{"id":1920,"date":"2023-12-27T04:11:29","date_gmt":"2023-12-26T22:41:29","guid":{"rendered":"https:\/\/envisionmathanswerkey.com\/?p=1920"},"modified":"2023-12-28T14:28:47","modified_gmt":"2023-12-28T08:58:47","slug":"envision-math-common-core-grade-7-answer-key-topic-5","status":"publish","type":"post","link":"https:\/\/envisionmathanswerkey.com\/envision-math-common-core-grade-7-answer-key-topic-5\/","title":{"rendered":"enVision Math Common Core Grade 7 Answer Key Topic 5 Solve Problems Using Equations and Inequalities"},"content":{"rendered":"

Go through the\u00a0enVision Math Common Core Grade 7 Answer Key<\/a>\u00a0Topic 5 Solve Problems Using Equations and Inequalities<\/strong>\u00a0regularly and improve your accuracy in solving questions.<\/p>\n

enVision Math Common Core 7th Grade Answers Key Topic 5 Solve Problems Using Equations and Inequalities<\/h2>\n

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

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

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

Question 1.
\nA statement that contains the symbols
\n<, >, \u2264, or \u2265 is called a(n)
\nAnswer:
\nInequality,<\/p>\n

Explanation:
\nAn open sentence that contains the symbol
\n< , \u2264 , > , or \u2265 is called an inequality.
\n<\/span>Inequalities can be solved the same way as equations.<\/span><\/p>\n

Question 2.
\nProperties that state that performing the same
\noperation on both sides of an equation will keep
\nthe equation true are called
\nAnswer:
\nProperties of equation,<\/p>\n

Explanation:
\nThe first four properties of equality–those that
\ndeal with operations–allow us to add, subtract,
\nmultiply and divide variables.
\nThey also formally express the idea that when we
\nperform the same operations on both sides of an equation,
\nthe two sides are still equivalent.
\nIn other words, that we can perform the same
\noperation on both sides without changing
\nthe values of the variables these are called as
\nproperties of equation.<\/p>\n

Question 3.
\nAddition and subtraction have a(n) ___
\nbecause they can “undo” each other.
\nAnswer:
\nInverse relationship,<\/p>\n

Explanation:
\nAn inverse operation are two operations that undo each other,
\ne.g. addition and subtraction or multiplication and division.
\nCan perform the same inverse operation on each side of
\nan equivalent equation without changing\u00a0the\u00a0equality.<\/p>\n

Question 4.
\nTerms that have the same variable are called ___
\nAnswer:
\nLiketerms,<\/p>\n

Explanation:
\nTerms that have the same variable are called like terms as
\nlike terms are terms that have the same variables and
\npowers example ax + bx here there are two terms ax, bx
\nboth have x a same variable x and power also same.<\/p>\n

\"Using<\/p>\n

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

Use properties to solve each equation for x.<\/p>\n

Question 5.
\nx + 9.8 = 14.2
\nAnswer:
\nx = 4.4,<\/p>\n

Explanation:
\nGiven x + 9.8 = 14.2,
\nafter subtracting 9.8 both sides we get
\nx + 9.8 – 9.8 = 14.2 – 9.8,
\nx + 0 = 4.4, therefore x = 4.4.<\/p>\n

Question 6.
\n14x = 91
\nAnswer:
\nx = 6 remainder 7,<\/p>\n

Explanation:
\nGiven 14x = 91,
\ndividing both sides by 14 we get
\n14x \u00f714 = 91 \u00f7 14,
\nx = 91 \u00f714,
\n6<\/span>
\n14)91(
\n\u00a0 \u00a0 \u00a084
\n7
\n<\/span>therefore x = 6 with remainder 7.<\/p>\n

Question 7.
\n\\(\\frac{1}{3}\\)x = 24
\nAnswer:
\nx = 72,<\/p>\n

Explanation:
\nGiven \\(\\frac{1}{3}\\)x = 24,
\nmultiplying both sides by 3 we get
\n\\(\\frac{1}{3}\\)x X 3 = 24 X 3,
\ntherefore x= 24 X 3= 72.<\/p>\n

