nIf the perimeter of the triangle is 32 centimeters, what is the length of each of the two sides? Write and solve an equation.
    \nAnswer:
    \nIt is given that
    \nThe perimeter of the triangle is 32 centimeters
    \nNow,
    \nThe given figure is:
    \n\"Envision
    \nNow,
    \nWe know that,
    \nIn an Isosceles triangle, 2 sides lengths are the same
    \nThe “Perimeter” of any figure is the sum of all the side lengths in a given figure
    \nNow,
    \nLet x be the side length of the given triangle
    \nSo,
    \nx + x + 8 = 32
    \n2x + 8 = 32
    \n2x = 32 – 8
    \n2x = 24
    \nx = \\(\\frac{24}{2}\\)
    \nx = 12 cm
    \nHence, from the above,
    \nWe can conclude that
    \nThe length of one side in the given triangle is: 12 cm
    \nThe length of the other side in the given triangle is: 12 cm<\/p>\n

    Question 27.
    \nKelsey and her 4 sisters spent an equal amount of time cleaning their home. Their parents added their times. They found that each of the 5 girls spent 3 hours cleaning. Let c be the total number of hours the girls spent cleaning. Write and solve a division equation to find the total number of hours the girls spent cleaning.
    \nAnswer:
    \nIt is given that
    \nKelsey and her 4 sisters spent an equal amount of time cleaning their home. Their parents added their times. They found that each of the 5 girls spent 3 hours cleaning. Let c be the total number of hours the girls spent cleaning.
    \nSo,
    \nAccording to the given information,
    \n(The total number of hours the girls spent cleaning) \u00f7 (The total number of girls) = (The time took each girl spent cleaning)
    \nc \u00f7 5 = 3
    \nc = 3 \u00d7 5
    \nc = 15 hours
    \nHence, from the above,
    \nWe can conclude that
    \nThe total number of hours the five girls spent cleaning is: 15 hours<\/p>\n

    Question 28.
    \nHigher Order Thinking<\/strong> Veronica traveled 562 miles to Venice, Florida. She drove 85 miles every day. On the last day of her trip she only drove 52 miles. Write and solve an equation to find the number of days Veronica traveled. Explain each step of your problem-solving strategy.
    \nAnswer:
    \nIt is given that
    \nVeronica traveled 562 miles to Venice, Florida. She drove 85 miles every day. On the last day of her trip she only drove 52 miles
    \nNow,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe number of days Veronica traveled is: 7 days<\/p>\n

    Question 29.
    \nGeneralize<\/strong> A movie theater sells 11,550 tickets for 50 sold-out showings of the same movie. Write a division equation that you can use to find the number of people who bought tickets for each showing. Use what you know about dividing with larger numbers to solve the equation.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nA movie theater sells 11,550 tickets for 50 sold-out showings of the same movie.
    \nNow,
    \nLet the number of people who bought tickets for each showing be: p
    \nSo,
    \nAccording to the given information,
    \n(The total number of tickets sold by a movie theater) \u00f7 (The number of people who bought tickets for each showing) = (The number of showings)
    \n11,550 \u00f7 p = 50
    \np = \\(\\frac{11,550}{50}\\)
    \np = 231 people
    \nHence, from the above,
    \nWe can conclude that
    \nThe number of people who bought tickets for each showing is: 231 people<\/p>\n

    Assessment Practice<\/strong><\/p>\n

    Question 30.
    \nIn October, Calvin’s school used 4,920 pounds of sand to protect the building against flooding during different tropical storms. One bag contains 40 pounds of sand.
    \nWhich of the following equations can be used to find how many bags of sand, b, Calvin’s school used in October?
    \nA. 4,920b = 40
    \nB. b \u00f7 40 = 4,920
    \nC. 40b = 4,920
    \nD. b \u00f7 4,920 = 40
    \nAnswer:
    \nIt is given that
    \nIn October, Calvin’s school used 4,920 pounds of sand to protect the building against flooding during different tropical storms. One bag contains 40 pounds of sand.
    \nNow,
    \nLet b be the number of bags of sand
    \nSo,
    \nAccording to the given information,
    \n(The number of bags) \u00d7 (The amount of sand in each bag) = (The total amount of sand)
    \n40b = 4,920
    \n\\(\\frac{4,920}{b}\\) = 40
    \nHence, from the above,
    \nWe can conclude that
    \nThe equation that can be used to find how many bags of sand, b, Calvin’s school used in October is:
    \n\"\"<\/p>\n

    Lesson 4.5 Write and Solve Equations with Rational Numbers<\/h3>\n

    Explore It!<\/strong>
    \nThe cost of T-shirts for four different soccer teams are shown below.
    \nI can… write and solve equations that involve rational numbers.
    \n\"Envision<\/p>\n

    A. Lorna is on Team A. Ben is on another team. They paid a total of $21.25 for both team T-shirts. Write an equation represent the cost of Ben’s shirt
    \nAnswer:
    \nIt is given that
    \nThe cost of T-shirts for four different soccer teams are shown below.
    \nNow,
    \nThe given figure is:
    \n\"Envision
    \nNow,
    \nLet the cost of Ben’s shirt be: x
    \nSo,
    \nAccording to the given information,
    \n(The total cost of both T-shirts) = (The cost of Lorna’s shirt which is in Team A) + (The cost of Ben’s shirt)
    \n$21.25 = $12.50 + x
    \nSo,
    \nx = $21.25 – $12.50
    \nx = $8.75
    \nHence, from the above,
    \nWe can conclude that
    \nThe cost of Ben’s shirt is: $8.75<\/p>\n

    B. Dario also plays soccer and he says that, based on the price of Ben’s T-shirt, Ben is on Team B. is Dario correct explain.
    \nAnswer:
    \nIt is given that
    \nDario said that based on the price of Ben’s T-shirt, Ben is on Team B
    \nNow,
    \nThe given figure is:
    \n\"Envision
    \nNow,
    \nWe know that,
    \nFrom Part A,
    \nThe cost of Ben’s T-shirt is: $8.75
    \nNow,
    \nFrom the given figure,
    \nWe can observe that
    \nThe T-shirt costs $8.75 that belongs to Team C
    \nHence, from the above,
    \nWe can conclude that
    \nDario is not correct<\/p>\n

    Focus on math practices<\/strong>
    \nGeneralize<\/strong> How is solving for unknowns involving money like solving for unknowns involving whole numbers?
    \nWe know that,
    \nThe representation of money is as the same as the rational numbers
    \nSo,
    \nWhatever operations we are performing on the whole numbers, we can perform the same operations on the unknowns involving money<\/p>\n

    Essential Question<\/strong>
    \nHow can you write and solve equations involving rational numbers?
    \nAnswer:
    \nConvert the rational number to decimals, then solve it as a decimal instead, (with the optional choice of re-converting it back to rational number).<\/p>\n

    Try It!<\/strong><\/p>\n

    Suppose you cut the shorter piece of fruit snack from the example above into two pieces. The longer of the two pieces is 1\\(\\frac{3}{8}\\) feet long. Complete the bar diagram to represent the equation. Then find the length of the shorter piece.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nyou cut the shorter piece of fruit snack from the example above into two pieces. The longer of the two pieces is 1\\(\\frac{3}{8}\\) feet long.
    \nNow,
    \nFrom Example 1,
    \nThe length of the shorter piece is: 2\\(\\frac{1}{4}\\) feet
    \nNow,
    \nThe length of the longer piece in the given shorter piece is: 1\\(\\frac{3}{8}\\) feet
    \nNow,
    \nLet the length of the shorter piece in the given shorter piece be: x
    \nNow,
    \nThe representation of the given information in the form of a bar diagram is:
    \n\"\"
    \nNow,
    \nFrom the given diagram,
    \nWe can observe that
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe shorter piece of the shorter piece is: \\(\\frac{7}{8}\\) feet<\/p>\n

    Convince Me!<\/strong> How does the equation change if you know the length of the shorter piece is \\(\\frac{7}{8}\\) foot and you want to know the length of the longer of the two pieces?
    \nAnswer:
    \nIf we know
    \nThe length of the shorter piece is: \\(\\frac{7}{8}\\) foot
    \nSo,
    \nThe equation in the above problem will change into
    \n\\(\\frac{7}{8}\\) + x = 2\\(\\frac{1}{4}\\)
    \nWhere,
    \nx is the length of the other piece
    \nNow,
    \nx = 2\\(\\frac{1}{4}\\) – \\(\\frac{7}{8}\\)
    \nx = 1\\(\\frac{3}{4}\\) feet
    \nSo,
    \n\\(\\frac{7}{8}\\) < 1\\(\\frac{3}{4}\\)
    \nHence, from the above,
    \nWe can conclude that
    \nThe length of the longer piece of the shorter piece is: 1\\(\\frac{3}{4}\\) feet<\/p>\n

    Try It!<\/strong><\/p>\n

    Solve \\(\\frac{5}{9}\\)y = 25 for y.
    \nAnswer:
    \nThe given equation is:
    \n\\(\\frac{5}{9}\\)y = 25
    \nNow,
    \nMultiply with \\(\\frac{9}{5}\\) on both sides
    \nSo,
    \n\\(\\frac{5}{9}\\) \u00d7 \\(\\frac{9}{5}\\)y = 25 \u00d7 \\(\\frac{9}{5}\\)
    \ny = \\(\\frac{25 \u00d7 9}{5}\\)
    \ny = 45
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of y for the given equation is: 45<\/p>\n

    Try It!<\/strong><\/p>\n

    Molly also buys a bag of 8 apples for $3.60. Write and solve an equation to find how much Molly paid for each apple.
    \nAnswer:
    \nIt is given that
    \nMolly also buys a bag of 8 apples for $3.60
    \nSo,
    \nThe cost of each apple = (The total cost of apples) \u00f7 (The total number of apples)
    \n= \\(\\frac{$3.60}{8}\\)
    \n= $0.45
    \nHence, from the above,
    \nWe can conclude that
    \nThe cost of each apple is: $0.45<\/p>\n

    Try It!<\/strong><\/p>\n

    Carmen spent $12.50 for a new notebook and a compass. The notebook cost $6.35. Write and solve an equation to find c, the cost of the compass.
    \nAnswer:
    \nIt is given that
    \nCarmen spent $12.50 for a new notebook and a compass. The notebook cost $6.35
    \nNow,
    \nLet the cost of compass be c
    \nSo,
    \nThe total amount of money Carmen spent on a new notebook and a compass = (The cost of a compass) + (The cost of a new notebook)
    \n$12.50 = c + $6.35
    \nSo,
    \nc = $12.50 – $6.35
    \nc = $6.15
    \nHence, from the above,
    \nWe can conclude that
    \nThe cost of a compass is: $6.15<\/p>\n

    KEY CONCEPT<\/strong>
    \nYou can solve equations using properties of equality and inverse relationships.
    \n\"Envision<\/p>\n

    Do You Understand?<\/strong>
    \nQuestion 1.
    \nEssential Question<\/strong> How can you write and solve equations involving rational numbers?
    \nAnswer:
    \nConvert the rational number to decimals, then solve it as a decimal instead, (with the optional choice of re-converting it back to rational number).<\/p>\n

    Question 2.
    \nConstruct Arguments<\/strong> Why are inverse relationships important for solving equations?
    \nAnswer:
    \nMathematically, Inverse operations are opposite operations. Addition is the opposite of subtraction; division is the opposite of multiplication, and so on. Inverse operations are used to solve simple algebraic equations to more difficult equations that involve exponents, logarithms, and trigonometry<\/p>\n

    Question 3.
    \nCritique Reasoning<\/strong> Johnny says that he solved the equation x – 3.5 = 7.2 by adding 3.5 to the left side of the equation. Explain whether Johnny is correct.
    \nAnswer:
    \nIt is given that
    \nJohnny says that he solved the equation x – 3.5 = 7.2 by adding 3.5 to the left side of the equation.
    \nNow,
    \nThe given equation is:
    \nx – 3.5 = 7.2
    \nNow,
    \nWe know that,
    \nAccording to the “Addition Property of Equality”,
    \nAdd the same amount on both sides to make the equation balanced
    \nSo,
    \nFor the given equation,
    \nAdd 3.5 on both sides
    \nSo,
    \nx – 3.5 + 3.5 = 7.2 – 3.5
    \nHence, from the above,
    \nWe can conclude that
    \nJohnny is not correct<\/p>\n

    Question 4.
    \nGeneralize<\/strong> When solving an equation involving a mixed number, such as y + \\(\\frac{3}{4}\\) = 4\\(\\frac{1}{2}\\), what do you need to do to the mixed number?
    \nAnswer:
    \nThe given equation is:
    \ny + \\(\\frac{3}{4}\\) = 4\\(\\frac{1}{2}\\)
    \nNow,
    \nTo solve the given equation,
    \nWe have to convert the mixed number into a proper fraction
    \nHence, from the above,
    \nWe can conclude that
    \nWe have to convert the given mixed number into a proper fraction<\/p>\n

    Question 5.
    \nConstruct Arguments<\/strong> How is solving an equation with fractions like solving an equation with whole numbers? How is it different?
    \nAnswer:
    \nSolving an equation with a fraction isn’t much different than solving an equation full of whole numbers.\u00a0You group the variable terms on one side, the constants on the other, and then simplify.<\/p>\n

    Do You Know How?<\/strong>
    \nIn 6-14, solve each equation.<\/strong>
    \nQuestion 6.
    \nt – \\(\\frac{2}{3}\\) = 25\\(\\frac{3}{4}\\)
    \nAnswer:
    \nThe given equation is:
    \nt – \\(\\frac{2}{3}\\) = 25\\(\\frac{3}{4}\\)
    \nNow,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of t for the given expression is: \\(\\frac{317}{12}\\)<\/p>\n

    Question 7.
    \n\\(\\frac{f}{2}\\) = \\(\\frac{5}{8}\\)
    \nAnswer:
    \nThe given equation is:
    \n\\(\\frac{f}{2}\\) = \\(\\frac{5}{8}\\)
    \nNow,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of f for the given equation is: \\(\\frac{5}{4}\\)<\/p>\n

    Question 8.
    \n13.27 = t – 24.45
    \nAnswer:
    \nThe given equation is:
    \n13.27 = t – 24.45
    \nNow,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of t for the given equation is: \\(\\frac{943}{25}\\)<\/p>\n

    Question 9.
    \nr \u00f7 5.5 = 18.2
    \nAnswer:
    \nThe given equation is:
    \nr \u00f7 5.5 = 18.2
    \nNow,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of r for the given equation is: \\(\\frac{1,001}{10}\\)<\/p>\n

    Question 10.
    \n\\(\\frac{7}{10}=x – [latex]\\frac{3}{5}\\)
    \nAnswer:
    \nThe given equation is:
    \n\\(\\frac{7}{10}=x-[latex]\\frac{3}{5}\\)
    \nNow,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of x for the given equation is: \\(\\frac{13}{10}\\)<\/p>\n

    Question 11.
    \n1.8x = 40.14
    \nAnswer:
    \nThe given equation is:
    \n1.8x = 40.14
    \nNow,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of x for the given equation is: \\(\\frac{223}{10}\\)<\/p>\n

    Question 12.
    \n17.3 + v = 22.32
    \nAnswer:
    \nThe given equation is:
    \n17.3 + v = 22.32
    \nNow,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of v for the given equation is: \\(\\frac{251}{50}\\)<\/p>\n

    Question 13.
    \n9 = \\(\\frac{3}{8}\\)y
    \nAnswer:
    \nThe given equation is:
    \n9 = \\(\\frac{3}{8}\\)y
    \nNow,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of y for the given equation is: 24<\/p>\n

    Question 14.
    \n1\\(\\frac{3}{4}+z=2[latex]\\frac{2}{3}\\)
    \nAnswer:
    \nThe given equation is:
    \n1\\(\\frac{3}{4}+z=2[latex]\\frac{2}{3}\\)
    \nNow,
    \nConvert the given mixed numbers into the fraction form
    \nSo,
    \n1\\(\\frac{3}{4}\\) = \\(\\frac{7}{4}\\)
    \n2\\(\\frac{2}{3}\\) = \\(\\frac{8}{3}\\)
    \nNow,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of y for the given equation is: \\(\\frac{11}{12}\\)<\/p>\n

    Practice & Problem Solving<\/strong><\/p>\n

    Leveled Practice
    \nIn 15-22, solve each equation.<\/strong>
    \nQuestion 15.
    \nw – 3.2 = 5.6
    \nw – 3.2 + _____ = 5.6 + _____
    \nw = _____
    \nAnswer:
    \nThe given equation is:
    \nw – 3.2 = 5.6
    \nNow,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of w for the given equation is: 8.8<\/p>\n

    Question 16.
    \n9.6 = 1.6y
    \n9.6 \u00f7 _____ = 1.6y \u00f7 ______
    \n_______ = y
    \nAnswer:
    \nThe given equation is:
    \n9.6 = 1.6y
    \nNow,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of y for the given equation is: 6<\/p>\n

    Question 17.
    \n48.55 + k = 61.77
    \n48.55 + k – _____ = 61.77 – _____
    \nk = _____
    \nAnswer:
    \nThe given equation is:
    \n48.55 + k = 61.77
    \nNow,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of k for the given equation is: 13.22<\/p>\n

    Question 18.
    \nm \u00f7 3.54 = 1.5
    \nm \u00f7 3.54 \u00d7 _____ = 1.5 \u00d7 _____
    \nm = _____
    \nAnswer:
    \nThe given equation is:
    \nm \u00f7 3.54 = 1.5
    \nNow,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of m for the given equation is: 5.31<\/p>\n

    Question 19.
    \n\\(7 \\frac{1}{9}=2 \\frac{4}{5}+m\\)
    \nAnswer:
    \nThe given equation is:
    \n\\(7 \\frac{1}{9}=2 \\frac{4}{5}+m\\)
    \nNow,
    \nConvert the given mixed numbers into fractions
    \nSo,
    \n7\\(\\frac{1}{9}\\) = \\(\\frac{64}{9}\\)
    \n2\\(\\frac{4}{5}\\) = \\(\\frac{14}{5}\\)
    \nNow,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of m for the given equation is: \\(\\frac{194}{45}\\)<\/p>\n

    Question 20.
    \n\\(a+3 \\frac{1}{4}=5 \\frac{2}{9}\\)
    \nAnswer:
    \nThe given equation is:
    \n\\(a+3 \\frac{1}{4}=5 \\frac{2}{9}\\)
    \nNow,
    \nConvert the given mixed numbers into the fractions
    \nSo,
    \n3\\(\\frac{1}{4}\\) = \\(\\frac{13}{4}\\)
    \n5\\(\\frac{2}{9}\\) = \\(\\frac{47}{9}\\)
    \nNow,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of a for the given equation is: \\(\\frac{71}{36}\\)<\/p>\n

    Question 21.
    \n\\(\\frac{1}{8}\\) \u2022 y=4
    \nAnswer:
    \nThe given equation is:
    \n\\(\\frac{1}{8}\\) \u2022 y=4
    \nNow,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of y for the given equation is: 32<\/p>\n

    Question 22.
    \n\\(k-6 \\frac{3}{8}=4 \\frac{6}{7}\\)
    \nAnswer:
    \nThe given equation is:
    \n\\(k-6 \\frac{3}{8}=4 \\frac{6}{7}\\)
    \nNow,
    \nConvert the given mixed numbers into the fractions
    \nSo,
    \n6\\(\\frac{3}{8}\\) = \\(\\frac{51}{8}\\)
    \n4\\(\\frac{6}{7}\\) = \\(\\frac{34}{7}\\)
    \nNow,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of k for the given equation is: \\(\\frac{629}{56}\\)<\/p>\n

    Question 23.
    \nMr. Marlon buys these tickets for his family to visit the water park. The total cost is $210. Write and solve an equation to find the cost of each ticket.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nMr. Marlon buys these tickets for his family to visit the water park. The total cost is $210
    \nNow,
    \nThe given figure is:
    \n\"Envision
    \nNow,
    \nFrom the given figure,
    \nWe can observe that
    \nThe number of tickets Mr. Marlon bought is: 4
    \nNow,
    \nThe cost of each ticket = (The total cost of tickets) \u00f7 (The total number of people)
    \n= \\(\\frac{210}{4}\\)
    \n= $ 52.50
    \nHence, from the above,
    \nWe can conclude that
    \nThe cost of each ticket is: $52.50<\/p>\n

    Question 24.
    \nHigher Order Thinking<\/strong> Without solving, tell which equation below has a greater solution. Explain.
    \n\\(\\frac{5}{8}\\)m = 2\\(\\frac{3}{4}\\)
    \n\\(\\frac{5}{9}\\)m = 2\\(\\frac{3}{4}\\)
    \nAnswer:
    \nThe given equations are:
    \n\\(\\frac{5}{8}\\)m = 2\\(\\frac{3}{4}\\)
    \n\\(\\frac{5}{9}\\)m = 2\\(\\frac{3}{4}\\)
    \nNow,
    \nWe know that,
    \nIn the given fraction,
    \nThe numerators are the same and to find whether which fraction is greater,
    \nWe can compare by using the denominators
    \nIf the denominator is a small number when the numerators are the same then the other denominator, then the fraction that contains the smallest value of denominator is the greater fraction
    \nNow,
    \nFrom the given fractions,
    \nWhen we compare the denominators,
    \n8 < 9
    \nHence, from the above,
    \nWe can conclude that
    \nThe equation which is greater is:
    \n\\(\\frac{5}{8}\\)m = 2\\(\\frac{3}{4}\\)<\/p>\n

    Question 25.
    \nMake Sense and Persevere<\/strong> A high school track team’s long jump record is 21 feet 2\\(\\frac{1}{4}\\) inches. This year, Tim’s best long jump is 20 feet 9\\(\\frac{1}{2}\\) inches. If long jumps are measured to the nearest quarter inch, how much farther must Tim jump to break the record?
    \nAnswer:
    \nIt is given that
    \nA high school track team’s long jump record is 21 feet 2\\(\\frac{1}{4}\\) inches. This year, Tim’s best long jump is 20 feet 9\\(\\frac{1}{2}\\) inches
    \nNow,
    \nWe know that,
    \n1 foot = 12 inches
    \nSo,
    \n21 feet 2\\(\\frac{1}{4}\\) inches = (21 \u00d7 12) + 2\\(\\frac{1}{4}\\) inches
    \n= 252 + 2\\(\\frac{1}{4}\\) inches
    \n= 252 + 2.25 inches
    \n= 254.25 inches
    \nSo,
    \n20 feet 9\\(\\frac{1}{2}\\) inches = (20 \u00d7 12) + 9\\(\\frac{1}{2}\\) inches
    \n= 240 + 9\\(\\frac{1}{2}\\) inches
    \n= 240 + 9.5 inches
    \n= 249.5 inches
    \nSo,
    \nThe distance Tim has to cross to break the record = 254.25 – 249.5
    \n= 4.75 inches
    \nHence, from the above,
    \nWe can conclude that
    \nThe distance Tim has to cross to break the record is: 4.75 inches<\/p>\n

    Question 26.
    \nMake Sense and Persevere<\/strong> About how many gallons of fuel does it take to move the space shuttle 3 miles from its hangar to the Vehicle Assembly Building?
    \n\"Envision
    \nAnswer:
    \nThe given figure is:
    \n\"Envision
    \nNow,
    \nFrom the given figure,
    \nWe can observe that
    \n1 mile = 5,280 ft
    \nIt will take 1 gallon of fuel to move space shuttle 42 feet
    \nNow,
    \nThe distance of 3 miles traveled by the space shuttle in fuel = (The number of feet required for 1 mile) \u00d7 3
    \n= 3 \u00d7 5,280
    \n= 15,840 feet
    \nNow,
    \nThe number of gallons of fuel required to move the space shuttle 3 miles = (The total distance of 3 miles in feet) \u00f7 (The distance covered by 1 gallon of fuel)
    \n= \\(\\frac{15,840}{42}\\)
    \n= 377 gallons
    \nHence, from the above,
    \nWe can conclude that
    \nIt will take 377 gallons of fuel to move the space shuttle 3 miles from its hangar to the Vehicle Assembly Building<\/p>\n

    Question 27.
    \nIs the solution of b \u00d7 \\(\\frac{5}{6}\\) = 25 greater than or less than 25? How can you tell before computing?
    \nAnswer:
    \nThe given equation is:
    \nb \u00d7 \\(\\frac{5}{6}\\) = 25
    \nNow,
    \nWe know that,
    \nWhen a multiplication is converted into a division, then the fraction after \u00d7 or \u00f7 symbols will have to be reciprocal and vice-versa
    \nSo,
    \nThe reciprocal of \\(\\frac{5}{6}\\) is: \\(\\frac{6}{5}\\)
    \nSo,
    \nb = 25 \u00d7 \\(\\frac{6}{5}\\)
    \nNow,
    \nWe know that,
    \nThe value of \\(\\frac{6}{5}\\) is greater than 1
    \nSo,
    \nThe solution for the given equation will also be greater than 25
    \nHence, from the above,
    \nWe can conclude that
    \nThe solution of b \u00d7 \\(\\frac{5}{6}\\) = 25 is greater than 25<\/p>\n

    Question 28.
    \nWhat is the width of a rectangle with a length of \\(\\frac{3}{7}\\) ft and an area of 2 ft2<\/sup>? Write an equation to show your work.
    \nAnswer:
    \nIt is given that
    \nThe area of a rectangle is: 2 ft\u00b2
    \nThe length of a rectangle is: \\(\\frac{3}{7}\\) ft
    \nNow,
    \nLet the width of the rectangle be x
    \nNow,
    \nWe know that,
    \nThe area of a rectangle = Length \u00d7 Width
    \nSo,
    \n2 = \\(\\frac{3}{7}\\) \u00d7 x
    \nNow,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe width of the given rectangle is: \\(\\frac{14}{3}\\) ft<\/p>\n

    Question 29.
    \nModel with Math<\/strong> Helen is filling the pool shown for her little brother. She can carry 1\\(\\frac{7}{8}\\) gallons of water each trip. Write and solve an equation to find how many trips Helen needs to make.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nHelen is filling the pool shown for her little brother. She can carry 1\\(\\frac{7}{8}\\) gallons of water each trip
    \nNow,
    \nThe given figure is:
    \n\"Envision
    \nSo,
    \nThe number of trips Helen needs to make = (The total amount of water held in the pool) \u00f7 (The amount of water Helen can carry each trip)
    \n= 10\\(\\frac{1}{2}\\) \u00f7 1\\(\\frac{7}{8}\\)
    \nNow,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe number of trips Helen needs to make is about 6 trips<\/p>\n

    Question 30.
    \nAfter the pool was full, Helen’s little brother and his friend splashed g gallons of water out of the pool. There are 7\\(\\frac{7}{8}\\) gallons still left in the pool. Write and solve an equation to find how much water was splashed out of the pool.
    \nAnswer:
    \nIt is given that
    \nAfter the pool was full, Helen’s little brother and his friend splashed g gallons of water out of the pool. There are 7\\(\\frac{7}{8}\\) gallons still left in the pool
    \nNow,
    \nThe given figure is:
    \n\"Envision
    \nNow,
    \nLet the amount of water splashed from the pool be: g gallons
    \nSo,
    \n(The total amount of water present in the pool) – (The amount of water splashed out of the pool) = (The amount of water still left in the pool)
    \n10\\(\\frac{1}{2}\\) – g = 7\\(\\frac{7}{8}\\)
    \ng = 10\\(\\frac{1}{2}\\) – 7\\(\\frac{7}{8}\\)
    \nNow,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe amount of water that splashed out from the pool is: 2\\(\\frac{5}{8}\\) gallons<\/p>\n

    Question 31.
    \nGrace solved the equation 2\\(\\frac{1}{2}\\)y = \\(\\frac{5}{8}\\). Her steps for the solution are shown in the table but are all mixed up. Write her steps in the correct order on the right side of the table.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nGrace solved the equation 2\\(\\frac{1}{2}\\)y = \\(\\frac{5}{8}\\). Her steps for the solution are shown in the table but are all mixed up.
    \nNow,
    \nThe given equation is:
    \n2\\(\\frac{1}{2}\\)y = \\(\\frac{5}{8}\\)
    \nNow,
    \n\"\"
    \nHence,from the above,
    \nWe can conclude that
    \nHer steps in the correct order on the right side of the table is:
    \n\"\"<\/p>\n

    Question 32.
    \nThe scientific name for the little bumps on your tongue is fungiform papillae. Each bump can contain many taste buds. The number of taste buds a person has varies. There are three general classifications of taste: supertaster, medium taster, and non-taster. Suppose a supertaster has 8,640 taste buds. Solve the equation 4.5n = 8,640 to find the number of taste buds, n, a non-taster may have.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nThe scientific name for the little bumps on your tongue is fungiform papillae. Each bump can contain many taste buds. The number of taste buds a person has varies. There are three general classifications of taste: supertaster, medium taster, and non-taster. Suppose a supertaster has 8,640 taste buds
    \nNow,
    \nThe given figure is:
    \n\"Envision
    \nThe given equation according to the given information is:
    \n4.5n = 8,640
    \nNow,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe number of taste buds a non-taster may have is: 1,920 taste buds<\/p>\n

    Question 33.
    \nModel with Math<\/strong> In one study, the number of women classified as supertasters was 2.25 times the number of men classified as supertasters. Suppose 72 women were classified as supertasters. Write an equation that represents the number of men, m, who were classified as supertasters. Then solve the equation. How many men were classified as supertasters?
    \nAnswer:
    \nIt is given that
    \nIn one study, the number of women classified as supertasters was 2.25 times the number of men classified as supertasters. Suppose 72 women were classified as supertasters
    \nNow,
    \nFrom the given information,
    \nThe number of women who are supertasters = 2.25 \u00d7 (The numberof men who are supertasters)
    \n72 = 2.25 \u00d7 (The number of men who are supertasters)
    \nThe number of men who are supertasters = \\(\\frac{72}{2.25}\\)
    \n= 32
    \nHence, from the above,
    \nWe can conclude that
    \nThe number of men who are classified as supertasters is: 32 men<\/p>\n

    Question 34.
    \nUse Structure<\/strong> A fraction, f, multiplied by 5 equals \\(\\frac{1}{8}\\). Write an algebraic sentence to show the equation. Then solve the equation and explain how you solved it.
    \nAnswer:
    \nIt is given that
    \nA fraction, f, multiplied by 5 equals \\(\\frac{1}{8}\\).
    \nNow,
    \nThe representation of the given information in the form of an equation is:
    \nf \u00d7 5 = \\(\\frac{1}{8}\\)
    \n5f = \\(\\frac{1}{8}\\)
    \nNow,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of f for the given equation is: \\(\\frac{1}{40}\\)<\/p>\n

    Question 35.
    \nYelena needs to swim a total of 8 miles this week. So far, she swam 5\\(\\frac{3}{8}\\) miles. Use the equation 5\\(\\frac{3}{8}\\) + m = 8 to find how many more miles Yelena needs to swim.
    \nAnswer:
    \nIt is given that
    \nYelena needs to swim a total of 8 miles this week. So far, she swam 5\\(\\frac{3}{8}\\) miles.
    \nNow,
    \nThe givene quation according to the given information is:
    \n5\\(\\frac{3}{8}\\) + m = 8
    \nNow,
    \nConvert the given mixed number into a fraction
    \nSo,
    \n\\(\\frac{43}{8}\\) + m = 8
    \nNow,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe number of miles Yelena needs to swim is: \\(\\frac{21}{8}\\) miles<\/p>\n

    Question 36.
    \nCan any equation that is written using addition be written as an equivalent equation using subtraction? Explain your reasoning and give an example containing decimals that shows your reasoning.
    \nAnswer:
    \nYes, any equation that is written using addition can be written as an equivalent equation using subtraction
    \nEx:
    \n3.35 + y = 6.88
    \ny = 6.88 – 3.35
    \ny = 3.53
    \nNow,
    \nBy using Subtraction:
    \nSubtract any number let say 5 from both sides of 3.35 + y = 6.88
    \nNow,
    \n3.35 + y – 5 = 6.88 – 5
    \ny – 1.65 = 1.88
    \ny = 3.53<\/p>\n

    Question 37.
    \nCritique Reasoning<\/strong> Oscar is 12 years old and his little sister is 6. Oscar uses a to represent his age. He says that he can use the expression a \u00f7 2 to always know his sister’s age. Do you agree? Explain.
    \nAnswer:
    \nIt is given that
    \nOscar is 12 years old and his little sister is 6. Oscar uses a to represent his age. He says that he can use the expression a \u00f7 2 to always know his sister’s age
    \nNow,
    \nLet the age of Oscar be: 18 years
    \nThen,
    \nThe age of his sister will be: 8 – 6 = 12 years
    \nSo,
    \nThe relationship between the ages of Oscar and his sister is:
    \nThe age of Oscar = The age of his sister + 6
    \nHence, from the above,
    \nWe can conclude that
    \nWe can not agree with Oscar<\/p>\n

    Assessment Practice<\/strong><\/p>\n

    Question 38.
    \nWhich value for y makes the equation 0.26y = 0.676 true?
    \nA. y = 0.17576
    \nB. y = 0.26
    \nC. y = 2.6
    \nD. y = 26
    \nAnswer:
    \nThe given equation is:
    \n0.26y = 0.676
    \nNow,
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of y that makes the given equation true is:
    \n\"\"<\/p>\n

    Question 39.
    \nWhich value for x makes the equation 0.435 + x = 0.92 true?
    \nA. x = 1.355
    \nB. x = 0.595
    \nC. x = 0.495
    \nD. x = 0.485
    \nAnswer:
    \nThe given equation is:
    \n0.435 + x = 0.92
    \nNow,
    \nx = 0.92 – 0.435
    \nx = 0.485
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of x that makes the givene quation true is:
    \n\"\"<\/p>\n

    Topic 4 Mid-Topic Checkpoint<\/h3>\n

    Question 1.
    \nVocabulary<\/strong> Describe the relationship between equations and the properties of equality. Lessons 4-1 and 4-2
    \nAnswer:
    \nThe property of equality states that when we do arithmetic operations on both sides of an equation by the same number, the two sides remain equal and the property of equality is pplicable only on equations<\/p>\n

    In 2-4, write an equation for the situation. Then solve the equation.<\/strong>
    \nQuestion 2.
    \nA fraction f multiplied by 4 equals \\(\\frac{1}{2}\\). Lesson 4-5
    \nAnswer:
    \nThe given situation is:
    \nA fraction f multiplied by 4 equals \\(\\frac{1}{2}\\)
    \nNow,
    \nThe representation of the given situation in the form of an equation is:
    \n4f = \\(\\frac{1}{2}\\)
    \nNow,
    \nDivide the equation by 4 on both sides
    \nSo,
    \nf = \\(\\frac{1}{2 \u00d7 4}\\)
    \nf = \\(\\frac{1}{8}\\)
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of f for the given equation is: \\(\\frac{1}{8}\\)<\/p>\n

    Question 3.
    \nWhen 832 is divided by n, the result is 16. Lesson 4-4
    \nAnswer:
    \nThe given situation is:
    \nWhen 832 is divided by n, the result is 16
    \nNow,
    \nThe representation of the given situation in the form of an equation is:
    \n\\(\\frac{832}{n}\\) = 16
    \nNow,
    \nMultiply the equation with n on both sides
    \nSo,
    \nn = \\(\\frac{832}{16}\\)
    \nn = 52
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of n for the given equation is: 52<\/p>\n

    Question 4.
    \nWhen 10 is subtracted from x, the result is 6. Lesson 4-3
    \nAnswer:
    \nThe given situation is:
    \nWhen 10 is subtracted from x, the result is 6.
    \nNow,
    \nThe representation of the given situation in the form of an equation is:
    \nx – 10 = 6
    \nNow,
    \nAdd 10 on both sides for the equation
    \nSo,
    \nx – 10 + 10 = 6 + 10
    \nx = 16
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of x for the given equation is: 16<\/p>\n

    Question 5.
    \nSelect all the equations that are equivalent to n-9 = 12. Lesson 4-2
    \n\u2610 n – n – 9 = 12 – n
    \n\u2610 n – 9 + 12 = 12 – 9
    \n\u2610 n – 9 + 9 = 12 + 9
    \n\u2610 n – 9 – n = 12 – n
    \n\u2610 n – 9 + 9 = 12 \u2013 12
    \nAnswer:
    \nThe given equation is: n – 9 = 12
    \nNow,
    \nWe know that,
    \nAccording to the “Property of Equality”,
    \nWe have to do the same arithmetic operation with the same number on both sides of the equation to keep the equation balanced
    \nHence, from the above,
    \nWe can conclude that
    \nThe equations that are equivalent to n – 9 = 12 are:
    \n\"\"<\/p>\n

    Question 6.
    \nSelect all the values ford that make the equation 9 = 18 \u00f7 d true. Lesson 4-1
    \n\u2610 2
    \n\u2610 0.5
    \n\u2610 \\(\\frac{10}{5}\\)
    \n\u2610 162
    \n\u2610 \\(\\frac{1}{4}\\)
    \nAnswer:
    \nThe given equation is:
    \n9 = 18 \u00f7 d
    \nNow,
    \nMultiply with d on both sides fro the equation
    \nSo,
    \nd = \\(\\frac{18}{9}\\)
    \nd = 2
    \nHence,f rom the above,
    \nWe can conclude that
    \nThe values of d that make the given equation true are:
    \n\"\"<\/p>\n

    Question 7.
    \nThe area, A, of a triangle is 15.3 square cm. Its base, b, is 4.5 cm. The formula for finding the area of a triangle is A = \\(\\frac{1}{2}\\)bh. Write and solve an equation to find the height, h, of the triangle. Lessons 4-4 and 4-5
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nThe area, A, of a triangle is 15.3 square cm. Its base, b, is 4.5 cm. The formula for finding the area of a triangle is A = \\(\\frac{1}{2}\\)bh.
    \nNow,
    \nThe given figure is:
    \n\"Envision
    \nNow,
    \nAccording to the given formula in the given informatio,
    \n15.3 = \\(\\frac{1}{2}\\) \u00d7 4.5 \u00d7 h
    \n15.3 = 2.25 \u00d7 h
    \nh = \\(\\frac{15.3}{2.25}\\)
    \nh = 6.8 cm
    \nHence,f rom the above,
    \nWe can conclude that
    \nThe value of h of the given triangle is: 6.8 cm<\/p>\n

    Topic 4 Mid-Topic Performance Task<\/h3>\n

    Ronald carved 3\\(\\frac{3}{8}\\) feet of a totem pole. He says that the totem pole is \\(\\frac{3}{4}\\) complete.
    \n\"Envision<\/p>\n

    PART A
    \nIf h represents the height, in feet, of the finished totem pole, then \\(\\frac{3}{4}\\)h = 3\\(\\frac{3}{8}\\) represents this situation. Which equations show the use of a reciprocal to write an equivalent equation that can be used to solve for h? Select all that apply.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nRonald carved 3\\(\\frac{3}{8}\\) feet of a totem pole. He says that the totem pole is \\(\\frac{3}{4}\\) complete.
    \nNow,
    \nThe given figure is:
    \n\"Envision
    \nNow,
    \nIf h represents the height, in feet, of the finished totem pole, then
    \n\\(\\frac{3}{4}\\)h = 3\\(\\frac{3}{8}\\) represents this situation
    \nNow,
    \nWe know that,
    \nThe term “Reciprocal” is applicable only for multiplication and division
    \nNow,
    \nWe know that,
    \nThe reciprocal of \\(\\frac{3}{4}\\) is: \\(\\frac{4}{3}\\)
    \nNow,
    \nWe know that,
    \nAccording to the “Property of Equality”,
    \nThe arithmetic operation we performed on the left side of th equation with a number will have to be the same to the right side of the equation also
    \nHence, from the above,
    \nWe can conclude that
    \nThe equations that show the use of a reciprocal to write an equivalent equation that can be used to solve for h are:
    \n\"\"<\/p>\n

    PART B
    \nUse the equation in Part A to determine the height of the finished totem pole. Then write and solve an equation to find the height, s, of the section that has not been carved.
    \nAnswer:
    \nFrom Part A,
    \nThe equation to determine the height h of the finished totem pole is:
    \n\\(\\frac{3}{4}\\)h = 3\\(\\frac{3}{8}\\)
    \nNow,
    \nThe given figure is:
    \n\"Envision
    \nNow,
    \nFrom the given figure,
    \nWe can observe that
    \nThe \\(\\frac{1}{4}\\) of the section has not been carved
    \nNow,
    \nh = 3\\(\\frac{3}{8}\\) \u00d7 \\(\\frac{4}{3}\\)
    \nh = \\(\\frac{27}{8}\\) \u00d7 \\(\\frac{4}{3}\\)
    \nh = \\(\\frac{27 \u00d7 4}{8 \u00d7 3}\\)
    \nh = \\(\\frac{9}{2}\\) ft
    \nNow,
    \nThe height of the section that ahs not been carved (s) = \\(\\frac{h}{4}\\)
    \n= h \u00d7 \\(\\frac{1}{4}\\)
    \n= \\(\\frac{9}{2}\\) \u00d7 \\(\\frac{1}{4}\\)
    \n= \\(\\frac{9}{8}\\) ft
    \nHence, from the above,
    \nWe can conclude that
    \nThe height of the section that has not been carved is: \\(\\frac{9}{8}\\) ft<\/p>\n

    PART C
    \nRonald spent $10.50 on tools and x dollars on the wood for the totem pole. His total cost for the totem pole is $19.35. The equation $10.50 + x = $19.35 represents this situation. What is the cost of the wood Ronald used?
    \nAnswer:
    \nIt is given that
    \nRonald spent $10.50 on tools and x dollars on the wood for the totem pole. His total cost for the totem pole is $19.35. The equation $10.50 + x = $19.35 represents this situation
    \nNow,
    \nThe given equation that represents the given situation is:
    \n$10.50 + x = $19.35
    \nx = $19.35 – $10.50
    \nx = $8.85
    \nHence, from the above,
    \nWe can conclude that
    \nThe cost of Wood Ronald used is: $8.85<\/p>\n

    PART D
    \nTo make the same totem pole with wood that costs y dollars, Ronald would have to spend a total of $35.19. Explain which property of equality Ronald could use to solve the equation $10.50 + y = $35.19 and why that property can be used. Then show how to use that property to solve for y.
    \nAnswer:
    \nIt is given that
    \nTo make the same totem pole with wood that costs y dollars, Ronald would have to spend a total of $35.19.
    \nNow,
    \nThe given equation that is used to represent the given situation is:
    \n$10.50 + y = $35.19
    \nNow,
    \nWe know that,
    \nAccording to the “Subtraction Property of Equality”,
    \nSubtract the same amount from both sides to make the equation balanced
    \nNow,
    \nSubtract $10.50 from the given equation by using the “Subtraction Property of Equality”
    \nSo,
    \n$10.50 + y – $10.50 = $35.19 – $10.50
    \ny = $24.69
    \nHence, from the above,
    \nWe can conclude that
    \nThe cost of wood that is used to make the totem pole is: $24.69<\/p>\n

    Lesson 4.6 Understand and Write Inequalities<\/h3>\n

    Solve & Discuss It!<\/strong>
    \nThe record time for the girls’ 50-meter freestyle swimming competition is 24.49 seconds. Camilla has been training and wants to break the record. What are some possible times Camilla would have to swim to break the current record?<\/p>\n

    I can… understand and write an inequality that describes a real-world situation.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nThe record time for the girls’ 50-meter freestyle swimming competition is 24.49 seconds. Camilla has been training and wants to break the record”
    \nNow,
    \nThe given table is:
    \n\"\"
    \nNow,
    \nFrom the given table,
    \nWe can observe that
    \nTo break the record,
    \nCamilia has to swim greater than the time it is taken for the girls’ 50-meter freestyle swimming competition
    \nSo,
    \nThe possible times Camilia would have to swim to break the current record is: 26.56 seconds, 25.14 seconds, and 25.32 seconds
    \nHence, from the above,
    \nWe can conclude that
    \nThe possible times Camilia would have to swim to break the current record is:
    \na. 26.56 seconds b. 25.14 seconds c. 25.32 seconds<\/p>\n

    Reasoning<\/strong>
    \nHow do Camilla’s possible times compare to 24.49 seconds?
    \nAnswer:
    \nFrom the above Exercise,
    \nWe know that,
    \nCamilia wants to break the record
    \nSo,
    \nThe possible times of Camilia must be greater than 24.49 seconds
    \nHence, from the above,
    \nWe can conclude that
    \nCamilia’s possible times are greater than 24.49 seconds<\/p>\n

    Focus on math practices<\/strong>
    \nBe Precise<\/strong> Fran won a blue ribbon for growing the heaviest pumpkin. It weighed 217 pounds. What could be the weights of other pumpkins in the contest? How could you show the weights of the other pumpkins using a mathematical statement? Explain.
    \nAnswer:
    \nIt is given that
    \nFran won a blue ribbon for growing the heaviest pumpkin. It weighed 217 pounds.
    \nNow,
    \nWe know that,
    \nThe heaviest weight means there is no other weight greater than that weight
    \nNow,
    \nFrom the given information,
    \n217 pounds is the heaviest weight
    \nSo,
    \nOther weights other than 217 pounds are lesser
    \nNow,
    \nLet the weight of all the other pumpkins be x
    \nSo,
    \nAccording to the given information,
    \nThe weights of all the pumpkins present in the contest have lesser weight than the heaviest pumpkin
    \nSo,
    \nThe mathematical representation of the above statement is:
    \nx < 217 pounds
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the give information in the form of a mathematical statement is:
    \nThe weights of all the pumpkins present in the contest have lesser weight than the heaviest pumpkin. i.e.,
    \nx < 217 pounds<\/p>\n

    Essential Question<\/strong>
    \nHow can you write an inequality to describe a situation?
    \nAnswer:
    \nThe steps to write an inequality to describe a situation are:
    \na. Read through the entire problem.
    \nb. Highlight the important information and key words that you need to solve the problem.
    \nc. Identify your variables.
    \nd. Write the equation or inequality.
    \ne. Solve.
    \nf. Write your answer in a complete sentence.
    \ng. Check or justify your answer<\/p>\n

    Try It!<\/strong><\/p>\n

    Use the number line to show some of the ages of people who do not need to be accompanied by an adult. Write an inequality to represent the ages of people, n, who do not need to be accompanied by an adult.
    \n\"Envision
    \nAnswer:
    \nLet the ages of the people who do not need to be accompanied by an adult be: n
    \nNow,
    \nThe given figure is:
    \n\"\"
    \nNow,
    \nFrom the given figure,
    \nWe can observe that
    \nThe people who are less than 8 years have to be accompanied by an adult
    \nNow,
    \nThe representation of the people who do not need to be accompanied by an adult in the given number line is:
    \n\"\"
    \nSo,
    \nThe inequality that represents ages of people ,n, who do not need to be accompanied by an adult is:
    \nn > 8 years
    \nHence, from the above,
    \nWe can conclude that
    \nThe inequality that represents ages of people ,n, who do not need to be accompanied by an adult is:
    \nn > 8 years<\/p>\n