Like Terms<\/strong>
\nCombine like terms in each expression.<\/p>\n

Question 8.
\n\\(\\frac{1}{4}\\)k + \\(\\frac{1}{4}\\)m –
\n\\(\\frac{2}{3}\\)k + \\(\\frac{5}{9}\\)m
\nAnswer:
\n\\(\\frac{1}{4}\\)k + \\(\\frac{1}{4}\\)m –
\n\\(\\frac{2}{3}\\)k + \\(\\frac{5}{9}\\)m =
\n– \\(\\frac{5}{12}\\)k + \\(\\frac{29}{36}\\)m,<\/p>\n

Explanation:
\nGiven \\(\\frac{1}{4}\\)k + \\(\\frac{1}{4}\\)m –
\n\\(\\frac{2}{3}\\)k + \\(\\frac{5}{9}\\)m,
\nwe combine like terms in expression as
\n(\\(\\frac{1}{4}\\)k – latex]\\frac{2}{3}[\/latex]k) +
\n(\\(\\frac{1}{4}\\)m + \\(\\frac{5}{9}\\)m) =
\n(\\(\\frac{1}{4}\\) – \\(\\frac{2}{3}\\))k +
\n(\\(\\frac{1}{4}\\) + \\(\\frac{5}{9}\\))m,
\nbefore subtracting or adding we make common
\ndenominators for the both terms so
\n\\(\\frac{1 X 3 – 2 X 4}{12}\\)k +
\n\\(\\frac{1 X 9 + 5 X 4}{36}\\)m =
\n\\(\\frac{3 – 8}{12}\\)k + \\(\\frac{ 9+ 20}{36}\\)m = \\(\\frac{- 5}{12}\\)k + \\(\\frac{29}{36}\\)m =
\n–\\(\\frac{5}{12}\\)k + \\(\\frac{29}{36}\\)m.<\/p>\n

Question 9.
\n-4b + 2w +(-4b) + 8w
\nAnswer:
\n-8b + 10w,<\/p>\n

Explanation:
\nGiven -4b + 2w +(-4b) + 8w we combine like terms as
\n– 4b + (-4b) and 2 w + 8w =
\n– 4b + (-4b) + 2w + 8w =
\n– 8b + 10w, therefore -4b + 2w +(-4b) + 8w = -8b + 10w.<\/p>\n

Question 10.
\n6 – 5z + 8 – 4z + 1
\nAnswer:
\n-9z + 15 or 15 – 9z,<\/p>\n

Explanation:
\nGiven 6 – 5z + 8 – 4z + 1 we combine like terms as
\n(6 + 8 + 1) and (-5z – 4z) as
\n6 + 8 + 1 – 5z – 4z = 15 – 9z,
\ntherefore 6 – 5z + 8 – 4z + 1 = 15 – 9z or -9z + 15.<\/p>\n

Inequalities<\/strong><\/p>\n

Question 11.
\nWrite an inequality that represents the situation:
\nA large box of golf balls has more than 12 balls.
\nDescribe how your inequality represents the situation.
\nAnswer:
\n12 > x is Inequality,<\/p>\n

Explanation:
\nGiven a large box of golf balls has more than 12 balls means
\n12 > x is inequality, where x\u00a0= the number of golf balls,
\ninequality states that the number of golf balls is over 12.<\/p>\n

Language Development<\/strong><\/p>\n

Fill in the Venn diagram to compare and contrast equations and inequalities.
\n\"Envision
\nIn the box below, draw pictures to represent the
\nterms and phrases in the overlap section of your diagram.
\nAnswer:
\n\"Envision<\/p>\n

PICK A PROJECT<\/strong><\/p>\n

PROJECT 5A<\/strong><\/p>\n

How many different ways could you sort a basket of vegetables?
\nPROJECT: COMPARING WITH A VENN DIAGRAM
\n\"Envision
\n\"Envision<\/p>\n

Explanation:
\nWays of sorting a basket of vegetables as per
\n1. A part of a plant used as a food (spinach),
\n2. Do not contain seeds (potatoes),
\n3. Distinct in taste – can be sweet (sweet potato), salty,
\nsore or bitter (bitter gourd),
\n4. Mostly green in color (cucumber),
\n5. Supply fiber, vitamins, minerals and trace elements,
\n6. Involved in vegetative reproduction.<\/p>\n

\"Using<\/p>\n

PROJECT 5B<\/strong>
\nWhich character would you be from your favorite play? Why?
\nPROJECT: WRITE A PLAY
\n\"Envision<\/p>\n

Answer:
\nCharacter I would be from my favorite plays is Cinderella,
\nAlways Be Kind No Matter What –
\nA little kindness goes a long way, and no doubt
\nCinderella is a kind soul. Her kindness can make
\nher seem like a pushover (especially when her evil
\nstepmother and stepsisters are overloading her with housework!).<\/p>\n

Explanation:
\nPlay:<\/p>\n

    \n
  1. Cinderella is a princess.<\/li>\n
  2. She was born to 2 lovely parents.<\/li>\n
  3. Her mother dies early, and her father remarries.<\/li>\n
  4. But the stepmother and Cinderella\u2019s two
    \nstepsisters treated her like a servant.<\/li>\n
  5. Cinderella meets the Prince once in the forest.<\/li>\n
  6. She goes to the royal ball ceremony with the
    \nfairy godmother\u2019s help and dances with the prince.<\/li>\n
  7. The carriage, driver, horse, dress, etc.,
    \nwere all magical and would have returned
    \nto normal just after the clock struck midnight.<\/li>\n
  8. In a hurry to leave the palace before midnight,
    \nCinderella forgets her slippers.<\/li>\n
  9. The prince traces her back by those slippers and marries her.<\/li>\n
  10. They live happily ever after.
    \nAt the end Cinderella kind heart wins,
    \nso it’s my favorite play.<\/li>\n<\/ol>\n

    PROJECT 5C<\/strong>
    \nIf you could live in another country, where would you live, and why?
    \nPROJECT: EXCHANGE SOUVENIRS
    \n\"Envision
    \nAnswer:
    \nI would live in Canada,
    \nMany people have the dream of living in another country for at least some time during their life. And with good reason. Their are so many benefits.<\/p>\n

    Moving overseas can dramatically changes my life for the better.
    \nIn fact, there are a whole host of reasons why I decided to move to another country.
    \nLiving overseas can offer new opportunities, new lifestyles,
    \nnew careers and a new direction, financially better prospects.<\/p>\n

    Explanation:
    \nI would prefer living in canada because
    \nIt\u2019s an absolutely stunning place,
    \nGreat job opportunities,
    \nPeople are family friendly,
    \nThe best of modern metropolitan living,
    \nFamiliar culture and language,
    \nFree health care,
    \nThe Canadian sense of humor,
    \nCanadian crime rates are incredibly low,
    \nCanada is an \u2018education superpower\u2019 country,
    \nWorld leaders in quantum computing,
    \nmedical research and space science,
    \nCanada has a wonderfully varied climate,
    \nCanadians breathe some of the cleanest air on this planet,
    \nGreat food, Niagara Falls, Cheap living costs,
    \nA country you can be proud of living.<\/p>\n

    \"Using<\/p>\n

    PROJECT 5D<\/strong>
    \nHow would you prepare for being on a game show?
    \nPROJECT: SOLVE RANDOMIZED EQUATIONS AND INEQUALITIES
    \n\"Envision<\/p>\n

    Preparing for being on a game show
    \n1. Studying the format of the show,
    \n2.Taking it seriously and practice,
    \n3. Don’t be guilt – No matter how much I am
    \nlosing by keep myself cool and dignity,
    \n4. Stay positive,
    \n5. Enjoy it.<\/p>\n