    Convince Me!<\/strong> How do you know to which group an 8-year-old belongs, those who must be accompanied by an adult or those who do not need to be accompanied by an adult? Explain.
    \nAnswer:
    \nThe given figure is:
    \n\"\"
    \nNow,
    \nLet the number of children who must be accompanied by an adult be: x
    \nLet the number of children who do not need to be accompanied by an adult be: y
    \nNow,
    \nFrom the given figure,
    \nWe can observe that
    \nThe children under the age of 8 must be accompanied by an adult
    \nSo,
    \nThe representation of an inequality that represents the ages of the children who must be accompanied by an adult is:
    \nx < 8
    \nThe representation of an inequality that represents the ages of the children who need not to be accompanied by an adult is:
    \ny > 8
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of an inequality that represents the ages of the children who must be accompanied by an adult is:
    \nx < 8
    \nThe representation of an inequality that represents the ages of the children who need not to be accompanied by an adult is:
    \ny > 8<\/p>\n

    Try It!<\/strong><\/p>\n

    Write an inequality to represent each situation.<\/strong>
    \na. Harry is taller than 60 inches.
    \nAnswer:
    \nThe given situation is:
    \nHarry is taller than 60 inches.
    \nNow,
    \nLet h be the height of Harry
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the inequality that represents the given situation is:
    \nh > 60 inches<\/p>\n

    b. Sherry is not 4 years old.
    \nAnswer:
    \nThe given situation is:
    \nSherry is not 4 years old
    \nNow,
    \nLet a be the age of Sherry
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the inequality that represents the given situation is:
    \na \u2260 4 years<\/p>\n

    c. Hank has at least $7.50.
    \nAnswer:
    \nThe given situation is:
    \nHank has at least $7.50
    \nNow,
    \nLet s be the amount of money Hank has
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the inequality that represents the given situation is:
    \ns \u2265 $7.50<\/p>\n

    KEY CONCEPT<\/strong>
    \nInequality symbols can be used to describe situations that have more than one possible solution.
    \n\"Envision<\/p>\n

    Do You Understand?<\/strong>
    \nQuestion 1.
    \nEssential Question<\/strong> How can you write an inequality to describe a situation?
    \nAnswer:
    \nThe steps to write an inequality to describe a situation are:
    \na. Read through the entire problem.
    \nb. Highlight the important information and key words that you need to solve the problem.
    \nc. Identify your variables.
    \nd. Write the equation or inequality.
    \ne. Solve.
    \nf. Write your answer in a complete sentence.
    \ng. Check or justify your answer<\/p>\n

    Question 2.
    \nGeneralize<\/strong> What is the difference between an equation with a variable and an inequality with a variable?
    \nAnswer:
    \nAn equation is a mathematical statement that shows the equal value of two expressions while an inequality is a mathematical statement that shows that an expression is lesser than or more than the other. An equation shows the equality of two variables while an inequality shows the inequality of two variables.<\/p>\n

    Question 3.
    \nWould it be more efficient to use an inequality or to list all of the quantities less than 6? Explain.
    \nAnswer:
    \nWithout using “Inequalities”,
    \nThe qualities less than 6 are: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, and 5
    \nBy using the “Inequalities”,
    \nThe quantities less than 6 are: x < 6
    \nWhere,
    \nx is the list of numbers that are less than 6<\/p>\n

    Question 4.
    \nGeneralize<\/strong> How are the symbols for greater than (>) and greater than or equal to (\u2265) related?
    \nAnswer:
    \n\u201cGreater than or equal to\u201d and \u201cless than or equal to\u201d are just the applicable symbol\u00a0with half an equal sign under it. For example, 4 or 3 \u2265 1 shows us a greater sign over half an equal sign, meaning that 4 or 3 are greater than or equal to 1. It works the other way, too<\/p>\n

    Do You Know How?<\/strong>
    \nIn 5-12, write an inequality for each situation.<\/strong>
    \nQuestion 5.
    \nA number, n, is greater than 22.
    \nAnswer:
    \nThe given situation is:
    \nA number, n, is greater than 22.
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the given situation in the form of an inequality is:
    \nn > 22<\/p>\n

    Question 6.
    \nThe value, v, does not equal 2\\(\\frac{1}{2}\\).
    \nAnswer:
    \nThe given situation is:
    \nThe value, v, does not equal 2\\(\\frac{1}{2}\\).
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the given situation in the form of an inequality is:
    \nv \u2260 2\\(\\frac{1}{2}\\)<\/p>\n

    Question 7.
    \nSally’s age, a, is at most 15.
    \nAnswer:
    \nThe given situation is:
    \nSally’s age, a, is at most 15.
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the given situation in the form of an inequality is:
    \na \u2264 15<\/p>\n

    Question 8.
    \nThe width of the picture, w, is shorter than 8.5 inches.
    \nAnswer:
    \nThe given situation is:
    \nThe width of the picture, w, is shorter than 8.5 inches.
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the given situation in the form of an inequality is:
    \nw < 8.5 inches<\/p>\n

    Question 9.
    \nSteve’s height, h, is at least 48 inches.
    \nAnswer:
    \nThe given situation is:
    \nSteve’s height, h, is at least 48 inches.
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the given situation in the form of an inequality is:
    \nh \u2265 48 inches<\/p>\n

    Question 10.
    \nVera’s baby brother’s age, b, is not 24 months.
    \nAnswer:
    \nThe given situation is:
    \nVera’s baby brother’s age, b, is not 24 months.
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the given situation in the form of an inequality is:
    \nb \u2260 24 months<\/p>\n

    Question 11.
    \nThe number of quarters, q, in the jar is less than 75.
    \nAnswer:
    \nThe given situation is:
    \nThe number of quarters, q, in the jar is less than 75.
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the given situation in the form of an inequality is:
    \nq < 75<\/p>\n

    Question 12.
    \nThe length of the fish a fisherman catches, f, must be at least 10 inches for him to keep it.
    \nAnswer:
    \nThe given situation is:
    \nThe length of the fish a fisherman catches, f, must be at least 10 inches for him to keep it
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the given situation in the form of an inequality is:
    \nf \u2265 10 inches<\/p>\n

    Practice & Problem Solving<\/strong><\/p>\n

    In 13-22, write an inequality for each situation.<\/strong>
    \nQuestion 13.
    \nUp to 12 people, p, can ride in the van.
    \nAnswer:
    \nThe given situation is:
    \nUp to 12 people, p, can ride in the van.
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the given situation in the form of an inequality is:
    \np \u2264 12 people<\/p>\n

    Question 14.
    \nA number of days, d, of sunshine is not 28.
    \nAnswer:
    \nThe given situation is:
    \nA number of days, d, of sunshine is not 28.
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the given situation in the form of an inequality is:
    \nd \u2260 28<\/p>\n

    Question 16.
    \nThe value, v, of the bracelet is less than $85.25.
    \nAnswer:
    \nThe given situation is:
    \nThe value, v, of the bracelet is less than $85.25.
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the given situation in the form of an inequality is:
    \nv < $85.25<\/p>\n

    Question 15.
    \nThe distance of the race, r, is farther than 6.2 miles.
    \nAnswer:
    \nThe given situation is:
    \nThe distance of the race, r, is farther than 6.2 miles.
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the given situation in the form of an inequality is:
    \nr > 6.2 miles<\/p>\n

    Question 17.
    \nThe number of people, p, that a restaurant can seat at one time is no more than 171.
    \nAnswer:
    \nThe given situation is:
    \nThe number of people, p, that a restaurant can seat at one time is no more than 171.
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the given situation in the form of an inequality is:
    \np \u2264 171 people<\/p>\n

    Question 18.
    \nThe time, t, a customer has left on a parking meter is at least 25 minutes.
    \nAnswer:
    \nThe given situation is:
    \nThe time, t, a customer has left on a parking meter is at least 25 minutes.
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the given situation in the form of an inequality is:
    \nt \u2265 25 minutes<\/p>\n

    Question 19.
    \nThe bill, b, was less than $45.
    \nAnswer:
    \nThe given situation is:
    \nThe bill, b, was less than $45.
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the given situation in the form of an inequality is:
    \nb < $45<\/p>\n

    Question 20.
    \nThe girls live b blocks apart; they do not live 7\\(\\frac{1}{2}\\) blocks apart.
    \nAnswer:
    \nThe given situation is:
    \nThe girls live b blocks apart; they do not live 7\\(\\frac{1}{2}\\) blocks apart.
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the given situation in the form of an inequality is:
    \nb \u2260 7\\(\\frac{1}{2}\\) blocks<\/p>\n

    Question 21.
    \nThe speed of the truck, s, must be no less than 34 miles per hour.
    \nAnswer:
    \nThe given situation is:
    \nThe speed of the truck, s, must be no less than 34 miles per hour.
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the given situation in the form of an inequality is:
    \ns \u2264 34 miles per hour<\/p>\n

    Question 22.
    \nThe number of baseball games, x, that Karen went to last year is more than 5.
    \nAnswer:
    \nThe given situation is:
    \nThe number of baseball games, x, that Karen went to last year is more than 5.
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the given situation in the form of an inequality is:
    \nx \u2264 5 games<\/p>\n

    Question 23.
    \nMia is taller than Gage. If m represents Mia’s height and g represents Gage’s height, write an inequality that shows the relationship between their heights.
    \nAnswer:
    \nIt is given that
    \nMia is taller than Gage and m represents Mia’s height and g represents Gage’s height
    \nNow,
    \nAccording to the given information,
    \nThe representation of the inequality for the given situation is:
    \nm > g
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the inequality for the given situation is:
    \nm > g<\/p>\n

    Question 24.
    \nTaryn sold gift-wrapping paper for a school fund-raiser. She sold at least 15 rolls of paper. Write an inequality to represent the amount of money, d, she earned for the fund-raiser.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nTaryn sold gift-wrapping paper for a school fund-raiser. She sold at least 15 rolls of paper
    \nNow,
    \nThe given figure is:
    \n\"Envision
    \nNow,
    \nLet the number of rolls of paper be: x
    \nSo,
    \nThe representation of the inequality for representing the rolls of paper is:
    \nx \u2265 15 rolls
    \nNow,
    \nThe amount of money, d, Taryn earned for the fundraiser = (The total number of rolls) \u00d7 (The cost of each roll)
    \n= 15 \u00d7 $8
    \n= $120
    \nSo,
    \nThe representation of the inequality for representing the amount of money Taryn earned for the fundraiser is:
    \nd \u2265 $120
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the inequality for representing the amount of money Taryn earned for the fundraiser is:
    \nd \u2265 $120<\/p>\n

    Question 25.
    \nA city in New England just experienced its greatest 1-day snowfall. Write an inequality to represent a snowfall that would beat this record.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nA city in New England just experienced its greatest 1-day snowfall
    \nNow,
    \nThe given figure is:
    \n\"Envision
    \nNow,
    \nLet the snowfall that would beat the given record be: t
    \nFrom the given figure,
    \nWe can observe that
    \nThe representation of the inequality to represent a snowfall that would beat the given record is:
    \nt > 19.7 inches of snow<\/p>\n

    Question 26.
    \nThe first bookcase, a, in a library can hold 1 less book than the second bookcase. The second bookcase holds 2,492 books. Write an inequality to represent the number of books the first bookcase can hold.
    \nAnswer:
    \nIt is given that
    \nThe first bookcase, a, in a library can hold 1 less book than the second bookcase. The second bookcase holds 2,492 books
    \nSo,
    \nThe number of books the first bookcase can hold = (The number of books the second bookcase can hold) – 1
    \n= 2,492 – 1
    \n= 2,491 books
    \nNow,
    \nLet the number of books the first bookcase can hold be: x
    \nLet the number of books the second bookcase can hold be: y
    \nSo,
    \nFrom the above,
    \nWe can observe that
    \nx < y
    \nHence, from the above,
    \nWe can conclude that
    \nThe number of books the first bookcase can hold < The number of books the second bookcase can hold<\/p>\n

    Question 27.
    \nA certain airplane must carry no more than 134 passengers during a flight. Write an inequality to represent the number of passengers, p, that would NOT be allowed during this flight.
    \nAnswer:
    \nIt is given that
    \nA certain airplane must carry no more than 134 passengers during a flight
    \nNow,
    \nLet p be the number of passengers during a flight
    \nSo,
    \nThe representation of the inequality to represent the number of passengers that would not be allowed during this flight is:
    \np > 134 passengers
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the inequality to represent the number of passengers that would not be allowed during this flight is:
    \np > 134 passengers<\/p>\n

    Question 28.
    \nHigher Order Thinking<\/strong> To ride a certain roller coaster, a rider must be at least 42 inches tall. To represent this situation, Elias wrote h > 42 and Nina wrote h > 42. Who is correct? Explain.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nTo ride a certain roller coaster, a rider must be at least 42 inches tall. To represent this situation, Elias wrote h > 42 and Nina wrote h > 42.
    \nNow,
    \nLet the height of the rider be: h inches
    \nSo,
    \nThe representation of the inequality that represents the height of a rider to ride a certain roller coaster is:
    \nh \u2265 42
    \nHence, from the above,
    \nWe can conclude that
    \nElias is correct<\/p>\n

    Assessment Practice<\/strong><\/p>\n

    Question 29.
    \nMiguel earns extra money working two weekends with his dad. He is saving to buy a new bike that costs $140.
    \nHeather says that Miguel needs to earn more than $6 for each hour that he works to have enough money to buy the bike. Her work is shown below. Write an inequality to explain why she is incorrect.
    \n\"Envision
    \n$140 \u00f7 23 hours > $6.00 per hour
    \nMiguel has to earn more than $6.00 per hour.
    \nAnswer:
    \nIt is given that
    \nMiguel earns extra money working two weekends with his dad. He is saving to buy a new bike that costs $140.
    \nHeather says that Miguel needs to earn more than $6 for each hour that he works to have enough money to buy the bike. Her work is shown below
    \nNow,
    \nThe given figure is:
    \n\"\"
    \nNow,
    \nFrom the given figure,
    \nWe can observe that
    \nMiguel’s work record in Weekend 1 is: 9 hours
    \nMiguel’s work record in Weekend 2 is: 7 hours
    \nSo,
    \nThe total number of hours in 2 Weekends for Migul = 9 + 7
    \n= 16 hours
    \nSo,
    \nThe amount of money Migul will earn per hour = (The amount of money Migul saved to buy a new bike) \u00f7 (The total number of hours in 2 Weekends for Migul)
    \n= \\(\\frac{$140}{16}\\)
    \n= $8.75
    \nHence, from the above,
    \nWe can conclude that
    \nThe inequality that represents Migul’s work is:
    \nW > $8.75 per hour<\/p>\n

    Lesson 4.7 Solve Inequalities<\/h3>\n

    Solve & Discuss It!<\/strong>
    \nHenry is thinking of a number that is less than 17. What number could he be thinking of?
    \nI can… write and represent solutions of inequalities.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nHenry is thinking of a number that is less than 17
    \nNow,
    \nLet x be the numbers less than 17
    \nWhere,
    \nx may be an integer, decimal number (or) a whole number that is less than 17
    \nSo,
    \nThe representation of the inequality that represents the number that is less than 17 that Henry is thinking is:
    \nx < 17
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the inequality that represents the number that is less than 17 that Henry is thinking is:
    \nx < 17<\/p>\n

    Use Appropriate Tools<\/strong>
    \nHow can you use a number line to show all the numbers that are less than 17?
    \n\"Envision
    \nAnswer:
    \nWe know that,
    \nThe numebrs that are 17 will be an integer, decimal number or a Whole number
    \nNow,
    \nThe whole numbers that are less than 17 are: 0, 1, 2, 3, …….. 16
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the list of all the Whole numbers that are less than 17 in the given number line is:
    \n\"\"<\/p>\n

    Focus on math practices<\/strong>
    \nReasoning<\/strong> Could Henry be thinking of 17? Explain.
    \nAnswer:
    \nNo, Henry should not be thinking of 17 because it is clearly mentioned that Henry is thinking of a number that is less than 17
    \nSo,
    \nThe list of the numbers that Henry may be thinking is: 0, 1, 2, 3 ….. 16<\/p>\n

    Essential Question<\/strong>
    \nHow can you represent the solutions of an inequality?
    \nAnswer:
    \nThe rules for Solving Inequalities are:
    \na. Add the same number on both sides.
    \nb. From both sides, subtract the same number.
    \nc. By the same positive number, multiply both sides.
    \nd. By the same positive number, divide both sides.
    \ne. Multiply the same negative number on both sides and reverse the sign.
    \nA\u00a0closed, or shaded, circle is used to represent the inequalities greater than or equal to (\u2265) or less than or equal to (\u2264) . The point is part of the solution. An open circle is used for greater than (>) or less than (<). The point is not part of the solution<\/p>\n

    Try It!<\/strong><\/p>\n

    Graph all of the solutions of x < 8.
    \n\"Envision
    \nTo graph x < 8, draw a(n) ________ circle at 8 on the number line.
    \n7 and 4 are two of the many possible solutions of the inequality.
    \nShade the solutions to the ________ of the ________ circle you drew at 8.
    \nAnswer:
    \nThe given inequality is: x < 8
    \nNow,
    \nThe representation of the given inequality in the given number line is:
    \n\"\"
    \nNow,
    \n\"\"
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe solutions for the given inequality in the given number line is:
    \n\"\"<\/p>\n

    Convince Me!<\/strong> How does the graph of the inequality change when the less than sign is changed to a greater than sign? How does it stay the same?
    \nAnswer:
    \nWe know that multiplying or dividing an inequality by a negative number changes the direction of the inequality. In other words, a greater-than symbol becomes a less-than symbol, and vice versa
    \nNow,
    \nWhen the inequality has the symbol > (or) \u2265, the numbers to the right of this inequality will become the solutions
    \nWhen the inequality has the symbol < (or) \u2264, the numbers to the left of this inequality will become the solutions<\/p>\n

    Try It!<\/strong><\/p>\n

    There are no menu items on the children’s menu at the Flamingo Restaurant that cost more than $8.50. What are all the possible costs of the items on the children’s menu?
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nThere are no menu items on the children’s menu at the Flamingo Restaurant that cost more than $8.50
    \nNow,
    \nLet c be the cost of the menu items on the children’s menu at the Flamingo Restaurant
    \nSo,
    \nThe representation of the inequality that represents the given situation is:
    \nc < $8.50
    \nHence,
    \nThe representation of all the possible costs of the items on the children’s menu on the given number line is:
    \n\"\"<\/p>\n

    Try It!<\/strong><\/p>\n

    Which athletes, if any, would qualify for the finals if the length of a jump that qualifies for the finals were at least 20\\(\\frac{1}{2}\\) feet?
    \nAnswer:
    \nIt is given that
    \nThe length of a jump that qualifies for the finals were at least 20\\(\\frac{1}{2}\\) feet
    \nNow,
    \nThe given table is:
    \n\"\"
    \nNow,
    \nLet the length of a jump be: d feet
    \nNow,
    \nThe representation of the given situation in teh form of an inequality is:
    \nd \u2265 20\\(\\frac{1}{2}\\) feet
    \nNow,
    \nFrom the given table,
    \nWe can observe that
    \nThe values that are greater than 20\\(\\frac{1}{2}\\) feet are:
    \n20\\(\\frac{1}{2}\\) = 20\\(\\frac{1}{2}\\)
    \n20\\(\\frac{1}{2}\\) > 22\\(\\frac{1}{3}\\)
    \nHence, from the above,
    \nWe can conclude that
    \nAmir and Ryan would qualify for the finals if the length of a jump that qualifies for the finals were at least 20\\(\\frac{1}{2}\\) feet<\/p>\n

    KEY CONCEPT<\/strong>
    \nAn inequality uses these symbols: <, >, \u2264 , or \u2265 to compare two expressions.
    \n\"Envision<\/p>\n

    Do You Understand?<\/strong>
    \nQuestion 1.
    \nEssential Question<\/strong> How can you represent the solutions of an inequality?
    \nAnswer:
    \nThe rules for Solving Inequalities are:
    \na. Add the same number on both sides.
    \nb. From both sides, subtract the same number.
    \nc. By the same positive number, multiply both sides.
    \nd. By the same positive number, divide both sides.
    \ne. Multiply the same negative number on both sides and reverse the sign.
    \nA\u00a0closed, or shaded, circle is used to represent the inequalities greater than or equal to (\u2265) or less than or equal to (\u2264) . The point is part of the solution. An open circle is used for greater than (>) or less than (<). The point is not part of the solution<\/p>\n

    Question 2.
    \nIn Example 1, why is 9 a solution of x > 5?
    \nAnswer:
    \nThe given equation is: x > 5
    \nNow,
    \nFrom the given equation,
    \nWe can observe that
    \nSince x > 5,
    \nThe solutions to the given equation are all the numbers that are greater than 5
    \nSo,
    \nThe given equation will have infinitely many solutions that are greater than 5
    \nHence, from the above,
    \nWe can conclude that
    \n9 is a solution of x > 5<\/p>\n

    Question 3.
    \nExplain why 2 is NOT a solution of x > 5.
    \nAnswer:
    \nThe given equation is: x > 5
    \nNow,
    \nFrom the given equation,
    \nWe can observe that
    \nSince x > 5,
    \nThe solutions to the given equation are all the numbers that are greater than 5
    \nSo,
    \nThe given equation will have infinitely many solutions that are greater than 5
    \nHence, from the above,
    \nWe can conclude that
    \n2 is not a solution of x > 5<\/p>\n

    Question 4.
    \nHow many solutions does the inequality x > 12 have? Explain.
    \nAnswer:
    \nThe given equation is: x > 12
    \nNow,
    \nFrom the given equation,
    \nWe can observe that
    \nSince x > 12,
    \nThe solutions to the given equation are all the numbers that are greater than 12
    \nSo,
    \nThe given equation will have infinitely many solutions that are greater than 12
    \nHence, from the above,
    \nWe can conclude that
    \nx > 12 will have infinitely many solutions that are greater than 12<\/p>\n

    Question 5.
    \nGeneralize<\/strong> How do the graphs of the solutions of inequalities involving greater than (>) and greater than or equal to (\u2265) compare?
    \nAnswer:
    \nA\u00a0closed, or shaded, circle is used to represent the inequalities greater than or equal to (\u2265) or less than or equal to (\u2264) . The point is part of the solution. An open circle is used for greater than (>) or less than (<). The point is not part of the solution<\/p>\n

    Do You Know How?<\/strong>
    \nIn 6 and 7, write the inequality that each graph represents.<\/strong>
    \nQuestion 6.
    \n\"Envision
    \nAnswer:
    \nThe given number line is:
    \n\"\"
    \nNow,
    \nFrom the given number line,
    \nWe can observe that
    \nThere is an open circle on 14.
    \nSo,
    \n14 will not be included and the selected numbers are to the left side of 14 in the given number line
    \nSo,
    \nThe representation of the inequality for the given number line is:
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the inequality for the given number line is:
    \nz < 14<\/p>\n

    Question 7.
    \n\"Envision
    \nAnswer:
    \nThe given number line is:
    \n\"\"
    \nNow,
    \nFrom the given number line,
    \nWe can observe that
    \nThere is a closed circle on 18.
    \nSo,
    \n18 will be included and the selected numbers are to the right side of 18 in the given number line
    \nSo,
    \nThe representation of the inequality for the given number line is:
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the inequality for the given number line is:
    \nd \u2265 18<\/p>\n