    Lesson 5.1 Write Two-Step Equations<\/h3>\n

    Marley collects golf balls. His neighbor Tucker
    \ncollects 3 more than twice as many golf balls as Marley.
    \n\"Envision
    \nA. How can you use a table to represent the number of golf balls in Marley’s collection, m, and the number of golf balls in Tucker’s collection?
    \nB. How can you use an algebraic expression to represent
    \nthe number of golf balls in Tucker’s collection?
    \na.\"Envision<\/p>\n

    b. Algebraic expression is t = 2m + 3,<\/p>\n

    Explanation:
    \nGiven Marley collects golf balls. His neighbor Tucker
    \ncollects 3 more than twice as many golf balls as Marley so
    \na. The table to represent the number of golf balls in
    \nMarley’s collection m and the number of golf balls
    \nin Tucker’s collection are if Marley’s has 10 golf balls
    \nthen Tucker will have 2 X 10 + 3 = 20 + 3 = 23,
    \nif Marley’s has 12 golf balls then Tucker will have
    \n2 X 12 + 3 = 24 + 3 = 27, if Marley’s has 15 golf balls
    \nthen Tucker will have 2 X 15 + 3 = 30 + 3 = 33,
    \nif Marley’s has 18 golf balls
    \nthen Tucker will have 2 X 18 + 3 = 36 + 3 = 39 and
    \nif Marley’s has 20 golf balls
    \nthen Tucker will have 2 X 20 + 3 = 40 + 3 = 43 golf balls
    \nas shown above in the table.
    \nb. As Marley collects golf balls, His neighbor Tucker
    \ncollects 3 more than twice as many golf balls as Marley so
    \nthe algebraic expression to represent
    \nthe number of golf balls in Tucker’s collection is
    \nt = 2m + 3 here t is to represent Tucker and m
    \nis to represent Marley.<\/p>\n

    Focus on math practices<\/strong>
    \nLook for Relationships How do the terms of the
    \nexpression you wrote in Part B relate to the values in the table?
    \nAnswer:
    \nYes, the terms of the expression I wrote in Part B
    \nrelate to the values in the table,<\/p>\n

    Explanation:
    \nAs the values in the table of Marley’s collection of
    \ngolf balls and Tucker’s collection of golf balls are related
    \nwith the terms of expression as Tucker’s collection
    \nof golf balls are dependent on collection of
    \nMarley’s golf balls as twice plus 3 more so values
    \nof table are m = m and t = 2m + 3,
    \ntherefore the terms of expression I wrote in Part B
    \nrelate to the values in the table.<\/p>\n

    Essential Question<\/strong>
    \nHow does an equation show the relationship between variables and
    \nother quantities in a situation?
    \nAnswer:
    \nYes, an equation show the relationship between
    \nvariables and other quantities in a situation with
    \nequality and inequalities signs,<\/p>\n

    Explanation:
    \nAn equation is distinct because it has an equals sign and
    \nthat in itself creates a\u00a0relationship.
    \nUsually, it’s relating the left side\u00a0to\u00a0the right side in
    \nterms\u00a0of\u00a0that fact that they’re equal\u00a0to\u00a0each\u00a0other and
    \nunlike inequalities which form more\u00a0of\u00a0a\u00a0relation between\u00a0two\u00a0quantities\u00a0that can be greater than or less than.<\/p>\n

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

    Cole buys a new laptop for $335. He makes a down
    \npayment of $50 and pays the rest in 6 equal monthly payments, p.
    \nWhat equation represents the relationship between the cost of the laptop and Cole’s payments?
    \n\"EnvisionAnswer:
    \n\"Envision<\/p>\n

    Explanation:
    \nGiven Cole buys a new laptop for $335. He makes a down
    \npayment of $50 and pays the rest in 6 equal monthly payments, p.
    \nThe equation cost = $50 + 6 X p represents
    \nthe relationship between the cost of the laptop and
    \nCole’s payments as shown above.<\/p>\n

    Convince Me!<\/strong>
    \nWhy are both multiplication and addition used in the equation that
    \nrepresents Cole’s monthly payments?
    \nAnswer:
    \nCole’s makes a down payment of $50 first and pays
    \nthe rest in 6 equal monthly payments, p not all at same time,
    \nSo, both multiplication and addition are used in the equation,<\/p>\n