    In 8-11, substitute each given value of the variable to find which, if any, is a solution of the inequality.<\/strong>
    \nQuestion 8.
    \nw < 8\u00a0 \u00a0 \u00a0 \u00a0 \u00a0w = 4.3, 5.3, 8.3, 9
    \nAnswer:
    \nThe given inequality is: w < 8
    \nNow,
    \nThe given list of numbers are: 4.3, 5.3, 8.3, and 9
    \nNow,
    \nWhen we observe the given list of numbers,
    \n4.3, and 5.3 are less than 8
    \nHence, from the above,
    \nWe can conclude that
    \nThe solutions for the given inequality are: 4.3, 5.3<\/p>\n

    Question 9.
    \nt > 25\u00a0 \u00a0 \u00a0 t = 24, 25, 25.1, 27
    \nAnswer:
    \nThe given inequality is: t > 25
    \nNow,
    \nThe given list of numbers are: 24, 25, 25.1, and 27
    \nNow,
    \nWhen we observe the given list of numbers,
    \n25.1 and 27 are greater the 25
    \nHence, from the above,
    \nWe can conclude that
    \nThe solutions for the given inequality are: 25.1, 27<\/p>\n

    Question 10.
    \ng \u2264 4\u00a0 \u00a0 \u00a0 \u00a0 g = 0, 4, 5, 6
    \nAnswer:
    \nThe given inequality is: g \u2264 4
    \nNow,
    \nThe given list of numbers are: 0, 4, 5, and 6
    \nNow,
    \nWhen we observe the given list of numbers,
    \n0 and 4 are less than or equal to 4
    \nHence, from the above,
    \nWe can conclude that
    \nThe solutions for the given inequality are: 0, 4<\/p>\n

    Question 11.
    \ny \u2265 28\u00a0 \u00a0 \u00a0 \u00a0 y = 4, 5, 6, 7
    \nAnswer:
    \nThe given inequality is: y \u2265 8
    \nNow,
    \nThe given list of numbers are: 4, 5, 6, and 7
    \nNow,
    \nWhen we observe the given list of numbers,
    \nThe given list of numbers are less than 28
    \nHence, from the above,
    \nWe can conclude that
    \nThere are no solutions for the given inequality from the given list of numbers<\/p>\n

    Practice & Problem Solving<\/strong><\/p>\n

    In 12-15, write the inequality that each graph represents.<\/strong>
    \nQuestion 12.
    \n\"Envision
    \nAnswer:
    \nThe given number line is:
    \n\"\"
    \nNow,
    \nFrom the given number line,
    \nWe can observe that
    \nThere is an open circle on 7.
    \nSo,
    \n7 will not be included and the selected numbers are to the left side of 7 in the given number line
    \nSo,
    \nThe representation of the inequality for the given number line is:
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the inequality for the given number line is:
    \ny < 7<\/p>\n

    Question 13.
    \n\"Envision
    \nAnswer:
    \nThe given number line is:
    \n\"\"
    \nNow,
    \nFrom the given number line,
    \nWe can observe that
    \nThere is an open circle on 0.
    \nSo,
    \n0 will not be included and the selected numbers are to the right side of 0 in the given number line
    \nSo,
    \nThe representation of the inequality for the given number line is:
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the inequality for the given number line is:
    \nb > 0<\/p>\n

    Question 14.
    \n\"Envision
    \nAnswer:
    \nThe given number line is:
    \n\"\"
    \nNow,
    \nFrom the given number line,
    \nWe can observe that
    \nThere is an open circle on 3.
    \nSo,
    \n3 will not be included and the selected numbers are to the right side of 3 in the given number line
    \nSo,
    \nThe representation of the inequality for the given number line is:
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the inequality for the given number line is:
    \nx > 3<\/p>\n

    Question 15.
    \n\"Envision
    \nAnswer:
    \nThe given number line is:
    \n\"\"
    \nNow,
    \nFrom the given number line,
    \nWe can observe that
    \nThere is a closed circle on 5.
    \nSo,
    \n5 will be included and the selected numbers are to the left side of 5 in the given number line
    \nSo,
    \nThe representation of the inequality for the given number line is:
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the inequality for the given number line is:
    \nt \u2264 5<\/p>\n

    In 16-19, graph each inequality on a number line.<\/strong>
    \nQuestion 16.
    \nh \u2265 9
    \n\"Envision
    \nAnswer:
    \nThe given inequality is: h \u2265 9
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the given inequality on the number line is:
    \n\"\"<\/p>\n

    Question 17.
    \np < 3
    \n\"Envision
    \nAnswer:
    \nThe given inequality is: p < 3
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the given inequality on the number line is:
    \n\"\"<\/p>\n

    Question 18.
    \nt \u2264 6
    \n\"Envision
    \nAnswer:
    \nThe given inequality is: t \u2264 6
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the given inequality on the number line is:
    \n\"\"<\/p>\n

    Question 19.
    \ns > 1
    \n\"Envision
    \nAnswer:
    \nThe given inequality is: s > 1
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the given inequality on the number line is:
    \n\"\"<\/p>\n

    In 20-27, name three solutions of each inequality.<\/strong>
    \nQuestion 20.
    \nx > 10.5
    \nAnswer:
    \nThe given inequality is: x > 10.5
    \nNow,
    \nWe know that,
    \nIf the inequality has a “>” symbol, then the values that are greater than the given number will be the solutions for the equation
    \nHence, from the above,
    \nWe can conclude that
    \nThe three solutions for the given inequality are: 11, 12, and 13<\/p>\n

    Question 21.
    \nr < 19
    \nAnswer:
    \nThe given inequality is: r < 19
    \nNow,
    \nWe know that,
    \nIf the inequality has a “<” symbol, then the values that are lesser than the given number will be the solutions for the equation
    \nHence, from the above,
    \nWe can conclude that
    \nThe three solutions for the given inequality are: 15, 16, and 17<\/p>\n

    Question 22.
    \ny \u2265 200
    \nAnswer:
    \nThe given inequality is: y \u2265 200
    \nNow,
    \nWe know that,
    \nIf the inequality has a “\u2265” symbol, then the values that are greater than or equal to the given number will be the solutions for the equation
    \nHence, from the above,
    \nWe can conclude that
    \nThe three solutions for the given inequality are: 200, 205, and 210<\/p>\n

    Question 23.
    \nm \u2264 82
    \nAnswer:
    \nThe given inequality is: m \u2264 82
    \nNow,
    \nWe know that,
    \nIf the inequality has a “\u2264” symbol, then the values that are lesser than or equal to the given number will be the solutions for the equation
    \nHence, from the above,
    \nWe can conclude that
    \nThe three solutions for the given inequality are: 75, 80, and 81<\/p>\n

    Question 24.
    \nx \u2265 12
    \nAnswer:
    \nThe given inequality is: x \u2265 12
    \nNow,
    \nWe know that,
    \nIf the inequality has a “\u2265” symbol, then the values that are greater than or equal to the given number will be the solutions for the equation
    \nHence, from the above,
    \nWe can conclude that
    \nThe three solutions for the given inequality are: 12, 13, and 14<\/p>\n

    Question 25.
    \nq \u2264 3.5
    \nAnswer:
    \nThe given inequality is: q \u2264 3.5
    \nNow,
    \nWe know that,
    \nIf the inequality has a “\u2264” symbol, then the values that are lesster than or equal to the given number will be the solutions for the equation
    \nHence, from the above,
    \nWe can conclude that
    \nThe three solutions for the given inequality are: 0, 1.5, and 3.5<\/p>\n

    Question 26.
    \nv > 35
    \nAnswer:
    \nThe given inequality is: v > 35
    \nNow,
    \nWe know that,
    \nIf the inequality has a “>” symbol, then the values that are greater than the given number will be the solutions for the equation
    \nHence, from the above,
    \nWe can conclude that
    \nThe three solutions for the given inequality are: 40, 45, and 50<\/p>\n

    Question 27.
    \nm < 2.5
    \nAnswer:
    \nThe given inequality is: m < 2.5
    \nNow,
    \nWe know that,
    \nIf the inequality has a “<” symbol, then the values that are lesser than the given number will be the solutions for the equation
    \nHence, from the above,
    \nWe can conclude that
    \nThe three solutions for the given inequality are: 0, 1, and 2<\/p>\n

    Question 28.
    \nThe inequality w \u2264 1,500 describes the maximum weight in pounds, w, allowed by law in a freight elevator. Is a total weight of either 1,505 pounds or 1,600 pounds allowed in a freight elevator? Explain.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nThe inequality w \u2264 1,500 describes the maximum weight in pounds, w, allowed by law in a freight elevator.
    \nNow,
    \nThe given equation is: w \u2264 1,500
    \nNow,
    \nFrom the given inequality,
    \nWe can observe that
    \nThe solutions for the given inequality will be less than or equal to 1,500 but not greater than 1,500
    \nSo,
    \n1,505 > 1,500 and 1,600 > 1,500
    \nHence, from the above,
    \nWe can conclude that
    \nThe total weight of either 1,505 pounds or 1,600 pounds will not be allowed in a freight elevator<\/p>\n

    Question 29.
    \nReasoning<\/strong> Graph the inequalities x > 2 and x < 2 on the same number line. What value, if any, is not a solution of either inequality? Explain.
    \n\"Envision
    \nAnswer:
    \nThe given inequalities are: x > 2 and x < 2
    \nNow,
    \nThe representation of x > 2 on the given number line is:
    \n\"\"
    \nThe representtaion of x < 2 on the given number line is:
    \n\"\"
    \nNow,
    \nWhen we observe the number lines of x > 2 and x < 2,
    \nWe can say that
    \n2 is not a solution for either of the number lines
    \nHence, from the above,
    \nWe can conclude that
    \n2 is not a solution of either inequality<\/p>\n

    Question 30.
    \nModel with Math<\/strong> Death Valley is the hottest place in the United States. The highest temperature ever recorded there was 134\u00b0F. The lowest temperature recorded there was 15\u00b0F. Write two inequalities that would describe the temperature, in degrees Fahrenheit, in Death Valley at any time since temperatures have been recorded.
    \nAnswer:
    \nIt is given that
    \nDeath Valley is the hottest place in the United States. The highest temperature ever recorded there was 134\u00b0F. The lowest temperature recorded there was 15\u00b0F
    \nNow,
    \nLet the highest temperature recorded in the Death Valley be: H
    \nLet the lowest temperature recorded in the Death Valley be: L
    \nSo,
    \nThe inequality that represents the highest temperature in \u00b0F in the Death Valley is: H < 134\u00b0 F
    \nThe inequality that represents the lowest temperature in \u00b0F in the Death Valley is: L > 15\u00b0 F
    \nHence, from the above,
    \nWe can conclude that
    \nThe two inequalities that describe the temperature in the Death Vally at any time since the temperatures have been recorded is:
    \nThe inequality that represents the highest temperature in \u00b0F in the Death Valley is: H < 134\u00b0 F
    \nThe inequality that represents the lowest temperature in \u00b0F in the Death Valley is: L > 15\u00b0 F<\/p>\n

    Question 31.
    \nThe number line below represents the solutions of the inequality x > 7. Is 7.1 a solution? Is 7.01 a solution? Explain.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nThe number line below represents the solutions of the inequality x > 7
    \nNow,
    \nThe given number that represents the solution for the given inequality x > 7 is:
    \n\"Envision
    \nNow,
    \nFrom the above number line,
    \nWe can observe that
    \nThe solutions for x > 7 will be all the numbers that are greater than 7 and will be infinite
    \nSo,
    \n7.1 > 7 and 7.01 > 7
    \nHence, from the above,
    \nWe can conclude that
    \n7.1 and 7.01 are the solutions of the given inequality x > 7<\/p>\n

    Question 32.
    \nThe temperature in a greenhouse should be 65 degrees or higher. Write an inequality to describe the allowable temperature in the greenhouse.
    \nAnswer:
    \nIt is given that
    \nThe temperature in a greenhouse should be 65 degrees or higher.
    \nNow,
    \nLet the temperature of the greenouse be: t
    \nSo,
    \nThe inequality that describes the allowable temperature in the greenhouse is:
    \nt \u2265 65\u00b0
    \nHence, from the above,
    \nWe can conclude that
    \nThe inequality that describes the allowable temperature in the greenhouse is:
    \nt \u2265 65\u00b0<\/p>\n

    Question 33.
    \nHigher Order Thinking<\/strong> Francine received a gift card to buy cell phone apps. She says that the card’s value is enough to buy any of the apps shown at the right. Let v be the dollar value of the gift card. Write an inequality that best describes the value of the gift card.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nFrancine received a gift card to buy cell phone apps. She says that the card’s value is enough to buy any of the apps shown at the right. Let v be the dollar value of the gift card.
    \nNow,
    \nThe given figure is:
    \n\"Envision
    \nNow,
    \nWhen we observe the given figure,
    \nWe can say that
    \nThe maximum vaue of the given gift card is: $12
    \nSo,
    \nThe costs of the given apps should be less than $12
    \nSo,
    \nThe representation of the inequality that describes the value of the gift card is:
    \nv \u2264 $12
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the inequality that describes the value of the gift card is:
    \nv \u2264 $12<\/p>\n

    Question 34.
    \nThe maximum load on a small plane is 400 pounds. Let w represent the weight on the plane. Write an inequality to describe the allowable weight on the plane.
    \nAnswer:
    \nIt is given that
    \nThe maximum load on a small plane is 400 pounds. Let w represent the weight on the plane
    \nNow,
    \nSince the maximum load is 400 pounds, the remaining weights will be less than or equal to 400 pounds
    \nSo,
    \nThe representation of the inequalty that describes teh allowable weight on the plane is:
    \nw \u2264 400 pounds
    \nHence, from the above,
    \nWe can conclude that
    \nThe representation of the inequalty that describes teh allowable weight on the plane is:
    \nw \u2264 400 pounds<\/p>\n

    Question 35.
    \nJillian is thinking of a whole number that is greater than 21. What numbers, if any, make the inequality n > 21 true for n = 0, 1, 2, 3, 4, …?
    \nAnswer:
    \nIt is given that
    \nJillian is thinking of a whole number that is greater than 21
    \nNow,
    \nThe given inequality is: n > 21
    \nNow,
    \nWe know that,
    \nIf any inequality has “>” symbol, then the solutions for that inequality will be greater than thenumber and the solutions will be infinite
    \nSo,
    \nThe solutions for the given inequality will be: 22, 23, 24,……..
    \nHence, from the above,
    \nWe can conclude that
    \nThe numbers that make the inequality n > 21 true are: 22, 23, 24, …..<\/p>\n

    Assessment Practice<\/strong><\/p>\n

    Question 36.
    \nSelect all the given values of y that make the inequality 3y < 25 true.
    \n\u2610 6.5
    \n\u2610 7
    \n\u2610 8
    \n\u2610 8.5
    \n\u2610 9
    \nAnswer:
    \nThe given inequality is: 3y < 25
    \nNow,
    \nDivide the given nequality be 3 on both sides
    \nSo,
    \n\\(\\frac{3y}{3}\\) < \\(\\frac{25}{3}\\)
    \ny < 8.3
    \nHence, from the above,
    \nWe can conclude that
    \nAll the values of y that make the inequality 3y < 25 true are:
    \n\"\"<\/p>\n

    Question 37.
    \nTania started a graph to show the inequality y < 3.7. Finish labeling the number line and draw the graph.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nTania started a graph to show the inequality y < 3.7
    \nNow,
    \nThe given inequality is: y < 3.7
    \nNow,
    \nThe representation of the given number line and the solution for thegiven inequality is:
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe completed number line and the solutions for y < 3.7 on the completed number line is:
    \n\"\"<\/p>\n

    3-ACT MATH<\/strong><\/p>\n

    3-Act Mathematical Modeling: Checking a Bag
    \n\"Envision<\/p>\n

    \u0410\u0421\u0422 1<\/strong>
    \nQuestion 1.
    \nAfter watching the video, what is the first question that comes to mind?
    \nAnswer:<\/p>\n

    Question 2.
    \nWrite the Main Question you will answer.
    \nAnswer:<\/p>\n

    Question 3.
    \nConstruct Arguments Predict an answer to this Main Question. Explain your prediction.
    \n\"Envision
    \nAnswer:<\/p>\n

    Question 4.
    \nOn the number line below, write a number that is too small to be the answer. Write a number that is too large.
    \n\"Envision
    \nAnswer:<\/p>\n

    Question 5.
    \nPlot your prediction on the same number line.
    \nAnswer:<\/p>\n

    ACT 2<\/strong>
    \nQuestion 6.
    \nWhat information in this situation would be helpful to know? How would you use that information?
    \n\"Envision
    \nAnswer:<\/p>\n

    Question 7.
    \nUse Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.
    \nAnswer:<\/p>\n

    Question 8.
    \nModel with Math Represent the situation using mathematics. Use your representation to answer the Main Question.
    \nAnswer:<\/p>\n

    Question 9.
    \nWhat is your answer to the Main Question? is it higher or lower than your prediction? Explain why.
    \n\"Envision
    \nAnswer:<\/p>\n

    ACT 3<\/strong>
    \nQuestion 10.
    \nWrite the answer you saw in the video.
    \nAnswer:<\/p>\n

    Question 11.
    \nReasoning Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?
    \n\"Envision
    \nAnswer:<\/p>\n

    Question 12.
    \nMake Sense and Persevere Would you change your model now that you know the answer? Explain.
    \n\"Envision
    \nAnswer:<\/p>\n

    Reflect<\/strong>
    \nQuestion 13.
    \nModel with Math Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?
    \nAnswer:<\/p>\n

    Question 14.
    \nWas an equation or an inequality more useful to answer the Main Question? Explain.
    \nAnswer:<\/p>\n

    SEQUEL<\/strong>
    \nQuestion 15.
    \nBe Precise A different airline has a weight limit of 40 pounds for a checked bag. Explain how the answer would change for this airline.
    \n\"Envision
    \nAnswer:<\/p>\n

    Lesson 4.8 Understand Dependent and Independent Variables<\/h3>\n

    Explain It!<\/strong>
    \nMax is shipping a present to his grandmother.
    \nI can… identify dependent and independent variables.
    \n\"Envision<\/p>\n

    A. What are three factors that will affect the weight of the box? What are three factors that will not affect the weight of the box?
    \nAnswer:
    \nIt is given that
    \nMax is shipping a present to his grandmother.
    \nNow,
    \nThe given figure is:
    \n\"Envision
    \nNow,
    \nFrom the given figure,
    \nWe can observe that
    \nThe three factors that will affect the weight of the box is:
    \na. Size of the box
    \nb. The material of the box
    \nc. Contents inside the box
    \nThe three factors that will not affect the weight of the box is:
    \na. The color of the box
    \nb. The shape of the box
    \nc. The design of the box<\/p>\n

    B. How might the size of the box and the contents of the box affect the weight of the box?
    \nAnswer:
    \nWe know that,
    \nWhen we increase the size of the box, the contents of the box will also increase in the box and also the weight of the box will increase
    \nSo,
    \nThe relationship between the size of the box and the contents of the box and the weight of the box is:
    \nSize of the box \u221d Contents of the box \u221d Weight of the box<\/p>\n

    Focus on math practices<\/strong>
    \nModel with Math<\/strong> Describe another situation in which changing one factor results in changes to another factor.
    \nAnswer:
    \nThe example of a situation in which changing one factor results in changes to another factor is:
    \nThe distance a car travels, d, is dependent on the speed, s, at which it travels. So, speed is the independent variable and distance is the dependent variable
    \nSo,
    \nHere,
    \nThe distance a car travels, d, is dependent on the speed, s, at which it travels. So, speed is the independent variable and distance is the dependent variable<\/p>\n

    Essential Question<\/strong>
    \nWhat does it mean for one variable to be dependent on another variable?
    \nAnswer:
    \nA dependent variable is\u00a0a variable whose value will change depending on the value of another variable, called the independent variable. In a scientific experiment, it is the variable being tested, and therefore, it is called the dependent variable<\/p>\n

    Try It!<\/strong><\/p>\n

    A baker used a certain number of cups of batter, b, to make p pancakes. Which variable, p, pancakes or b, batter is the dependent variable? Explain.
    \nAnswer:
    \nIt is given that
    \nA baker used a certain number of cups of batter, b, to make p pancakes
    \nNow,
    \nWe know that,
    \nAn “Independent variable” causes the dependent variable to change
    \nA “Dependent variable” change in response to another variable
    \nNow,
    \nFrom the given situation,
    \nWe can observe that
    \nThe independent variable is: b
    \nThe dependent variable is: p
    \nHence, from the above,
    \nWe can conclude that
    \nPancakes (p) is the dependent variable<\/p>\n

    Convince Me!<\/strong> If the baker doubles the number of cups of batter used, b, what would you expect to happen to the number of pancakes made, p? Explain.
    \nAnswer:
    \nIt is given that
    \nThe baker doubles the number of cups of batter used, b
    \nNow,
    \nWe know that,
    \nAn “Independent variable” causes the dependent variable to change
    \nA “Dependent variable” change in response to another variable
    \nSo,
    \nFrom the given situation,
    \nWe can observe that
    \nIf the baker doubles the number of cups of batter used, b, then the number of pancakes, p, would also be double in quantity
    \nHence, from the above,
    \nWe can conclude that
    \nIf the baker doubles the number of cups of batter used, b, then the number of pancakes, p, would also be double in quantity<\/p>\n

    Try It!<\/strong><\/p>\n

    Jenna wants to rent a mountain bike by the week. Identify the independent variables that affect the total rental cost.
    \nAnswer:
    \nIt is given that
    \nJenna wants to rent a mountain bike by the week
    \nNow,
    \nFrom Example 3,
    \nWe can observe that
    \nThe dependent variables and independent present in Example 3 are:
    \n\"\"
    \nNow,
    \nWe know that,
    \nAn “Independent variable” causes the dependent variable to change
    \nA “Dependent variable” change in response to another variable
    \nSo,
    \nThe independent variables that affect the total rental cost for the given situation is:
    \na. Price per day (d)
    \nb. Price per week (w)
    \nc. Length of rental (l)
    \nHence, from the above,
    \nWe can conclude that
    \nThe independent variables that affect the total rental cost for the given situation is:
    \na. Price per day (d)
    \nb. Price per week (w)
    \nc. Length of rental (l)<\/p>\n

    KEY CONCEPT<\/strong>
    \nA dependent variable changes in response to another variable, called an independent variable. An independent variable causes the change in a dependent variable. It is independent because its value is not affected by other variables.
    \nThe distance a car travels, d, is dependent on the speed, s, at which it travels. Speed is the independent variable, and distance is the dependent variable.<\/p>\n

    Do You Understand?<\/strong>
    \nQuestion 1.
    \nEssential Question<\/strong> What does it mean for one variable to be dependent on another variable?
    \nAnswer:
    \nA dependent variable is\u00a0a variable whose value will change depending on the value of another variable, called the independent variable. In a scientific experiment, it is the variable being tested, and therefore, it is called the dependent variable<\/p>\n

    Question 2.
    \nJake and Viola record the number of miles, m, they bike to help track the number of calories, s, they burn in an hour.
    \n\"Envision
    \nCritique Reasoning<\/strong> Viola says the number of calories, c, they burn is the dependent variable. Do you agree? Explain.
    \nAnswer:
    \nIt is given that
    \nJake and Viola record the number of miles, m, they bike to help track the number of calories, s, they burn in an hour.
    \nNow,
    \nFrom the given situation,
    \nWe can observe that
    \nThe number of calories they burn is dependent on the number of miles
    \nNow,
    \nWe know that,
    \nAn “Independent variable” causes the dependent variable to change
    \nA “Dependent variable” change in response to another variable
    \nSo,
    \nThe independent variable for the given situation is: Number of miles
    \nThe dependent variable for the given situation is: Number of calories they burn
    \nHence, from the above,
    \nWe can conclude that
    \nWe can agree with Viola<\/p>\n

    Question 3.
    \nReasoning<\/strong> in the biking problem above, identify at least one other independent variable that could affect the dependent variable.
    \nAnswer:
    \nThe given situation from Exercise 2 is:
    \n\"\"
    \nNow,
    \nFrom the given situation,
    \nWe can observe that
    \nThe number of calories they burn is dependent on the number of miles
    \nNow,
    \nWe know that,
    \nAn “Independent variable” causes the dependent variable to change
    \nA “Dependent variable” change in response to another variable
    \nSo,
    \nThe independent variable for the given situation is: Number of miles
    \nThe dependent variable for the given situation is: Number of calories they burn
    \nHence, from the above,
    \nWe can conclude that
    \nThe independent variable that could affect the dependent variale for the given situation is: Number of miles<\/p>\n

    Do You Know How?<\/strong>
    \nIn 4-11, identify the independent variable and the dependent variable.<\/strong>
    \nQuestion 4.
    \nThe amount of money, m, earned if t raffle tickets are sold
    \nAnswer:
    \nThe given situation is:
    \nThe amount of money, m, earned if t raffle tickets are sold
    \nNow,
    \nWe know that,
    \nAn “Independent variable” causes the dependent variable to change
    \nA “Dependent variable” change in response to another variable
    \nSo,
    \nThe independent variable for the given situation is: traffic tickets, t
    \nThe dependent variable for the given situation is: Amount of oney, m
    \nHence, from the above,
    \nWe can conclude that
    \nThe independent variable for the given situation is: traffic tickets, t
    \nThe dependent variable for the given situation is: Amount of money, m<\/p>\n

    Question 5.
    \nThe number of hours, h, worked and the amount of money, m, earned
    \nAnswer:
    \nThe given situation is:
    \nThe number of hours, h, worked and the amount of money, m, earned
    \nNow,
    \nWe know that,
    \nAn “Independent variable” causes the dependent variable to change
    \nA “Dependent variable” change in response to another variable
    \nSo,
    \nThe independent variable for the given situation is: Number of hours , h
    \nThe dependent variable for the given situation is: Amount of money, m
    \nHence, from the above,
    \nWe can conclude that
    \nThe independent variable for the given situation is: traffic tickets, t
    \nThe dependent variable for the given situation is: Amount of money, m<\/p>\n

    Question 6.
    \nThe number of shelves, s, in a bookcase and the number of books, b, the bookcase can hold
    \nAnswer:
    \nThe given situation is:
    \nThe number of shelves, s, in a bookcase and the number of books, b, the bookcase can hold
    \nNow,
    \nWe know that,
    \nAn “Independent variable” causes the dependent variable to change
    \nA “Dependent variable” change in response to another variable
    \nSo,
    \nThe independent variable for the given situation is: Number of shelves, s
    \nThe dependent variable for the given situation is: Number of books, b, the bookcase can hold
    \nHence, from the above,
    \nWe can conclude that
    \nThe independent variable for the given situation is: Number of shelves, s
    \nThe dependent variable for the given situation is: Number of books, b, the bookcase can hold<\/p>\n

    Question 7.
    \nThe number of pages, p, you read in your book in h hours
    \nAnswer:
    \nThe given situation is:
    \nThe number of pages, p, you read in your book in h hours
    \nNow,
    \nWe know that,
    \nAn “Independent variable” causes the dependent variable to change
    \nA “Dependent variable” change in response to another variable
    \nSo,
    \nThe independent variable for the given situation is: Number of pages, p
    \nThe dependent variable for the given situation is: Number of hours, h
    \nHence, from the above,
    \nWe can conclude that
    \nThe independent variable for the given situation is: Number of pages, p
    \nThe dependent variable for the given situation is: Number of hours, h<\/p>\n

    Question 8.
    \nThe number of gallons, g, of water a garden hose produces after running for m minutes
    \nAnswer:
    \nThe given situation is:
    \nThe number of gallons, g, of water a garden hose produces after running for m minutes
    \nNow,
    \nWe know that,
    \nAn “Independent variable” causes the dependent variable to change
    \nA “Dependent variable” change in response to another variable
    \nSo,
    \nThe independent variable for the given situation is: Number of gallons, g, of water
    \nThe dependent variable for the given situation is: Number of minutes, m
    \nHence, from the above,
    \nWe can conclude that
    \nThe independent variable for the given situation is: Number of gallons, g, of water
    \nThe dependent variable for the given situation is: Number of minutes, m<\/p>\n

    Question 9.
    \nThe number of peaches, y, a farmer harvests in x bushels
    \nAnswer:
    \nThe given situation is:
    \nThe number of peaches, y, a farmer harvests in x bushels
    \nNow,
    \nWe know that,
    \nAn “Independent variable” causes the dependent variable to change
    \nA “Dependent variable” change in response to another variable
    \nSo,
    \nThe independent variable for the given situation is: Number of bushels, x
    \nThe dependent variable for the given situation is: Number of peaches, y
    \nHence, from the above,
    \nWe can conclude that
    \nThe independent variable for the given situation is: Number of bushels, x
    \nThe dependent variable for the given situation is: Number of peaches, y<\/p>\n

    Question 10.
    \nThe number of hours, h, you spend driving at a speed of r miles per hour
    \nAnswer:
    \nThe given situation is:
    \nThe number of hours, h, you spend driving at a speed of r miles per hour
    \nNow,
    \nWe know that,
    \nAn “Independent variable” causes the dependent variable to change
    \nA “Dependent variable” change in response to another variable
    \nSo,
    \nThe independent variable for the given situation is: Number of miles per hour, r
    \nThe dependent variable for the given situation is: Number of hours, h
    \nHence, from the above,
    \nWe can conclude that
    \nThe independent variable for the given situation is: Number of miles per hour, r
    \nThe dependent variable for the given situation is: Number of hours, h<\/p>\n

    Question 11.
    \nName at least two independent variables that could result in a change in a monthly electric bill.
    \nAnswer:
    \nThe two independent variables that could result in a change in a monthly electric bill is:
    \na. Outside temperature and humidity
    \nb. The frequency of the electrical appliances inside the house you use<\/p>\n

    Practice & Problem Solving<\/strong><\/p>\n

    In 12-15, identify the independent variable and the dependent variable.<\/strong>
    \nQuestion 12.
    \nThe pages, p, in a book and the weight, w, of the book
    \nAnswer:
    \nThe given situation is:
    \nThe pages, p, in a book and the weight, w, of the book
    \nNow,
    \nWe know that,
    \nAn “Independent variable” causes the dependent variable to change
    \nA “Dependent variable” change in response to another variable
    \nSo,
    \nThe independent variable for the given situation is: Number of pages, p, in a book
    \nThe dependent variable for the given situation is: Weight, w, of the book
    \nHence, from the above,
    \nWe can conclude that
    \nThe independent variable for the given situation is: Number of pages, p, in a book
    \nThe dependent variable for the given situation is: Weight, w, of the book<\/p>\n

    Question 13.
    \nThe number of hamburgers, h, sold and the dollar amount of sales, s, taken in
    \nAnswer:
    \nThe given situation is:
    \nThe number of hamburgers, h, sold and the dollar amount of sales, s, taken in
    \nNow,
    \nWe know that,
    \nAn “Independent variable” causes the dependent variable to change
    \nA “Dependent variable” change in response to another variable
    \nSo,
    \nThe independent variable for the given situation is: Number of hamburgers, h, sold
    \nThe dependent variable for the given situation is: The amount of sales, s, taken in
    \nHence, from the above,
    \nWe can conclude that
    \nThe independent variable for the given situation is: Number of hamburgers, h, sold
    \nThe dependent variable for the given situation is: The amount of sales, s, taken in<\/p>\n

    Question 14.
    \nThe pounds, p, of flour you buy and the number of bread loaves, b, you want to make
    \nAnswer:
    \nThe given situation is:
    \nThe pounds, p, of flour you buy and the number of bread loaves, b, you want to make
    \nNow,
    \nWe know that,
    \nAn “Independent variable” causes the dependent variable to change
    \nA “Dependent variable” change in response to another variable
    \nSo,
    \nThe independent variable for the given situation is: Number of pounds, p, of flour you buy
    \nThe dependent variable for the given situation is: Number of bread loaves, b, you want to make
    \nHence, from the above,
    \nWe can conclude that
    \nThe independent variable for the given situation is: Number of pounds, p, of flour you buy
    \nThe dependent variable for the given situation is: Number of bread loaves, b, you want to make<\/p>\n

    Question 15.
    \nThe temperature, t, of water and the number of minutes, m, the water is in the freezer
    \nAnswer:
    \nThe given situation is:
    \nThe temperature, t, of water and the number of minutes, m, the water is in the freezer
    \nNow,
    \nWe know that,
    \nAn “Independent variable” causes the dependent variable to change
    \nA “Dependent variable” change in response to another variable
    \nSo,
    \nThe independent variable for the given situation is: The temperature, t, of water
    \nThe dependent variable for the given situation is: Number of minutes, m, the water is in the freezer
    \nHence, from the above,
    \nWe can conclude that
    \nThe independent variable for the given situation is: The temperature, t, of water
    \nThe dependent variable for the given situation is: Number of minutes, m, the water is in the freezer<\/p>\n

    Question 16.
    \nWrite your own situation. Identify the independent and dependent variables.
    \nAnswer:
    \nIn an experiment in Psychology, on the impact of sleep deprivation on test performance,
    \na. Sleep deprivation would be the independent variable.
    \nb. The dependent variable is the variable that is measured by the experimenter<\/p>\n

    Question 17.
    \nName at least two independent variables that could result in a change in the price of a basket of grapefruits.
    \nAnswer:
    \nThe independent variables that could result in a change in the price of a basket of grapefruits is:
    \na. The weight of grapefruits
    \nb. The size of a basket<\/p>\n

    Question 18.
    \nCritique Reasoning<\/strong> You spend c dollars for p identical pairs of pants. A friend claims that because c increases if you increase p, and p increases if you increase c, either corp could be the independent variable. Is your friend right or wrong? Explain.
    \nAnswer:
    \nIt is given that
    \nYou spend c dollars for p identical pairs of pants. A friend claims that because c increases if you increase p, and p increases if you increase c, either corp could be the independent variable.
    \nNow,
    \nWe know that,
    \nAn “Independent variable” causes the dependent variable to change
    \nA “Dependent variable” change in response to another variable
    \nNow,
    \nFrom the given situation,
    \nWe can observe that
    \nThe independent variable for the given situation is: Identical pairs of pants, p
    \nThe dependent variable for the given situation is: Amount of money spent, c
    \nHence, from the above,
    \nWe can conclude that
    \nYour friend is wrong<\/p>\n

    Question 19.
    \nThe number of oranges in a bag and the cost of the bag of oranges are related. What is the independent variable in this relationship? Explain.
    \nAnswer:
    \nIt is given that
    \nThe number of oranges in a bag and the cost of the bag of oranges are related
    \nNow,
    \nWe know that,
    \nAn “Independent variable” causes the dependent variable to change
    \nA “Dependent variable” change in response to another variable
    \nNow,
    \nFrom the given situation,
    \nWe can observe that
    \nThe independent variable for the given situation is: Number of oranges
    \nThe dependent variable for the given situation is: The cost of oranges
    \nHence, from the above,
    \nWe can conclude that
    \nThe independent variable for the given situation is: Number of oranges<\/p>\n

    Question 20.
    \nThe dependent variable g represents the growth of a plant. What variables can represent independent variables in this situation?
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nThe dependent variable g represents the growth of a plant
    \nNow,
    \nWe know that,
    \nAn “Independent variable” causes the dependent variable to change
    \nA “Dependent variable” change in response to another variable
    \nNow,
    \nFrom the given figure,
    \nWe can observe that
    \nThe independent variable for the given situation is: The number of plants
    \nHence, from the above,
    \nWe can conclude that
    \nThe independent variable for the given situation is: The number of plants<\/p>\n

    In 21 and 22, use the table at the right.<\/strong>
    \n\"Envision
    \nQuestion 21.
    \nThe table shows distances driven by the Williams family each day of their vacation. What is an independent variable that would affect the total distance they drove each day?
    \nAnswer:
    \nIt is given that
    \nThe table shows distances driven by the Williams family each day of their vacation
    \nNow,
    \nThe given table is:
    \n\"Envision
    \nNow,
    \nFrom the given table,
    \nWe can observe that
    \nThe independent variable that would affect the total distance is: Number of days
    \nHence, from the above,
    \nWe can conclude that
    \nThe independent variable that would affect the total distance is: Number of days<\/p>\n

    Question 22.
    \nName at least two dependent variables that could affect the amount of money the Williams family spends on meals during their vacation.
    \nAnswer:
    \nThe given table is:
    \n\"Envision
    \nNow,
    \nFrom the given table,
    \nWe can observe that
    \nThe independent variables that could affect the amount of money the Williams family spends on meals during their vacation is:
    \na. The number of items present during meals
    \nb. The number of days
    \nHence, from the above,
    \nWe can conclude that
    \nThe independent variables that could affect the amount of money the Williams family spends on meals during their vacation is:
    \na. The number of items present during meals
    \nb. The number of days<\/p>\n

    Question 23.
    \nThe cost of a salad at a restaurant depends on many factors. List at least two independent variables that could affect the cost of a salad.
    \nAnswer:
    \nIt is given that
    \nThe cost of a salad at a restaurant depends on many factors
    \nSo,
    \nThe independent variables that could affect the cost of a salad for the given situation is:
    \na. The amount of a salad
    \nb. The type of restaurant
    \nHence, from the above,
    \nWe can conclude that
    \nThe independent variables that could affect the cost of a salad for the given situation is:
    \na. The amount of a salad
    \nb. The type of restaurant<\/p>\n

    Question 24.
    \nJulian drove from New York to Florida. List at least two independent variables that could affect the number of days Julian took to make the trip.
    \nAnswer:
    \nIt is given that
    \nJulian drove from New York to Florida
    \nSo,
    \nThe independent variables that could affect the number of days Julian took to make the trip is:
    \na. The distance between New York and Florida
    \nb.\u00a0 The number of hours she drove each day
    \nHence, from the above,
    \nWe can conclude that
    \nThe independent variables that could affect the number of days Julian took to make the trip is:
    \na. The distance between New York and Florida
    \nb.\u00a0 The number of hours she drove each day<\/p>\n

    Question 25.
    \nThe number of incorrect answers and the score on a math test are related. What is the dependent variable in this relationship? Explain.
    \nAnswer:
    \nIt is given that
    \nThe number of incorrect answers and the score on a math test are related
    \nNow,
    \nFrom the given situation,
    \nWe can observe that
    \nThe dependent variable for the given situation is: The score on a math test
    \nHence, from the above,
    \nWe can conclude that
    \nThe dependent variable for the given situation is: The score on a math test<\/p>\n

    Question 26.
    \nHigher Order Thinking<\/strong> Write a situation in which time, t, is an independent variable Then write a situation in which time, t, is a dependent variable.
    \nAnswer:
    \nIn the study of uniform motion, “Time” is an independent variable, denoted by t and represented on the horizontal axis, while position is a dependent variable, denoted by x and represented on the vertical axis.
    \n“Time” is usually viewed as the independent variable for the simple reason that it doesn’t depend on anything else. Time simply ticks by at the same rate wherever you are (in non-relativistic context), independent of other variables so it doesn’t make sense to express time as a dependent variable<\/p>\n

    Assessment Practice<\/strong><\/p>\n

    Question 27.
    \nJonas is concerned about the amount of water he uses to wash his laundry. He made a table to show the number of gallons of water used by different washing machines to complete a load of laundry.
    \n\"Envision<\/p>\n

    PART A
    \nUse variables to represent the independent and dependent quantities shown in the table.
    \nAnswer:
    \nIt is given that
    \nJonas is concerned about the amount of water he uses to wash his laundry. He made a table to show the number of gallons of water used by different washing machines to complete a load of laundry.
    \nNow,
    \nThe given table is:
    \n\"Envision
    \nNow,
    \nFrom the given table,
    \nWe can observe that
    \nThe independent variable for the given situation is: Type of Washing Machine
    \nThe dependent variable for the given situation is: Gallons of Water for a Washing Machine
    \nHence, from the above,
    \nWe can conclude that
    \nThe independent variable for the given situation is: Type of Washing Machine
    \nThe dependent variable for the given situation is: Gallons of Water for a Washing Machine<\/p>\n

    PART B
    \nUse variables to represent the dependent variable and the independent variable in this sentence.
    \nJonas records the total cost of the water he uses and the number of gallons of water he uses.
    \nAnswer:
    \nIt is given that
    \nJonas records the total cost of the water he uses and the number of gallons of water he uses.
    \nNow,
    \nFrom the given sentence,
    \nWe can observe that
    \nThe independent variable for the given sentence is: Gallons of water
    \nThe dependent variable for the given sentence is: The total cost of water
    \nHence, from the above,
    \nWe can conclude that
    \nThe independent variable for the given sentence is: Gallons of water
    \nThe dependent variable for the given sentence is: The total cost of water<\/p>\n

    Lesson 4.9 Use Patterns to Write and Solve Equations<\/h3>\n

    Solve & Discuss It!<\/strong>
    \nThe table below shows how many candles are in different numbers of boxes. Find a pattern that explains the relationship between the values of c and b. Use words and numbers to describe the pattern. How many candles will there be in 10 boxes?<\/p>\n

    I can… use patterns to write and solve equations with variables.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nThe table below shows how many candles are in different numbers of boxes.
    \nNow,
    \nThe given table is:
    \n\"Envision
    \nNow,
    \nFrom the given table,
    \nWe can observe that
    \nThe pattern that explains the relationship between the number of candles and the number of boxes is:
    \nThe number of boxes is one-fourth of the number of candles
    \nNow,
    \nThe mathematical representation for the above pattern is:
    \nThe number of boxes = \\(\\frac{1}{4}\\) \u00d7 (The number of candles)
    \nNow,
    \nFrom the above relationship,
    \nThe number of candles present in 10 boxes is:
    \nThe number of boxes = \\(\\frac{1}{4}\\) \u00d7 (The number of candles)
    \n10 = \\(\\frac{1}{4}\\) \u00d7 (The number of candles)
    \nThe number of candles = 10 \u00d7 4
    \nThe number of candles = 40
    \nHence, from the above,
    \nWe can conclude that
    \nThe pattern that explains the relationship between the number of candles and the number of boxes is:
    \nThe number of boxes is one-fourth of the number of candles
    \nThe number of candles present in 10 boxes is: 40 candles<\/p>\n

    Look for Relationships<\/strong>
    \nHow can you get from each value in the left column to its matching value in the right column?
    \n\"Envision
    \nAnswer:
    \nThe given table is:
    \n\"Envision
    \nNow,
    \nFrom the above Problem,
    \nThe mathematical representation for the pattern between the left column and the right column is:
    \nThe number of boxes = \\(\\frac{1}{4}\\) \u00d7 (The number of candles)
    \nSo,
    \nThe number of candles = 4 \u00d7 (The number of boxes)
    \nHence, from the above,
    \nWe can conclude that
    \nThe relationship between each value in the left column to its matching value in the right column is:
    \nThe number of candles = 4 \u00d7 (The number of boxes)<\/p>\n

    Focus on math practices<\/strong>
    \nUse Structure<\/strong> Write a rule that explains how you get from the values in the right column of the table above to the values in the left column.
    \nAnswer:
    \nThe given table is:
    \n\"Envision
    \nNow,
    \nThe rule that explains the relationship between the values in the right column of the table to the values in the left column is:
    \nThe number of boxes = \\(\\frac{1}{4}\\) \u00d7 (The number of candles)<\/p>\n

    Essential Question<\/strong>
    \nHow can you use a pattern to write and solve an equation?
    \nAnswer:
    \nProblems that are solved most easily by finding a pattern include those that ask students to extend a sequence of numbers or to make a prediction based on data.\u00a0 Finding a Pattern is an appropriate strategy to use to solve the problem. This is a pattern that is predictable and will be continuous<\/p>\n

    Try It!<\/strong><\/p>\n

    The table shows the number of yards, y, that a professional bicyclist rides in s seconds. Find a pattern that relates the variables. If the cyclist maintains this speed, how far would the cyclist ride in 8 seconds?
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nThe table shows the number of yards, y, that a professional bicyclist rides in s seconds.
    \nNow,
    \nThe given table is:
    \n\"Envision
    \nNow,
    \nFrom the given table,
    \nWe can observe that
    \nThe relationship between the number of seconds, s, and the number of yards, y is:
    \n12.2 \u00d7 s = y
    \n12.2s = y
    \nNow,
    \nThe number of yards traveled by the cyclist in 8 seconds is:
    \n12.2 \u00d7 8 = y
    \ny = 97.6 yards
    \nHence, from the above,
    \nWe can conclude that
    \nThe relationship between the number of seconds, s, and the number of yards, y is:
    \n12.2s = y
    \nThe number of yards traveled by the cyclist in 8 seconds is: 97.6 yards<\/p>\n

    Convince Me!<\/strong> How do you know that the equation you wrote describes the pattern in the table?
    \nAnswer:
    \nThe given table is:
    \n\"Envision
    \nNow,
    \nFrom the above Problem,
    \nThe relationship (or) pattern between the number of seconds, s, and the number of yards, y is:
    \n12.2s = y
    \nNow,
    \nSubstitute 2, 3, 5 in the place of s
    \nSo,
    \ny = 12.2 (2) = 24.4 yards
    \ny = 12.2 (3) = 36.6 yards
    \ny = 12.2 (5) = 61 yards
    \nHence, from the above,
    \nWe can conclude that
    \nThe equation that you wrote describes the pattern in the given table because the ratio between the number of seconds and the number of yards is constant<\/p>\n

    Try It!<\/strong><\/p>\n

    If Ethan continues to pay $5 per week, how many more weeks will he need to pay his mother after 12 weeks? Explain.
    \nAnswer:
    \nFrom Example 2,
    \nIt is given that
    \nEthan owes his mother $75 and he repays his mother a set amount of $5 each week
    \nNow,
    \nFrom Example 2,
    \nThe pattern for the given situation is: $75 – $5n
    \nWhere,
    \nn is the number of weeks
    \nNow,
    \nFor 12 weeks,
    \nThe amount Eathan owed to his mother = $75 – $5 (12)
    \n= $75 – $60
    \n= $15
    \nNow,
    \nWe know that,
    \nEathan has to pay a set amount of $5 each week
    \nSo,
    \nThe number of weeks more Eathan has to pay $5 for his mother = \\(\\frac{$15}{$5}\\)
    \n= 3 weeks
    \nHence, from the above,
    \nWe can conclude that
    \nThe number of more weeks will ethan need to pay for his mother after 12 weeks is: 3 weeks<\/p>\n

    KEY CONCEPT<\/strong>
    \nYou can use patterns in a table to write an equation that relates the independent and dependent variables.
    \n\"Envision<\/p>\n

    Do You Understand?<\/strong>
    \nQuestion 1.
    \nEssential Question<\/strong> How can you use a pattern to write and solve an equation?
    \nAnswer:
    \nProblems that are solved most easily by finding a pattern include those that ask students to extend a sequence of numbers or to make a prediction based on data.\u00a0 Finding a Pattern is an appropriate strategy to use to solve the problem. This is a pattern that is predictable and will be continuous<\/p>\n

    Question 2.
    \nMake Sense and Persevere<\/strong> How do you find a pattern that relates the values in a table?
    \nAnswer:
    \nTo determine the rule for a table, simply\u00a0look at one pair of input\/output numbers and figure out what number has been added, subtracted, multiplied, or divided by it. Then, make sure that pattern matches the rest of the numbers. Once you know the rule for a table, you can finish filling in the rest of it<\/p>\n