    Explanation:
    \nAs given Cole buys a new laptop for $335.
    \nHe makes a down payment of $50 first and
    \npays the rest in 6 equal monthly payments, p.
    \nSo first we write multiplication as 6 X p then we add
    \ninitial down payment $50 in the equation.<\/p>\n

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

    Marcia and Tamara are running a race. Marcia has run 4 kilometers.
    \nTamara has completed of the race and is 2.5 kilometers ahead of Marcia.
    \nWrite an equation that represents the relationship between the distances each girl has run.
    \nLet k represent the total length of the race in kilometers.
    \nAnswer:
    \nMarica = m = 4km,
    \nk = Tamara = t = 2. 5km + m\u00a0 = 2.5km + 4km = 6.5km,<\/p>\n

    Explanation:
    \nGiven Marcia and Tamara are running a race. Marcia has run 4 kilometers.
    \nTamara has completed of the race and is 2.5 kilometers ahead of Marcia.
    \nLet k represent the total length of the race in kilometers, m for Marica and t for Tamara and
    \nMarica m = 4 km and Tamara and total length of the race
    \nis t or k = 2.5km + 4km = 6.5km,
    \nEquation that represents the relationship between
    \nthe distances each girl has run is\u00a0 k = t = 2.5 km + m.<\/p>\n

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

    At the mall, Claire buys a hat that is 60% off and socks that are reduced to $5.49.
    \nShe spends a total of $9.49. Let x represent the cost of the hat. Which of the following
    \nequations correctly represents Claire’s shopping trip?
    \n\"Envision
    \nAnswer:
    \nEquation correctly representing Claire’s shopping trip is
    \n0.4x + 5.49 = 9.49,<\/p>\n

    Explanation:
    \nGiven at the mall, Claire buys a hat that is 60% off and socks that are reduced to $5.49.
    \nShe spends a total of $9.49.
    \nLet x represent the cost of the hat means and 60% off
    \nso 40%x = 0.4x,
    \nSocks = $5.49 and total = $9.49,
    \nSo the following equation <\/span>correctly represents
    \nClaire’s shopping trip is 0.4x + 5.49 = 9.49.<\/p>\n

    \"Using<\/p>\n

    KEY CONCEPT<\/strong>
    \nYou can write an equation with more than
    \none operation to represent a situation.
    \n\"EnvisionDo You Understand?<\/strong><\/p>\n

    Question 1.
    \nEssential Question How does an equation show the
    \nrelationship between variables and other quantities in a situation?
    \nAnswer:
    \nYes, an equation show the relationship between
    \nvariables and other quantities in a situation with
    \nequality and inequalities signs,<\/p>\n

    Explanation:
    \nAn equation is distinct because it has an equals sign and
    \nthat in itself creates a\u00a0relationship.
    \nUsually, it’s relating the left side\u00a0to\u00a0the right side in
    \nterms\u00a0of\u00a0that fact that they’re equal\u00a0to\u00a0each\u00a0other and
    \nunlike inequalities which form more\u00a0of\u00a0a\u00a0relation between\u00a0two\u00a0quantities\u00a0that
    \ncan be greater than or less than.<\/p>\n

    Question 2.
    \nUse Structure<\/strong>
    \nDo the equations \\(\\frac{1}{5}\\)x + 2 = 6 and
    \n\\(\\frac{1}{5}\\)(x + 2) represent the same situation? Explain.
    \nAnswer:
    \nEquations \\(\\frac{1}{5}\\)x + 2 = 6 and
    \n\\(\\frac{1}{5}\\)(x + 2) do not represent the same situation,<\/p>\n