    Question 3.
    \nReasoning<\/strong> In Example 2, what happens to the value of the dependent variable, a, the amount still owed, when the value of the independent variable, w, the number of weeks Ethan pays $5, is increased by 1?
    \nAnswer:
    \nFrom Example 2,
    \nIt is given that
    \nEthan owes his mother $75 and he repays his mother a set amount of $5 each week
    \nNow,
    \nFrom Example 2,
    \nThe pattern for the given situation is: $75 – $5n
    \nWhere,
    \nn is the number of weeks
    \nNow,
    \nIt is given that
    \nEthan owes his mother $75 and he repays his mothe a set amount of $s each week for 13 weeks
    \nNow,
    \nFor 13 weeks,
    \nThe amount Eathan owed to his mother = $75 – $5 (13)
    \n= $75 – $65
    \n= $10
    \nHence, from the baove,
    \nWe can conclude that
    \nThe amount Etan still owed to his mother after 13 weeks is: $10<\/p>\n

    Question 4.
    \nLook for Relationships<\/strong> Use the pattern in the table below to write an equation.
    \n\"Envision
    \nAnswer:
    \nThe given table is:
    \n\"Envision
    \nNow,
    \nFrom the given table,
    \nWe can observe that
    \nThe pattern between the values of x and y is:
    \ny = 5x + 2
    \nHence, from the above,
    \nWe can conclude that
    \nThe pattern in the given table to represent as an equation is:
    \ny = 5x + 2<\/p>\n

    Do You Know How?<\/strong>
    \nQuestion 5.
    \nThe table shows Brenda’s age, b, when Talia’s age, t, is 7, 9, and 10. Find the pattern and then write a rule and an equation that represents the pattern. Then find Brenda’s age when Talia is 12.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nThe table shows Brenda’s age, b, when Talia’s age, t, is 7, 9, and 10.
    \nNow,
    \nThe given table is:
    \n\"Envision
    \nNow,
    \nFrom the given table,
    \nWe can observe that
    \nThe pattern between Talia’s age and Brenda’s age is:
    \nBrenda’s age = Talia’s age – 5
    \nb = t – 5
    \nSo,
    \nBrenda’s age when Talia’s age is 12 is:
    \nb = 12 – 5
    \nb = 7 years
    \nHence, from the above,
    \nWe can conclude that
    \nThe pattern between Talia’s age and Brenda’s age is:
    \nBrenda’s age = Talia’s age – 5
    \nBrenda’s age when Talia’s age is 12 is: 7 years<\/p>\n

    In 6 and 7, use the table below.<\/strong>
    \n\"Envision
    \nQuestion 6.
    \nUse the equation y = 2x \u2013 7 to complete the table.
    \nAnswer:
    \nThe given table is:
    \n\"Envision
    \nNow,
    \nThe given pattern (or) equation that represents the given table is:
    \ny = 2x – 7
    \nNow,
    \nFor x = 7,
    \ny = 2 (7) – 7
    \n= 14 – 7
    \n= 7
    \nFor x = 8,
    \ny = 2 (8) – 7
    \n= 16 – 7
    \n= 9
    \nHence, from the above,
    \nWe can conclude that
    \nThe completed table is:
    \n\"\"<\/p>\n

    Question 7.
    \nState the rule for the pattern in words.
    \nAnswer:
    \nThe given pattern for the given table is:
    \ny = 2x – 7
    \nNow,
    \nThe given pattern in terms of words is:
    \nThe value of y is equal to the value of 7 less than the 2 times the value of x
    \nHence, from the above,
    \nWe can conclude that
    \nThe given pattern in terms of words is:
    \nThe value of y is equal to the value of 7 less than the 2 times the value of x<\/p>\n

    Practice & Problem Solving<\/strong><\/p>\n

    In 8 and 9, write a rule and an equation that represents the pattern in each table.<\/strong>
    \nQuestion 8.
    \n\"Envision
    \nAnswer:
    \nThe given table is:
    \n\"Envision
    \nNow,
    \nFrom the given table,
    \nWe can observe that
    \nThe representation of the pattern for the given table in the form of an equation is:
    \ny = x + 32
    \nHence, from the above,
    \nWe can conclude that
    \nThe equation that represents the pattern in the given table is:
    \ny = x + 32<\/p>\n

    Question 9.
    \n\"Envision
    \nAnswer:
    \nThe given table is:
    \n\"Envision
    \nNow,
    \nFrom the given table,
    \nWe can observe that
    \nThe representation of the pattern for the given table in the form of an equation is:
    \nn = 3m
    \nHence, from the above,
    \nWe can conclude that
    \nThe equation that represents the pattern in the given table is:
    \nn = 3m<\/p>\n

    In 10 and 11, write a rule and an equation that represents the pattern in each table. Then complete the table.<\/strong>
    \nQuestion 10.
    \n\"Envision
    \nAnswer:
    \nThe given table is:
    \n\"Envision
    \nNow,
    \nFrom the given table,
    \nWe can observe that
    \nThe representation of the pattern for the given table in the form of an equation is:
    \nk = g – 15
    \nSo,
    \nThe completed table by using the above pattern is:
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe equation that represents the pattern in the given table is:
    \nk = g – 15
    \nThe completed table by using the above pattern is:
    \n\"\"<\/p>\n

    Question 11.
    \n\"Envision
    \nAnswer:<\/p>\n

    The given table is:
    \n\"Envision
    \nNow,
    \nFrom the given table,
    \nWe can observe that
    \nThe representation of the pattern for the given table in the form of an equation is:
    \ny = \\(\\frac{x}{9}\\)
    \nSo,
    \nThe completed table by using the above pattern is:
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe equation that represents the pattern in the given table is:
    \ny = \\(\\frac{x}{9}\\)
    \nThe completed table by using the above pattern is:
    \n\"\"<\/p>\n

    Question 12.
    \nTo celebrate its 125th anniversary, a company produced 125 expensive teddy bears. These “125 Karat Teddy Bears” are made of gold thread and have diamonds for eyes. The table shows the approximate cost of different numbers of these bears. Write an equation that can be used to find c, the cost of n bears.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nTo celebrate its 125th anniversary, a company produced 125 expensive teddy bears. These “125 Karat Teddy Bears” are made of gold thread and have diamonds for eyes. The table shows the approximate cost of different numbers of these bears.
    \nNow,
    \nThe given table is:
    \n\"Envision
    \nNow,
    \nFrom the given table,
    \nWe can observe that
    \nThe pattern between the number of bears and the cost of bears is:
    \nThe cost of n Teddy bears = 47,000 \u00d7 (n Teddy Bears)
    \nSo,
    \nc = 47,000n
    \nHence, from the above,
    \nWe can conclude that
    \nThe equation that can be used to find c, the cost of n bears is:
    \nc = 47,000n<\/p>\n

    Question 13.
    \nAndrea attends the county fair. The fair charges for admission and for each ride is shown below.
    \n\"Envision
    \na. Use the pattern in the table to the find the cost for Andrea to ride 5 rides or 8 rides. Then write an equation for the pattern.
    \nAnswer:
    \nIt is given that
    \nAndrea attends the county fair. The fair charges for admission and for each ride is shown
    \nNow,
    \nThe given table is:
    \n\"\"
    \nNow,
    \nFrom the given table,
    \nWe can observe that
    \nThe pattern between the number of rides and the cost of rides is:
    \nThe cost of r rides = 2.5 \u00d7 (Tne number of rides) + 8
    \nSo,
    \nc = 2.5r + 8
    \nNow,
    \nThe completed table by using the above pattern is:
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe pattern to find the cost for Andrea to ride 5 rides or 8rides is:
    \nc = 2.5r + 8
    \nThe completed table by using the above pattern is:
    \n\"\"<\/p>\n

    b. Find the cost, c, for 12 rides.
    \nAnswer:
    \nFrom Part (a),
    \nThe pattern to find the cost of r rides is:
    \nc = 2.5r + 8
    \nSo,
    \nThe cost for 12 rides is:
    \nc = 2.5 (12) + 8
    \nc = $38
    \nHence, from the above,
    \nWe can conclude that
    \nThe cost for Andrea to ride 12 rides is: $38<\/p>\n

    In 14 and 15, write an equation that best describes the pattern in each table.<\/strong>
    \nQuestion 14.
    \n\"Envision
    \nAnswer:
    \nThe given table is:
    \n\"Envision
    \nNow,
    \nFrom the given table,
    \nWe can observe that
    \nThe pattern between w and z is:
    \nz = w – 2
    \nHence, from the above,
    \nWe can conclude that
    \nThe equation that represents the pattern in the given table is:
    \nz = w – 2<\/p>\n

    Question 15.
    \n\"Envision
    \nAnswer:
    \nThe given table is:
    \n\"Envision
    \nNow,
    \nFrom the given table,
    \nWe can observe that
    \nThe pattern between x and y is:
    \ny = 4x
    \nHence, from the above,
    \nWe can conclude that
    \nThe equation that represents the pattern in the given table is:
    \ny = 4x<\/p>\n

    In 16-19, use the equation to complete each table.<\/strong>
    \nQuestion 16.
    \nt = 5d + 5
    \n\"Envision
    \nAnswer:
    \nThe given equation is:
    \nt = 5d + 5
    \nNow,
    \nThe given table that is related to the given equation is:
    \n\"Envision
    \nNow,
    \nFor d = 3,
    \nt = 5 (3) + 5
    \n= 20
    \nFor d = 4,
    \nt = 5 (4) + 5
    \n= 25
    \nSo,
    \nThe completed table is:
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe completed by using the given equation is:
    \n\"\"<\/p>\n

    Question 17.
    \ny = \\(\\frac{1}{2}\\)x – 1
    \n\"Envision
    \nAnswer:
    \nThe given equation is:
    \ny = \\(\\frac{1}{2}\\)x – 1
    \nNow,
    \nThe given table that is related to the given equation is:
    \n\"Envision
    \nNow,
    \nFor x = 8,
    \ny = \\(\\frac{1}{2}\\) (8) – 1
    \n= 3
    \nFor x = 10,
    \ny = \\(\\frac{1}{2}\\) (10) – 1
    \n= 4
    \nSo,
    \nThe completed table is:
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe completed by using the given equation is:
    \n\"\"<\/p>\n

    Question 18.
    \ny = 2x + 1
    \n\"Envision
    \nAnswer:
    \nThe given equation is:
    \ny = 2x + 1
    \nNow,
    \nThe given table that is related to the given equation is:
    \n\"Envision
    \nNow,
    \nFor x = 2,
    \ny = 2 (2) + 1
    \n= 5
    \nFor x = 3,
    \ny = 2 (3) + 1
    \n= 7
    \nSo,
    \nThe completed table is:
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe completed by using the given equation is:
    \n\"\"<\/p>\n

    Question 19.
    \nb = \\(\\frac{a}{2}\\) – 2
    \n\"Envision
    \nAnswer:
    \nThe given equation is:
    \nb = \\(\\frac{a}{2}\\) – 2
    \nNow,
    \nThe given table that is related to the given equation is:
    \n\"Envision
    \nNow,
    \nFor a = 17,
    \nb = \\(\\frac{17}{2}\\) – 2
    \n= 6.5
    \nFor a = 14,
    \nb = \\(\\frac{14}{2}\\) – 2
    \n= 5
    \nFor a = 11,
    \nb = \\(\\frac{11}{2}\\) – 2
    \n= 3.5
    \nFor a = 8,
    \nb = \\(\\frac{8}{2}\\) – 2
    \n= 2
    \nFor a = 5,
    \nb = \\(\\frac{5}{2}\\) – 2
    \n= 0.5
    \nSo,
    \nThe completed table is:
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe completed by using the given equation is:
    \n\"\"<\/p>\n

    Question 20.
    \nHigher Order Thinking<\/strong> Maya wrote the equation h= d+ 22 to represent the relationship shown in the table. Is this equation correct? Explain.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nMaya wrote the equation h= d+ 22 to represent the relationship shown in the table.
    \nNow,
    \nThe given table is:
    \n\"Envision
    \nNow,
    \nFrom the given table,
    \nWe can observe that
    \nThe relationship between the values of h and d is:
    \nd = 11h
    \nSo,
    \nh = d + 22 and d = 11h are different relationships
    \nHence, from the above,
    \nWe can conclude that
    \nh = d + 22 is not correct<\/p>\n

    Assessment Practice<\/strong><\/p>\n

    Question 21.
    \nThe table below shows the total cost for the number of movie tickets purchased. Write an equation that represents the relationship between these two quantities. Use the equation to find the cost of 6 tickets.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nThe table below shows the total cost for the number of movie tickets purchased
    \nNow,
    \nThe given table is:
    \n\"Envision
    \nNow,
    \nFrom the given table,
    \nWe can purchase that
    \nThe relationship between the number of tickets and their cost is:
    \nCost of n tickets (c) = 8.75 \u00d7 (Number of tickets)
    \nSo,
    \nc = 8.75n
    \nWhere,
    \nc is teh cost of n tickets
    \nn is the number of tickets
    \nNow,
    \nThe cost of 6 tickets by using the above pattern is:
    \nc = 8.75 (6)
    \nc = $52.5
    \nHence, from the above,
    \nWe can conclude that
    \nThe equation that represents the relationship between the number of tickets and their cost is:
    \nc = 8.75n
    \nThe cost of 6 tickets by using the above equation is: $52.5<\/p>\n

    Lesson 4.10 Relate Tables, Graphs, and Equations<\/h3>\n

    Solve & Discuss It!<\/strong>
    \nNancy walks 4 blocks to Maria’s house. Together, they continue the walk. The walk can be described as n = m + 4, where n is the number of blocks Nancy walks and m is the number of blocks Maria walks. Describe Nancy’s house how the equation, data table, and graph reflect the walk.
    \nI can… analyze the relationship between dependent and independent variables in tables, graphs, and equations.
    \n\"Envision\"Envision
    \nAnswer:
    \nIt is given that
    \nNancy walks 4 blocks to Maria’s house. Together, they continue the walk. The walk can be described as n = m + 4, where n is the number of blocks Nancy walks and m is the number of blocks Maria walks.
    \nNow,
    \nThe given equation that represents the given situation is:
    \nn = m + 4
    \nWhere,
    \nn is the number of blocks Nancy walks
    \nm is the number of blocks Maria walks
    \nNow,
    \nFrom the given equation,
    \nWe can observe that
    \nn is the output
    \nm is the input
    \nNow,
    \nThe given data table and the graph for the given situation is:
    \n\"Envision
    \nNow,
    \nFrom the given data table,
    \nWe can observe that
    \nm is the input
    \nn is the output
    \nWe can take any value of input and calculate the value of output
    \nNow,
    \nFrom the given graph,
    \nWe can observe that
    \nThe graph can be drawn by taking the ordered pairs (m, n) and if we observe the graph, we can say that it is a straight line without passingthrough the origin<\/p>\n

    Look for Relationships<\/strong> How can you use the values in one row of the data table to describe the relationship shown in the equation, data table, and graph?
    \n\"Envision
    \nAnswer:
    \nFrom the above,
    \nWe know that,
    \nThe given equation that represents the given situation is:
    \nn = m + 4
    \nWhere,
    \nn is the number of blocks Nancy walks
    \nm is the number of blocks Maria walks
    \nNow,
    \nFrom the given equation,
    \nWe can observe that
    \nn is the output
    \nm is the input
    \nNow,
    \nThe given data table and the graph for the given situation is:
    \n\"Envision
    \nNow,
    \nFrom the given data table,
    \nWe can observe that
    \nm is the input
    \nn is the output
    \nWe can take any value of input and calculate the value of output
    \nNow,
    \nFrom the given graph,
    \nWe can observe that
    \nThe graph can be drawn by taking the ordered pairs (m, n) and if we observe the graph, we can say that it is a straight line without passingthrough the origin<\/p>\n

    Focus on math practices<\/strong>
    \nModel with Math<\/strong> Draw a line through the points on the graph. What ordered pair on the line includes m = 5? Explain what that ordered pair represents.
    \nAnswer:
    \nThe given data table is:
    \n\"\"
    \nNow,
    \nWe know that,
    \nThe representation of the ordered pair is: (Input, Output)
    \nNow,
    \nFrom the given data table,
    \nWe can observe that
    \nThe ordered pairs are: (1, ), (2, 6), (3, 7)
    \nSo,
    \nThe representation of the ordered pairs in the coordinate plane is:
    \n\"\"
    \nNow,
    \nFrom the above,
    \nWe can observe that
    \nThe equation that represents the above situation is:
    \nn = m + 4
    \nNow,
    \nFor m = 5,
    \nn = 5 + 4
    \n= 9
    \nSo,
    \nThe ordered pair when m = 5 is: (5, 9)
    \nThe above ordered pair represents that
    \nWhen Maria walks 5 blocks, Nncy walks 9 blocks
    \nHence, from the above,
    \nWe can conclude that
    \nThe ordered pair when m = 5 is: (5, 9)
    \nThe above ordered pair represents that
    \nWhen Maria walks 5 blocks, Nancy walks 9 blocks<\/p>\n

    Essential Question<\/strong>
    \nHow can you analyze the relationship between dependent and independent variables using tables, graphs, and equations?
    \nAnswer:
    \nIn a graph, the X-axis runs horizontally (side to side) and the Y-axis runs vertically (up and down). Typically, the\u00a0independent variable will be shown on the X axis and the dependent variable will be shown on the Y axis
    \nIn a table, the independent variable will be shown in the left side or the top of the column ans the dependent variable will be shown in the right side or the bottom of the column
    \nIn an equation, the independent variable will be shown at the right side of “=” and the dependent variable will be shown at the left side of “=”<\/p>\n

    Try It!<\/strong><\/p>\n

    The booster club now raises $0.41 for each pom pom they sell. Complete the table and graph. Write and solve an equation to find how many pom poms they need to sell to raise $50.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nThe booster club now raises $0.41 for each pom pom they sell.
    \nNow,
    \nFrom Example 1,
    \nWe can observe that
    \nThe representation of the equation that represents the amount of money raised to the number of pom poms sold is:
    \nr = 0.45n
    \nWhere,
    \nr is the amount of money raised
    \nn is the number of pom poms
    \nNow,
    \nAccording to the given situation,
    \nThe representation of the equation that represents the amount of money raised to the number of pom poms sold is:
    \nr = 0.41n
    \nNow,
    \nFor n = 150,
    \nr = 0.41 (150)
    \n= 61.5
    \nSo,
    \nThe completed table is:
    \n\"\"
    \nThe representation of the graph in the coordinate plane according to the above table is:
    \n\"\"
    \nNow,
    \nFor r = $50,
    \nr = 0.41n
    \n$50 = 0.41n
    \nn = \\(\\frac{50}{0.41}\\)
    \nn = 121.95
    \nn \u2245 122
    \nHence, from the above,
    \nWe can conclude that
    \nThe number of pom poms they needed to sell to raise $50 must be atleast 122 pom poms<\/p>\n

    Convince Me!<\/strong> How does finding three values for x and y help you represent the relationship between x and y?
    \nAnswer:
    \nWe know that,
    \nIn a graph,
    \nThe x-values represent on the horizontal axis and the y-values represent on the vertical axis
    \nNow,
    \nThe representation of x and y values in a graph gives us the information that whether the equation is linear or not<\/p>\n

    Try It!<\/strong><\/p>\n

    A company makes decorations for pens. All the supplies cost $5, and the company plans to sell the decorations for $2 apiece. Analyze the relationship between the number of decorations sold and the profit by completing the table and the graph. Use the table and the graph to write and solve an equation to find the number of decorations that must be sold for the company to make a $15 profit.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nA company makes decorations for pens. All the supplies cost $5, and the company plans to sell the decorations for $2 apiece
    \nNow,
    \nLet the number of decorations be x
    \nLet the profit be y
    \nSo,
    \nThe representation of the given situation in the form of an equation is:
    \nProfit = $2 \u00d7 (Number of decorations)
    \ny = 2x
    \nNow,
    \nFor x = 3,
    \ny = 2 (3)
    \n= 6
    \nFor x = 4,
    \ny = 2 (4)
    \n= 8
    \nFor x = 5,
    \ny = 2 (5)
    \n= 10
    \nSo,
    \nThe completed table for y = 2x is:
    \n\"\"
    \nSo,
    \nThe representation of the above table in the cordinate plane is:
    \n\"\"
    \nNow,
    \nThe number of decorations that must be sold for the company to make a $15 profit is:
    \ny = 2x
    \n$15 = 2 (x)
    \nx = \\(\\frac{15}{2}\\)
    \nx = 7.5
    \nx \u2245 8
    \nHence, from the above,
    \nWe can conclude that
    \nThe number of decorations that must be sold for the company to make a profit of $15 should be about 8 decorations<\/p>\n

    KEY CONCEPT<\/strong>
    \nYou can analyze the relationship between independent and dependent variables in tables and graphs. You can relate tables and graphs to an equation.
    \n\"Envision<\/p>\n

    Do You Understand?<\/strong>
    \nQuestion 1.
    \nEssential Question<\/strong> How can you analyze the relationship between dependent and independent variables using tables, graphs, and equations?
    \nAnswer:
    \nIn a graph, the X-axis runs horizontally (side to side) and the Y-axis runs vertically (up and down). Typically, the\u00a0independent variable will be shown on the X axis and the dependent variable will be shown on the Y axis
    \nIn a table, the independent variable will be shown in the left side or the top of the column ans the dependent variable will be shown in the right side or the bottom of the column
    \nIn an equation, the independent variable will be shown at the right side of “=” and the dependent variable will be shown at the left side of “=”<\/p>\n

    Question 2.
    \nReasoning<\/strong> Using the relationship in Example 1, how many pom poms must the booster club sell to raise $75 for charity? Explain.
    \nAnswer:
    \nFrom Example 1,
    \nWe know that,
    \nThe relationship between the amount of money raised and the number of pom poms sold is:
    \nr = 0.45n
    \nWhere,
    \nr is the amount of money raised
    \nn is the number of pom poms sold
    \nNow,
    \nFor r = $75,
    \n75 = 0.45n
    \nn = \\(\\frac{75}{0.45}\\)
    \nn = 166.66
    \nn \u2245 167
    \nHence, from the above,
    \nWe can conclude that
    \nThe number of pom poms must booster club sell to raise $75 for charity should be at least 167 pom poms<\/p>\n

    Question 3.
    \nConstruct Arguments<\/strong> For every 4 bananas a grocery store sells, it sells 2 apples. Mary wrote the equation 4b \u00d7 2 = a, where b= the number of bananas sold and a = the number of apples sold. Does Mary’s equation correctly represent the relationship of bananas sold to apples sold? Explain.
    \nAnswer:
    \nIt is given that
    \nFor every 4 bananas a grocery store sells, it sells 2 apples. Mary wrote the equation 4b \u00d7 2 = a, where b= the number of bananas sold and a = the number of apples sold
    \nNow,
    \nAccording to the given information,
    \nThe representation of the given information in the form of an equation is:
    \n4 \u00d7 (The number of bananas) = 2 \u00d7 (The number of apples)
    \nSo,
    \n4b = 2a
    \n4b \u00d7 2 = a and 4b = 2a are not the same
    \nHence, from the above,
    \nWe can conclude that
    \nMary’s equation does not correctly represent the relationship of bananas sold to apples sold<\/p>\n

    Do You Know How?<\/strong>
    \nIn 4-6, use the equation d = 4t.<\/strong>
    \n\"Envision
    \nQuestion 4.
    \nComplete the table.
    \nd = distance t = time
    \nAnswer:
    \nThe given equation is: d = 4t
    \nNow,
    \nFor t = 1,
    \nd = 4 (1) = 4
    \nFor t = 2,
    \nd = 4 (2) = 8
    \nFor t = 3,
    \nd = 4 (3) = 12
    \nSo,
    \nThe completed table for the given equation is:
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe completed table for d = 4t is:
    \n\"\"<\/p>\n

    Question 5.
    \nName four ordered pairs found on the line plotted using this equation.
    \nAnswer:
    \nThe given table is:
    \n\"\"
    \nNow,
    \nThe representation of the given table in the coordinate plane is:
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe ordered pairs found on the line plotted using the equation d = 4t are:
    \n(1, 4), (2, 8), (3, 12)<\/p>\n

    Question 6.
    \nDescribe the relationship between the variables.
    \nAnswer:
    \nThe given equation is: d = 4t
    \nNow,
    \nFrom the given equation,
    \nWe can observe that
    \nt is the independent variable (or) input
    \nd is the dependent variable (or) output
    \nNow,
    \nFrom Question 5,
    \nWe can observe that
    \nAs the value of t increases, the value of d also increases
    \nHence, from the above,
    \nWe can conclude that
    \nThe relationship between the variables d and t in the equation d = 4t is:
    \nAs the value of t increases, the value of d also increases<\/p>\n

    In 7, complete the table and graph to show the relationship between the variables in the equation d = 5 + 5t.
    \nQuestion 7.
    \nd= distance
    \nt = time
    \n\"Envision
    \nAnswer:
    \nThe given equation is: d = 5 + 5t
    \nNow,
    \nFor t = 0,
    \nd = 5 + 0 = 5
    \nFor t = 2,
    \nd = 5 + 5 (2)
    \n= 5 + 10
    \n= 15
    \nFor t = 4,
    \nd = 5 + 5 (4)
    \n= 5 + 20
    \n= 25
    \nSo,
    \nThe completed table for the equation d = 5 + 5t is:
    \n\"\"
    \nSo,
    \nThe representation of the above table in the coordinate plane is:
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe completed table and the graph to show the relationship between the variables in the equation d = 5 +5t are:
    \n\"\"\"\"<\/p>\n

    Practice & Problem Solving<\/strong><\/p>\n

    In 8 and 9, complete the table and graph to show the relationship between the variables in each equation.<\/strong>
    \nQuestion 8.
    \nA rectangle is \\(\\frac{1}{2}\\) inch longer than it is wide.
    \nLet w = width.
    \nLet l = length
    \nGraph l = w + \\(\\frac{1}{2}\\)
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nA rectangle is \\(\\frac{1}{2}\\) inch longer than it is wide.
    \nNow,
    \nThe given equation is:
    \nl = w + \\(\\frac{1}{2}\\)
    \nNow,
    \nFor w = 1,
    \nl = 1 + \\(\\frac{1}{2}\\)
    \n= 1 + 0.5
    \n= 1.5
    \nFor w = 2,
    \nl = 2 + \\(\\frac{1}{2}\\)
    \n= 2 + 0.5
    \n= 2.5
    \nFor w = 3,
    \nl = 3 + \\(\\frac{1}{2}\\)
    \n= 3 + 0.5
    \n= 3.5
    \nSo,
    \nThe completed table for the equation l = w + \\(\\frac{1}{2}\\) is:
    \n\"\"
    \nSo,
    \nThe representation of the above table in the coordinate plane is:
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe completed table and the graph to show the relationship between the variables in the equation l = w + \\(\\frac{1}{2}\\) are:
    \n\"\"\"\"<\/p>\n

    Question 9.
    \nThe sale price is $5 less than the regular price.
    \nLet s = the sale price.
    \nLet r = the regular price.
    \nGraph s = r – 5.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nThe sale price is $5 less than the regular price.
    \nNow,
    \nThe given equation is:
    \ns = r – 5
    \nNow,
    \nFor r = 10,
    \ns = 10 – 5 = 5
    \nFor r = 20,
    \ns = 20 – 5 = 15
    \nFor r = 30,
    \ns = 30 – 5 = 25
    \nSo,
    \nThe completed table for the equation s = r – 5 is:
    \n\"\"
    \nSo,
    \nThe representation of the above table in the coordinate plane is:
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe completed table and the graph to show the relationship between the variables in the equation l = w + \\(\\frac{1}{2}\\) are:
    \n\"\"\"\"<\/p>\n

    Question 10.
    \nThe points (2, 4) and (-2, -4) are plotted on the coordinate plane using the equation y = a \u2022 x. How can you use the coordinates to find the value of a?
    \nAnswer:
    \nIt is given that
    \nThe points (2, 4) and (-2, -4) are plotted on the coordinate plane using the equation y = a \u2022 x
    \nNow,
    \nCompare the given points with (x, y)
    \nNow,
    \nThe given equation is:
    \ny = a . x
    \nNow,
    \nFor (2, 4),
    \n4 = a . 2
    \na = \\(\\frac{4}{2}\\)
    \na = 2
    \nNow,
    \nFor (-2, -4),
    \n-4 = a . (-2)
    \n4 = a . 2
    \na = \\(\\frac{4}{2}\\)
    \na = 2
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of a for the equation y = a . x is: 2<\/p>\n

    Question 11.
    \nWithout using a table or graph, identify three other points that a graph of the equation in Exercise 10 will pass through.
    \nAnswer:
    \nFrom Question 10,
    \nWe know that,
    \nThe equation is: y = a . x
    \nNow,
    \nFrom Question 10,
    \nWe can observe that
    \nThe points that pss through y = a . x are: (2, 4), and (-2, -4)
    \nNow,
    \nFor x = 3,
    \ny = 2 . 3 = 6
    \nFor x = -3,
    \ny = 2 . -3 = -6
    \nFor x = 4,
    \ny = 2 . 4 = 8
    \nHence, from the above,
    \nWe can conclude that
    \nThe other three points that a graph of the equation y = a . x will pass through are: (3, 6), (-3, -6), and (4, 8)<\/p>\n

    Question 12.
    \nReasoning<\/strong> The Jackson family is planning a weekend vacation. They plan to rent a car from the ABC Car Rental Company. Let m represent the number of miles the family will drive. Let c represent the cost for renting a car. Write an equation that shows what the cost for renting a car will be.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nThe Jackson family is planning a weekend vacation. They plan to rent a car from the ABC Car Rental Company. Let m represent the number of miles the family will drive. Let c represent the cost for renting a car.
    \nNow,
    \nThe given figure is:
    \n\"Envision
    \nNow,
    \nAccording the given information,
    \nThe equation that represents the given situation is:
    \nc = ($40 + $0.10)m
    \nSo,
    \nc = $40.1m
    \nHence, from the above,
    \nWe can conclude that
    \nThe equation tht shows the cost for renting a car is:
    \nc = $40.1m<\/p>\n

    In 13, write an equation. Complete the table and graph to solve the problem.<\/strong>
    \n\"Envision
    \nQuestion 13.
    \nA puppy weighs 1 pound. What does the puppy weigh after 4 weeks?
    \nAnswer:
    \nIt is given that
    \nA puppy weighs 1 pound and the puppy gains \\(\\frac{1}{2}\\) each week
    \nNow,
    \nLet w be the number of weeks
    \nLet p be the weight of the puppy
    \nNow,
    \nThe completed table for the given information is:
    \n\"\"
    \nNow,
    \nThe representation of the above table in the coordinate plane is:
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe completed table and the completed graph for the given problem is:
    \n\"\"\"\"<\/p>\n

    Question 14.<\/p>\n

    Model with Math<\/strong> During a movie matinee, the film projector broke. The theater manager refunded the ticket price to everyone attending. Let n represent the number of people watching the movie. Let r represent the total amount of money refunded. Write an equation to represent the amount of money refunded.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nDuring a movie matinee, the film projector broke. The theater manager refunded the ticket price to everyone attending. Let n represent the number of people watching the movie. Let r represent the total amount of money refunded.
    \nNow,
    \nThe given figure is:
    \n\"Envision
    \nNow,
    \nThe amount of money collecting by selling the movie tickets =\u00a0 $5.00
    \nNow,
    \nThe representation of the quation for the given information is:
    \nThe total amount of money refunded = $5 (The number of people watching the movie)
    \nSo,
    \nr = 5n
    \nHence, from the above,
    \nWe can conclude that
    \nThe equation to represent the amount of money refunded is:
    \nr = 5n<\/p>\n

    Question 15.
    \nHigher Order Thinking<\/strong> Write an algebraic equation that matches the values shown in the table at the right. Explain how you solved the problem.
    \n\"Envision
    \nAnswer:
    \nThe given table is:
    \n\"Envision
    \nNow,
    \nFrom the given table,
    \nWe can observe that
    \nThe pattern that describes the relationship between x and y is:
    \ny = 3x + 5
    \nHence, from the above,
    \nWe can conclude that
    \nThe algebraic equation that matches the values shown in the table is:
    \ny = 3x + 5<\/p>\n

    Assessment Practice<\/strong><\/p>\n

    Question 16.
    \nFor every hour Sonia worked, she had made 2 seashell necklaces for her gift shop.
    \n\"Envision<\/p>\n

    PART A
    \nWrite an equation that describes the relationship shown in the graph on the right.
    \nAnswer:
    \nIt is given that
    \nFor every hour Sonia worked, she had made 2 seashell necklaces for her gift shop.
    \nNow,
    \nThe given graph that describes the given situation is:
    \n\"Envision
    \nNow,
    \nFrom the given graph,
    \nThe representation of the quation that represents the given situation is:
    \nThe number of seashell necklaces = 2 \u00d7 (Time in hours)
    \nSo,
    \nn = 2h
    \nHence, from the above,
    \nWe can conclude that
    \nThe equation that describes the relationship shown in the graph is:
    \nn = 2h<\/p>\n

    PART B
    \nDescribe the relationship between the variables in the graph and the equation.
    \nAnswer:
    \nFrom Part A,
    \nThe equation that describes the relationship shown in the graph is:
    \nn = 2h
    \nWhere,
    \nn is the number of seashell necklaces
    \nh is the number of hours
    \nNow,
    \nIn the graph,
    \nThe x-axis represents the number of hours, h
    \nThe y-axis represents the number of seashell necklaces, n<\/p>\n

    Topic 4 Review<\/h3>\n

    Topic Essential Question<\/strong>
    \nWhat procedures can be used to write and solve equations and inequalities?
    \nAnswer:
    \nThe three methods most commonly used to solve systems of equation are substitution, and elimination. Substitution and elimination are simple methods that can effectively solve most systems of two equations in a few straightforward steps
    \nThe steps to solve an inequality are:
    \na. Add the same number to both sides.
    \nb. Subtract the same number from both sides.
    \nc. Multiply both sides by the same positive number.
    \nd. Divide both sides by the same positive number.
    \ne. Multiply both sides by the same negative number and reverse the sign<\/p>\n

    Vocabulary<\/strong><\/p>\n

    Review Complete each definition with a vocabulary word.<\/strong>
    \nVocabulary<\/p>\n

      \n
    • dependent variable<\/li>\n
    • independent variable<\/li>\n
    • inequality<\/li>\n
    • equation<\/li>\n<\/ul>\n

      Question 1.
      \nIn the equation y = x + 9, the variable x is the ___________
      \nAnswer:
      \nWe know that,
      \nIn the equation y = x + 9,
      \nx is called the “Independent variable”
      \nHence, from the above,
      \nWe can conclude that
      \nThe best term that is suitable for the given sentence is: Independent variable<\/p>\n

      Question 2.
      \nA(n). __________ has an infinite number of solutions.
      \nAnswer:
      \nWe know that,
      \nAn “Inequality” has infinite number of solutions
      \nHence, from the above,
      \nWe can conclude that
      \nThe best term that is suitable for the given sentence is: Inequality<\/p>\n

      Question 3.
      \nIn the equation y = x -9, the variable y is the _________
      \nAnswer:
      \nWe know that,
      \nIn the equation y = x – 9,
      \ny is called the “Dependent variable”
      \nHence, from the above,
      \nWe can conclude that
      \nThe best term that is suitable for the given sentence is: Dependent variable<\/p>\n

      Draw a line from each equation to the property of equality it illustrates.<\/strong>
      \n\"Envision
      \nAnswer:
      \nThe equations that matched to the corresponding property of equality it illustrates is:
      \n\"\"<\/p>\n

      Use Vocabulary in Writing<\/strong>
      \nDescribe how to solve \\(\\frac{3}{7}\\)n = 27. Use vocabulary words in your explanation.
      \nAnswer:
      \nThe given equation is:
      \n\\(\\frac{3}{7}\\)n = 27
      \nNow,
      \nFrom the given equation,
      \nWe can observe that
      \nn is the variable
      \nNow,
      \nMultiply the given equation with \\(\\frac{7}{3}\\) i.e., the reciprocal of \\(\\frac{3}{7}\\) on both sides
      \nSo,
      \n\\(\\frac{3}{7}\\) \u00d7 \\(\\frac{7}{3}\\) \u00d7 n = 27 \u00d7 \\(\\frac{7}{3}\\)
      \nn = \\(\\frac{27 \u00d7 7}{3}\\)
      \nn = 9 \u00d7 7
      \nn = 63
      \nHence, from the above,
      \nWe can conclude that
      \nThe value of n for the given equation is: 63<\/p>\n

      Concepts and Skills Review<\/strong><\/p>\n

      Lesson 4.1 Understand Equations and Solutions<\/strong><\/p>\n

      Quick Review<\/strong>
      \nThe solution of an equation makes the equation true. Substitute each of the given values into the equation for the variable to determine which value, if any, is a solution of the equation.<\/p>\n

      Example<\/strong>
      \nWhich value of x is a solution of the equation?
      \n\"Envision<\/p>\n

      Practice<\/strong>
      \nTell which value of the variable, if any, is a solution of the equation.<\/strong>
      \nQuestion 1.
      \nd + 9 = 25
      \nd = 6, 14, 16, 21
      \nAnswer:
      \nThe given equation is:
      \nd + 9 = 25
      \nNow,
      \nFor d = 6,
      \n6 + 9 = 25
      \n15 \u2260 25
      \nFor d = 14,
      \n14 + 9 = 25
      \n23 \u2260 25
      \nFor d = 16,
      \n16 + 9 = 25
      \n25 = 25
      \nFor d = 21,
      \n21 + 9 = 25
      \n30 \u2260 25
      \nHence, from the above,
      \nWe can conclude that
      \nd = 16 is a solution of the givene quation<\/p>\n

      Question 2.
      \nC – 8 = 25
      \nC = 17, 28, 33, 35
      \nAnswer:
      \nThe givene quation is:
      \nc – 8 = 25
      \nNow,
      \nFor c = 17,
      \n17 – 8 = 25
      \n9 \u2260 25
      \nFor c = 28,
      \n28 – 8 = 25
      \n20 \u2260 25
      \nFor c = 33,
      \n33 – 8 = 25
      \n25 = 25
      \nFor c = 35,
      \n35 – 8 = 25
      \n27 \u2260 25
      \nHence, from the above,
      \nWe can conclude that
      \nc = 33 is a solution of the given equation<\/p>\n

      Question 3.
      \n2y = 30
      \ny = 10, 12, 24, 36
      \nAnswer:
      \nThe given equation is:
      \n2y = 30
      \nNow,
      \nNow,
      \nFor y = 10,
      \n2 (10) = 30
      \n20 \u2260 30
      \nFor y = 12,
      \n2 (12) = 30
      \n24 \u226030
      \nFor y = 24,
      \n2 (24) = 30
      \n48 \u2260 30
      \nFor y = 36,
      \n2 (36) = 30
      \n72 \u2260 30
      \nHence, from the above,
      \nWe can conclude that
      \nThe given equation does not have any solution from the given values of y<\/p>\n

      Question 4.
      \n150 \u00f7 h = 50
      \nh = 2, 3, 4, 5
      \nAnswer:
      \nThe given equation is:
      \n150 \u00f7 h = 50
      \nNow,
      \nFor h = 2,
      \n150 \u00f7 2 = 50
      \n75 \u2260 50
      \nFor h = 3,
      \n150 \u00f7 3 = 50
      \n50 = 50
      \nFor h = 4,
      \n150 \u00f7 4 = 50
      \n37.5 \u2260 50
      \nFor h = 5,
      \n150 \u00f7 5 = 50
      \n30 \u2260 50
      \nHence, from the above,
      \nWe can conclude that
      \nh = 3 is a solution of the given equation<\/p>\n

      Question 5.
      \nf – 13.2 = 28.9
      \nf= 38.7, 42.2, 45.8, 51.4
      \nAnswer:
      \nThe given equation is:
      \nf – 13.2 = 28.9
      \nNow,
      \nFor f = 38.7,
      \n38.7 – 13.2 = 28.9
      \n25.5 \u2260 28.9
      \nFor f = 42.2,
      \n42.2 – 13.2 = 28.9
      \n29 \u2260 28.9
      \nFor f = 45.8,
      \n45.8 – 13.2 = 28.9
      \n32.6 \u2260 28.9
      \nFor f = 51.4,
      \n51.4 – 13.2 = 28.9
      \n38.2 \u2260 28.9
      \nHence, from the above,
      \nWe can conclude that
      \nThere are solutions for the given equation from the given values of f<\/p>\n

      Lesson 4.2 Apply Properties of Equality<\/strong><\/p>\n

      Quick Review<\/strong>
      \nThe properties of equality allow you to apply the same operation with the same amount to both sides of an equation.<\/p>\n

      Example<\/strong>
      \nThe properties of equality are illustrated in the table.
      \n\"Envision<\/p>\n

      Practice<\/strong>
      \nQuestion 1.
      \nIf 6+2 = 8, does 6 + 2 + 3 = 8+ 3? Why or why not?
      \nAnswer:
      \nThe given equations are:
      \n6 + 2 = 8
      \n6 + 2 + 3 = 8 + 3
      \nNow,
      \nAccording to “Addition Property of Equality”,
      \nAdd the same amount of number on both sides to make the equation balanced
      \nSo,
      \nFrom the given equation,
      \nWe can observe that 3 is added on both sides
      \nHence, from the above,
      \nWe can conclude that
      \nThe given equations are the same by using the Addition Property of Equality<\/p>\n

      Question 2.
      \nIf 8 – 1 = 7, does 8 – 1 – 2 = 7 – 3? Why or why not?
      \nAnswer:
      \nThe given equations are:
      \n8 – 1 = 7
      \n8- 1 – 2 = 7 – 3
      \nNow,
      \nAccording to “Subtraction Property of Equality”,
      \nSubtract the same amount of number from both sides to make the equation balanced
      \nSo,
      \nFrom the given equation,
      \nWe can observe that
      \n2 is subtracted from one side and 3 is subtracted from other side
      \nHence, from the above,
      \nWe can conclude that
      \nThe given equations are not the same by using the Subtraction Property of Equality<\/p>\n

      Question 3.
      \nIf 4 + 6 = 10, does (4 + 6) \u00d7 3 = 10 \u00d7 3? Why or why not?
      \nAnswer:
      \nThe given equations are:
      \n4 + 6 = 10
      \n(4 + 6) \u00d7 3 = 10 \u00d7 3
      \nNow,
      \nAccording to “Multiplication Property of Equality”,
      \nMultiply the same amount of number on both sides to make the equation balanced
      \nSo,
      \nFrom the given equation,
      \nWe can observe that 3 is multiplied on both sides
      \nHence, from the above,
      \nWe can conclude that
      \nThe given equations are the same by using the Multiplication Property of Equality<\/p>\n

      Question 4.
      \nIf 5 + 4 = 9, does (5 + 4) \u00f7 3 = 9 \u00f7 4? Why or why not?
      \nAnswer:
      \nThe given equations are:
      \n5 + 4 = 9
      \n(5 + 4) \u00f7 3 = 9 \u00f7 4
      \nNow,
      \nAccording to “Division Property of Equality”,
      \nDivide the same amount of number on both sides to make the equation balanced
      \nSo,
      \nFrom the given equation,
      \nWe can observe that
      \n3 is divided on one side and 4 is divided on the other side
      \nHence, from the above,
      \nWe can conclude that
      \nThe given equations are not the same by using the Division Property of Equality<\/p>\n

      Lessons 4.3 AND 4.4 Write and Solve Addition, Subtraction, Multiplication, and Division Equations<\/strong><\/p>\n

      Quick Review<\/strong>
      \nUse the inverse relationship of addition and subtraction or multiplication and division to solve equations. To check, substitute your answer back into the original equation.<\/p>\n

      Example<\/strong>
      \n23 + y = 57
      \n23 + y – 23 = 57 – 23a
      \ny = 34<\/p>\n

      a – 12 = 16
      \na – 12 + 12 = 16 + 12
      \na = 28<\/p>\n

      9z = 63
      \n9z \u00f7 9 = 63 \u00f7 9
      \nz = 7<\/p>\n

      c \u00f7 4 = 24
      \nc \u00f7 4 \u00d7 4 = 24 \u00d7 4
      \nc = 96<\/p>\n

      Practice<\/strong>
      \nSolve for x.<\/strong>
      \nQuestion 1.
      \n8x = 64
      \nAnswer:
      \nThe given equation is:
      \n8x = 64
      \nNow,
      \nDivide by 8 on both sides
      \nSo,
      \n\\(\\frac{8x}{8}\\) = \\(\\frac{64}{8}\\)
      \nx = 8
      \nHence, from the above,
      \nWe can conclude that
      \nThe value of x for the given equation is: 8<\/p>\n

      Question 2.
      \nx + 2 = 11
      \nAnswer:
      \nThe given equation is:
      \nx + 2 = 11
      \nNow,
      \nSubtract 2 on both sides
      \nSo,
      \nx + 2 – 2 = 11 – 2
      \nx = 9
      \nHence, from the above,
      \nWe can conclude that
      \nThe value of x for the given equation is: 9<\/p>\n

      Question 3.
      \nx \u00f7 20 = 120
      \nAnswer:
      \nThe given equation is:
      \nx \u00f7 20 = 120
      \nNow,
      \nMultiply with 20 on both sides
      \nSo,
      \n(x \u00f7 120) \u00d7 20 = 120 \u00d7 20
      \nx = 2,400
      \nHence, from the above,
      \nWe can conclude that
      \nThe value of x for the given equation is: 2,400<\/p>\n

      Question 4.
      \nx – 17 = 13
      \nAnswer:
      \nThe given equation is:
      \nx – 17 = 13
      \nNow,
      \nAdd 17 on both sides
      \nSo,
      \nx – 17 + 17 = 13 + 17
      \nx = 30
      \nHence, from the above,
      \nWe can conclude that
      \nThe value of x for the given equation is: 30<\/p>\n

      Question 5.
      \nx \u00f7 12 = 2
      \nAnswer:
      \nThe given equation is:
      \nx \u00f7 12 = 2
      \nNow,
      \nMultiply with 12 on both sides
      \nSo,
      \n(x \u00f7 12) \u00d7 12 = 2 \u00d7 12
      \nx = 24
      \nHence, from the above,
      \nWe can conclude that
      \nThe value of x for the given equation is: 24<\/p>\n

      Question 6.
      \n8 + x = 25
      \nAnswer:
      \nThe given equation is:
      \n8 + x = 25
      \nNow,
      \nSubtract 8 on both sides
      \nSo,
      \n8 + x – 8 = 25 – 8
      \nx = 17
      \nHence, from the above,
      \nWe can conclude that
      \nThe value of x for the given equation is: 17<\/p>\n

      Question 7.
      \n7x = 77
      \nAnswer:
      \nThe givene quation is:
      \n7x = 77
      \nNow,
      \nDivide by 7 on both sides
      \nSo,
      \n\\(\\frac{7x}{7}\\) = \\(\\frac{77}{7}\\)
      \nx = 11
      \nHence, from the above,
      \nWe can conclude that
      \nThe value of x for the given equation is: 11<\/p>\n

      Question 8.
      \nx – 236 = 450
      \nAnswer:
      \nThe given equation is:
      \nx – 236 = 450
      \nNow,
      \nAdd 236 on both sides
      \nSo,
      \nx – 236 + 236 = 450 + 236
      \nx = 686
      \nHence, from the above,
      \nWe can conclude that
      \nThe value of x for the given equation is: 686<\/p>\n

      Question 9.
      \n26 = 13x
      \nAnswer:
      \nThe given equation is:
      \n26 = 13x
      \nNow,
      \nDivide by 13 on both sides
      \nSo,
      \n\\(\\frac{26}{13}\\) = \\(\\frac{13x}{13}\\)
      \n2 = x
      \nx = 2
      \nHence, from the above,
      \nWe can conclude that
      \nThe value of x for the given equation is: 2<\/p>\n

      Question 10.
      \nx + 21.9 = 27.1
      \nAnswer:
      \nThe given equation is:
      \nx + 21.9 = 27.1
      \nNow,
      \nSubtract 21.9 on both sides
      \nSo,
      \nx + 21.9 – 21.9 = 27.1 – 21.9
      \nx = 5.2
      \nHence, from the above,
      \nWe can conclude that
      \nThe value of x for the given equation is: 5.2<\/p>\n

      Question 11.
      \n2,448 \u00f7 48 = x
      \nAnswer:
      \nThe given equation is:
      \n2,448 \u00f7 48 = x
      \nNow,
      \nx = 2,448 \u00f7 48
      \nx = 51
      \nHence, from the above,
      \nWe can conclude that
      \nThe value of x for the given equation is: 51<\/p>\n

      Question 12.
      \nx + 15 = 31
      \nAnswer:
      \nThe given equation is:
      \nx + 15 = 31
      \nNow,
      \nSubtract with 15 on both sides
      \nSo,
      \nx + 15 – 15 = 31 – 15
      \nx = 16
      \nHence, from the above,
      \nWe can conclude that
      \nThe value of x for the given equation is: 16<\/p>\n

      Lesson 4.5 Write and Solve Equations with Rational Numbers<\/strong><\/p>\n

      Quick Review<\/strong>
      \nYou can use inverse relationships and properties of equality to solve each equation.<\/p>\n

      Example<\/strong>
      \nSolve w + 4\\(\\frac{1}{3}\\) = 7.
      \nSubtract 4\\(\\frac{1}{3}\\) from both sides.
      \nw + 4\\(\\frac{1}{3}\\) \u2013 4\\(\\frac{1}{3}\\) = 7 – 4\\(\\frac{1}{3}\\)
      \nw = 2\\(\\frac{2}{3}\\)<\/p>\n

      Solve \\(\\frac{3}{5}\\)n = \\(\\frac{2}{3}\\)
      \nMultiply both sides by the reciprocal of \\(\\frac{3}{5}\\).
      \n\\(\\frac{5}{3} \\times \\frac{3}{5} n=\\frac{5}{3} \\times \\frac{2}{3}\\)
      \n\\(n=\\frac{10}{9} \\text { or } 1 \\frac{1}{9}\\)<\/p>\n

      Practice<\/strong>
      \nIn 1-8, solve for x.<\/strong>
      \nQuestion 1.
      \nx + 3\\(\\frac{5}{8}\\) = 7\\(\\frac{1}{4}\\)
      \nAnswer:
      \nThe given equation is:
      \nx + 3\\(\\frac{5}{8}\\) = 7\\(\\frac{1}{4}\\)
      \nNow,
      \nx = 7\\(\\frac{1}{4}\\) – 3\\(\\frac{5}{8}\\)
      \nSo,
      \n\"\"
      \nHence, from the above,
      \nWe can conclude that
      \nThe value of x for the given equation is: 3\\(\\frac{5}{8}\\)<\/p>\n

      Question 2.
      \nx – \\(\\frac{4}{8}\\) = 4\\(\\frac{1}{4}\\)
      \nAnswer:
      \nThe given equation is:
      \nx – \\(\\frac{4}{8}\\) = 4\\(\\frac{1}{4}\\)
      \nNow,
      \nx = 4\\(\\frac{1}{4}\\) + \\(\\frac{4}{8}\\)
      \nSo,
      \n\"\"
      \nHence, from the above,
      \nWe can conclude that
      \nThe value of x for the given equation is: 4\\(\\frac{3}{4}\\)<\/p>\n

      Question 3.
      \nx \u00f7 15 = 8\\(\\frac{1}{3}\\)
      \nAnswer:
      \nThe given equation is:
      \nx \u00f7 15 = 8\\(\\frac{1}{3}\\)
      \nNow,
      \nx = 8\\(\\frac{1}{3}\\) \u00d7 15
      \nSo,
      \n\"\"
      \nHence, from the above,
      \nWe can conclude that
      \nThe value of x for the given equation is: 125<\/p>\n

      Question 4.
      \n\\(\\frac{4}{2}\\)x = 6
      \nAnswer:
      \nThe given equation is:
      \n\\(\\frac{4}{2}\\)x = 6
      \nNow,
      \nx = 6 \u00d7 \\(\\frac{2}{4}\\)
      \nx = \\(\\frac{6 \u00d7 2}{4}\\)
      \nx = \\(\\frac{12}{4}\\)
      \nx = 3
      \nHence, from the above,
      \nWe can conclude that
      \nThe value of x for the given equation is: 3<\/p>\n

      Question 5.
      \n\\(\\frac{x}{3}\\) = 9
      \nAnswer:
      \nThe given equation is:
      \n\\(\\frac{x}{3}\\) = 9
      \nNow,
      \nx = 9 \u00d7 3
      \nx = 27
      \nHence, from the above,
      \nWe can conclude that
      \nThe value of x for the given equation is: 27<\/p>\n

      Question 6.
      \n14x = 73.5
      \nAnswer:
      \nThe given equation is:
      \n14x = 73.5
      \nNow,
      \nx = \\(\\frac{73.5}{14}\\)
      \nx = 5.25
      \nHence, from the above,
      \nWe can conclude that
      \nThe value of x for the given equation is: 5.25<\/p>\n

      Question 7.
      \n12x = 19.2
      \nAnswer:
      \nThe given equation is:
      \n12x = 19.2
      \nNow,
      \nx = \\(\\frac{19.2}{12}\\)
      \nx = 1.6
      \nHence, from the above,
      \nWe can conclude that
      \nThe value of x for the given equation is: 1.6<\/p>\n

      Question 8.
      \n17.9 – x = 12.8
      \nAnswer:
      \nThe given equation is:
      \n17.9 – x = 12.8
      \nNow,
      \nx = 17.9 – 12.8
      \nx = 5.1
      \nHence, from the above,
      \nWe can conclude that
      \nThe value of x for the given equation is: 5.1<\/p>\n

      Question 9.
      \nTomas buys a bag of 5 peaches for $3.55. Write and solve an equation to find how much money, m, Tomas paid for each peach.
      \nAnswer:
      \nIt is given that
      \nTomas buys a bag of 5 peaches for $3.55
      \nNow,
      \nThe amount of money Tomas paid for each peach = (The amount of money Tomas paid for 5 peaches) \u00f7 (The number of peaches)
      \n= \\(\\frac{$3.55}{5}\\)
      \n= $0.71
      \nHence, from the above,
      \nWe can conclude that
      \nThe amount of money Tomas paid for each peach is: $0.71<\/p>\n

      Question 10.
      \nKrys has $1.54 and spends $0.76. Write and solve an equation to find how much money, m, Krys has left.
      \nAnswer:
      \nIt is given that
      \nKrys has $1.54 and spends $0.76.
      \nNow,
      \nLet the amount of money Krys has left be x
      \nSo,
      \nx = (The amount of money Krys has) – (The amount of money Krys spent)
      \nx = $1.54 – $0.76
      \nx = $0.78
      \nHence, from the above,
      \nWe can conclude that
      \nThe amount of money Krys has left is: $0.78<\/p>\n

      Lesson 4.6 Understand and Write Inequalities<\/strong><\/p>\n

      Quick Review<\/strong>
      \nAn inequality is a mathematical sentence that contains < (less than), > (greater than), < (less than or equal to), 2 (greater than or equal to), or = (not equal to).<\/p>\n

      Example<\/strong>
      \n\"Envision<\/p>\n

      Practice<\/strong>
      \nWrite an inequality for each situation.<\/strong>
      \nQuestion 1.
      \nUp to 5 people, p, visited Mary today.
      \nAnswer:
      \nThe given situation is:
      \nUp to 5 people, p, visited Mary today.
      \nSo,
      \nThe representation of the given situation in the form of an inequality is:
      \np \u2264 5
      \nHence, from the above,
      \nWe can conclude that
      \nThe representation of the given situation in the form of an inequality is:
      \np \u2264 5<\/p>\n

      Question 2.
      \nThe value, v, of the hat is less than $9.
      \nAnswer:
      \nThe given situation is:
      \nThe value, v, of the hat is less than $9.
      \nSo,
      \nThe representation of the given situation in the form of an inequality is:
      \nv < $9
      \nHence, from the above,
      \nWe can conclude that
      \nThe representation of the given situation in the form of an inequality is:
      \nv < $9<\/p>\n

      Question 3.
      \nThe number of guests, g, coming for dinner is not 8.
      \nAnswer:
      \nThe given situation is:
      \nThe number of guests, g, coming for dinner is not 8.
      \nSo,
      \nThe representation of the given situation in the form of an inequality is:
      \ng \u2260 8
      \nHence, from the above,
      \nWe can conclude that
      \nThe representation of the given situation in the form of an inequality is:
      \ng \u2260 8<\/p>\n

      Question 4.
      \nThe distance of the race, d, is at least 6 miles.
      \nAnswer:
      \nThe given situation is:
      \nThe distance of the race, d, is at least 6 miles.
      \nSo,
      \nThe representation of the given situation in the form of an inequality is:
      \nd \u2265 6 miles
      \nHence, from the above,
      \nWe can conclude that
      \nThe representation of the given situation in the form of an inequality is:
      \nd \u2265 6 miles<\/p>\n

      Question 5.
      \nThe time it takes to get to Grandma’s house, t, is longer than 2 hours.
      \nAnswer:
      \nThe given situation is:
      \nThe time it takes to get to Grandma’s house, t, is longer than 2 hours.
      \nSo,
      \nThe representation of the given situation in the form of an inequality is:
      \nt > 2 hours
      \nHence, from the above,
      \nWe can conclude that
      \nThe representation of the given situation in the form of an inequality is:
      \nt > 2 hours<\/p>\n

      Lesson 4.7 Solve Inequalities<\/strong><\/p>\n

      Quick Review<\/strong>
      \nTo graph the solutions of an inequality on a number line, use an open circle for < or > and a closed circle for \u2264 or \u2265. If the values of the variable are less than the given number, shade to the left on the number line. If the values of the variable are greater than the given number, shade to the right on the number line.<\/p>\n

      Example<\/strong>
      \n“Molly is less than 15 years old\u201d is represented by the inequality x < 15. Write three ages that could represent Molly’s age.
      \nTo graph the inequality on a number line, draw an open circle at 15 and shade to the left of 15 because x is less than 15. Draw an arrow to show all numbers less than 15.
      \n\"Envision
      \nThere are many solutions. Molly could be 10, 12, 14, or any age less than 15 years.<\/p>\n

      Practice<\/strong>
      \nWrite the inequality that each graph represents.<\/strong>
      \nQuestion 1.
      \n\"Envision
      \nAnswer:
      \nThe given number line is:
      \n\"Envision
      \nNow,
      \nLet the solutions of the given number line be x
      \nNow,
      \nFrom the given number line,
      \nWe can observe that
      \nThere is an open circle on 10 and the list of numbers are on the left side of 10
      \nSo,
      \nThe representation of the given number line in the form of an inequality is:
      \nx < 10
      \nHence, from the above,
      \nWe can conclude that
      \nThe representation of the given number line in the form of an inequality is:
      \nx < 10<\/p>\n

      Question 2.
      \n\"Envision
      \nAnswer:
      \nThe given number line is:
      \n\"Envision
      \nNow,
      \nLet the solutions of the given number line be x
      \nNow,
      \nFrom the given number line,
      \nWe can observe that
      \nThere is an open circle on 5 and the list of numbers are on the right side of 5
      \nSo,
      \nThe representation of the given number line in the form of an inequality is:
      \nx > 5
      \nHence, from the above,
      \nWe can conclude that
      \nThe representation of the given number line in the form of an inequality is:
      \nx > 5<\/p>\n

      Question 3.
      \n\"Envision
      \nAnswer:
      \nThe given number line is:
      \n\"Envision
      \nNow,
      \nLet the solutions of the given number line be x
      \nNow,
      \nFrom the given number line,
      \nWe can observe that
      \nThere is a closed circle on 3 and the list of numbers are on the right side of 3
      \nSo,
      \nThe representation of the given number line in the form of an inequality is:
      \nx \u2265 3
      \nHence, from the above,
      \nWe can conclude that
      \nThe representation of the given number line in the form of an inequality is:
      \nx \u2265 3<\/p>\n

      Question 4.
      \n\"Envision
      \nAnswer:
      \nThe given number line is:
      \n\"Envision
      \nNow,
      \nLet the solutions of the given number line be x
      \nNow,
      \nFrom the given number line,
      \nWe can observe that
      \nThere is a closed circle on 7 and the list of numbers are on the left side of 7
      \nSo,
      \nThe representation of the given number line in the form of an inequality is:
      \nx \u2264 7
      \nHence, from the above,
      \nWe can conclude that
      \nThe representation of the given number line in the form of an inequality is:
      \nx \u2264 7<\/p>\n

      Lesson 4.8 Understand Dependent and Independent Variables<\/strong><\/p>\n

      Quick Review<\/strong>
      \nThink about how the values of variables affect each other.
      \nTo identify the dependent variable, ask yourself which variable depends on the other.
      \nTo identify the independent variable, ask yourself which variable causes the change.<\/p>\n

      Example<\/strong>
      \nThe spirit squad is washing cars. The equation m=2c represents the money they make, m, for washing c cars. Identify the dependent variable and the independent variable.
      \nThe amount of money the spirit squad makes depends on the number of cars they wash. The dependent variable is m.
      \nThe number of cars washed changes the amount of money made. The independent variable is c.<\/p>\n

      Practice<\/strong>
      \nIdentify the dependent variable and the independent variable in each situation.<\/strong>
      \nQuestion 1.
      \nThe distance traveled, d, and the speed, s
      \nAnswer:
      \nThe given situation is:
      \nThe distance traveled, d, and the speed, s
      \nNow,
      \nTo identify the “Dependent variable” ask yourself which variable depends on the other
      \nTo identify the “Independent variable”, ask yourself which variable causes the change
      \nSo,
      \nThe independent variable for the given situation is: d
      \nThe dependent variable for the given situation is: s
      \nHence, from the above,
      \nWe can conclude that
      \nThe independent variable for the given situation is: d
      \nThe dependent variable for the given situation is: s<\/p>\n

      Question 2.
      \nThe calories, c, in a snack and the amount of the snack, a
      \nAnswer:
      \nThe given situation is:
      \nThe calories, c, in a snack and the amount of the snack, a
      \nNow,
      \nTo identify the “Dependent variable” ask yourself which variable depends on the other
      \nTo identify the “Independent variable”, ask yourself which variable causes the change
      \nSo,
      \nThe independent variable for the given situation is: c
      \nThe dependent variable for the given situation is: a
      \nHence, from the above,
      \nWe can conclude that
      \nThe independent variable for the given situation is: c
      \nThe dependent variable for the given situation is: a<\/p>\n

      Question 3.
      \nThe amount of money you have spent, s, and how much money you have left, m
      \nAnswer:
      \nThe given situation is:
      \nThe amount of money you have spent, s, and how much money you have left, m
      \nNow,
      \nTo identify the “Dependent variable” ask yourself which variable depends on the other
      \nTo identify the “Independent variable”, ask yourself which variable causes the change
      \nSo,
      \nThe independent variable for the given situation is: s
      \nThe dependent variable for the given situation is: m
      \nHence, from the above,
      \nWe can conclude that
      \nThe independent variable for the given situation is: s
      \nThe dependent variable for the given situation is: m<\/p>\n

      Question 4.
      \nThe number of apple slices remaining, r, and the number of apple slices eaten, e
      \nAnswer:
      \nThe given situation is:
      \nThe number of apple slices remaining, r, and the number of apple slices eaten, e
      \nNow,
      \nTo identify the “Dependent variable” ask yourself which variable depends on the other
      \nTo identify the “Independent variable”, ask yourself which variable causes the change
      \nSo,
      \nThe independent variable for the given situation is: e
      \nThe dependent variable for the given situation is: r
      \nHence, from the above,
      \nWe can conclude that
      \nThe independent variable for the given situation is: e
      \nThe dependent variable for the given situation is: r<\/p>\n

      Lesson 4.9 Use Patterns to Write and Solve Equations<\/strong><\/p>\n

      Quick Review<\/strong>
      \nLook for patterns between two related variables to find rules and write equations.<\/p>\n

      Example<\/strong>
      \nWrite a rule and an equation that represents the pattern. Then complete the table.
      \n\"Envision
      \nFind the rule and write an equation.
      \n12 is 3 \u00d7 4
      \n16 is 4 \u00d7 4
      \n20 is 5 \u00d7 4
      \nRule: The value of y is 4 times the value of x.
      \nEquation: y = 4x
      \nEvaluate the equation for x = 6 and x = 7.
      \ny = 4 \u00d7 6 = 24
      \ny = 4 \u00d7 7 = 28<\/p>\n

      Practice<\/strong>
      \nQuestion 1.
      \nFind the pattern and then write a rule and an equation that represents the pattern. Then complete the table.
      \n\"Envision
      \nAnswer:
      \nThe given table is:
      \n\"Envision
      \nNow,
      \nFrom the given table,
      \nWe can observe that
      \nThe relationship (or) pattern between x and y is:
      \ny = \\(\\frac{x}{2}\\)
      \nNow,
      \nFor x = 16,
      \ny = \\(\\frac{16}{2}\\)
      \ny = 8
      \nFor x = 20,
      \ny = \\(\\frac{20}{2}\\)
      \ny = 10
      \nSo,
      \nThe completed table by using the above relationship is:
      \n\"\"
      \nHence, from the above,
      \nWe can conclude that
      \nThe completed table by using the above relationship is:
      \n\"\"<\/p>\n

      Question 2.
      \nUse the equation to complete the table.
      \ny = 6x + 1
      \n\"Envision
      \nAnswer:
      \nThe given equation is:
      \ny = 6x + 1
      \nNow,
      \nFor x = 1,
      \ny = 6 (1) + 1
      \n= 7
      \nFor x = 2,
      \ny = 6 (2) + 1
      \n= 13
      \nFor x = 3,
      \ny = 6 (3) + 1
      \n= 19
      \nFor x = 4,
      \ny = 6 (4) + 1
      \n= 25
      \nFor x = 5,
      \ny = 6 (5) + 1
      \n= 31
      \nSo,
      \nThe completed table by uisng the above equation is:
      \n\"\"
      \nHence, from the above,
      \nWe can conclude that
      \nThe completed table by uisng the above equation is:
      \n\"\"<\/p>\n

      Lesson 4.10 Relate Tables, Graphs, and Equations<\/strong><\/p>\n

      Quick Review<\/strong>
      \nA table, equation, or graph can be used to analyze the relationship between dependent and independent variables. Ordered pairs that make an equation true can be used to graph the equation.<\/p>\n

      Example<\/strong>
      \nComplete the table and graph to show the relationship between the variables in the equation t = s + 1.
      \nA restaurant has a special that when you buy one sandwich you get a second sandwich for $1
      \nLet s = price of one sandwich.
      \nLet t = total price of two sandwiches.
      \nStep 1
      \nMake a table. Include at least three values.
      \n\"Envision<\/p>\n

      Step 2
      \nGraph each ordered pair on a coordinate plane. Then draw a line through the points.
      \n\"Envision<\/p>\n

      Practice<\/strong>
      \nQuestion 1.
      \nThe cross country team practices by jogging on the town’s streets. The average jogging rate is 6 miles per hour. One member jogged for 3.5 hours one weekend. How many miles did the team member jog?
      \na. Complete the table to relate the number of miles to the number of hours jogged.
      \n\"Envision
      \nAnswer:
      \nIt is given that
      \nThe cross country team practices by jogging on the town’s streets. The average jogging rate is 6 miles per hour. One member jogged for 3.5 hours one weekend.
      \nSo,
      \nFrom the given situation,
      \nWe can observe that
      \nThe independent variable is: The number of hours the team member jogged
      \nThe dependent variable is: Number of miles the team member jogged
      \nNow,
      \nLet the dependent variable be y
      \nLet the independent variable be y
      \nSo,
      \nThe relation between x and y from the given situation is:
      \ny = 6x
      \nNow,
      \nThe completed table by using the above equation is:
      \n\"\"
      \nHence, from the above,
      \nWe can conclude that
      \nThe completed table by using the above equation is:
      \n\"\"<\/p>\n

      b. Graph the ordered pairs on the coordinate plane.
      \n\"Envision
      \nAnswer:
      \nThe given table is:
      \n\"\"
      \nNow,
      \nFrom the given table,
      \nWe can observe that
      \nThe ordered pairs to draw in the coordinate plane is:
      \n(1, 6), (2, 12), (3, 18)
      \nHence,
      \nThe representtaion of the given ordered pairs in the coordinate plane is:
      \n\"\"<\/p>\n

      c. Write an equation that describes the relationship. Then solve the problem.
      \nAnswer:
      \nFrom part (a),
      \nWe know that,
      \nThe equation that describes the given situation is:
      \ny = 6x
      \nSo,
      \nThe number of miles did the team member jog is:
      \ny = 6x
      \n= 6 (3.5)
      \n= 21 miles
      \nHence, from the above,
      \nWe can conclude that
      \nThe number of miles did the team member jog is: 21 miles<\/p>\n

      Question 2.
      \nAlex is making puppets for a show. He bought all the string needed for $125. It costs $18 for the remaining materials to make each puppet. What is the total cost to make 50 puppets?
      \nAnswer:
      \nIt is given that
      \nAlex is making puppets for a show. He bought all the string needed for $125. It costs $18 for the remaining materials to make each puppet
      \nSo,
      \nThe total cost of the materials to make each puppet = $125 + $18
      \n= $143
      \nSo,
      \nThe total cost to make 50 puppets = 50 \u00d7 $143
      \n= $7,150
      \nHence, from the above,
      \nWe can conclude that
      \nThe total cost to make 50 puppets = 50 \u00d7 $143
      \n= $7,150<\/p>\n

      Topic 4 Fluency Practice<\/h3>\n

      Riddle Rearranging<\/strong>
      \nFind each quotient. Then arrange the answers in order from least to greatest. The letters will spell out the answer to the riddle below.
      \nI can… divide multi-digit numbers.
      \n\"Envision<\/p>\n","protected":false},"excerpt":{"rendered":"

      Go through the\u00a0enVision Math Common Core Grade 6 Answer Key\u00a0Topic 4 Represent And Solve Equations And Inequalities\u00a0regularly and improve your accuracy in solving questions. Envision Math Common Core 6th Grade Answers Key Topic 4 Represent And Solve Equations And Inequalities ?Topic Essential Question What procedures can be used to write and solve equations and inequalities? …<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts\/16989"}],"collection":[{"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/comments?post=16989"}],"version-history":[{"count":3,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts\/16989\/revisions"}],"predecessor-version":[{"id":27337,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts\/16989\/revisions\/27337"}],"wp:attachment":[{"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/media?parent=16989"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/categories?post=16989"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/tags?post=16989"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}