    Explanation:
    \nGiven equations are
    \n1. \\(\\frac{1}{5}\\)x + 2 = 6 and
    \n2. \\(\\frac{1}{5}\\)(x + 2),
    \n1. If we solve \\(\\frac{1}{5}\\)x + 2 = 6 we get
    \n\\(\\frac{1}{5}\\)x = 6 -2,
    \n\\(\\frac{1}{5}\\)x = 4,
    \nx = 4 X 5 = 20 and
    \n2. If we solve \\(\\frac{1}{5}\\)(x + 2) we get
    \n\\(\\frac{1}{5}\\) X x + \\(\\frac{1}{5}\\) X 2 =
    \n\\(\\frac{1}{5}\\) X x = \\(\\frac{5}{2}\\),
    \nx = \\(\\frac{5}{2}\\) X 5,
    \ntherefore x = \\(\\frac{25}{2}\\) as
    \n20 \u2260\u00a0<\/span>\\(\\frac{25}{2}\\),
    \nSo equations \\(\\frac{1}{5}\\)x + 2 = 6 and
    \n\\(\\frac{1}{5}\\)(x + 2) do not represent the same situation.<\/p>\n

    Question 3.
    \nHow do you decide which operations to use when writing an equation?
    \nAnswer:
    \nThe\u00a0order of operations\u00a0is a rule that tells the
    \ncorrect sequence\u00a0of steps for evaluating a\u00a0math\u00a0expression.
    \nWe can remember the\u00a0order\u00a0using\u00a0PEMDAS:
    \nParentheses, Exponents, Multiplication and
    \nDivision (from left to right), Addition and
    \nSubtraction (from left to right).<\/p>\n

    Explanation:
    \nThe order of operations\u00a0define the priority in which complex\u00a0equations\u00a0are solved.
    \nThe top priority is your parenthesis, then exponents,
    \nfollowed by multiplication and division, and
    \nfinally addition and subtraction (PEMDAS).<\/p>\n

    Do You Know How?<\/strong><\/p>\n

    Question 4.
    \nRita started the day with r apps. Then she deleted 5 apps and still had
    \ntwice as many apps as Cora has. Write an equation that represents the
    \nnumber of apps each girl has.
    \n\"Envision
    \nAnswer:
    \nEquation : r – 5 = 2c,
    \nCora = 36 apps,
    \nRita =77 apps,<\/p>\n

    Explanation:
    \nGiven Rita started the day with r apps.
    \nThen she deleted 5 apps and still had
    \ntwice as many apps as Cora has.
    \nLet us take Cora as c = 36 apps and
    \nRita has twice as Cora after deleting 5 apps
    \nmeans the equation is 2c = r – 5,
    \nand Rita has r = 2c + 5 = 2 X 36 + 5 = 72 + 5 = 77 apps,
    \ntherefore, Equation : r – 5 = 2c,
    \nCora = 36 apps,
    \nRita =77 apps respectively.<\/p>\n

    Question 5.
    \nWrite a problem that could be represented by the equation 5n – 6 = 19.
    \nAnswer:
    \nJim’s age is 6 years less than 5 times
    \nhis younger brother’s age,<\/p>\n

    Explanation:
    \nGiven to write a problem that could be represented
    \nby the equation 5n – 6 = 19, So, lets take Jim’s age at
    \npresent is 19 years which is 6 years less than 5 times
    \nhis younger brother’s age.<\/p>\n

    Question 6.
    \nKayleigh babysat for 11 hours this week. That was 5 fewer than \\(\\frac{2}{3}\\) as
    \nmany hours as she babysat last week, h.
    \nWrite an equation to represent the number of hours she babysat each week.
    \nAnswer:
    \nEquation to represent the number of hours she
    \nbabysat each week is \\(\\frac{2}{3}\\)h – 5 = 11,<\/p>\n

    Explanation:
    \nGiven Kayleigh babysat for 11 hours this week.
    \nThat was 5 fewer than \\(\\frac{2}{3}\\)
    \nas many hours as she babysat last week, h.
    \nThe equation to represent the number of hours she
    \nbabysat each week is \\(\\frac{2}{3}\\)h – 5 = 11.<\/p>\n

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

    Question 7.
    \nA farmer ships oranges in wooden crates. Suppose each orange weighs the same amount.
    \nThe total weight of a crate filled with g oranges is 24.5 pounds. Write an equation that
    \nrepresents the relationship between the weight of the crate and the number of oranges it contains.
    \n\"Envision
    \nAnswer:
    \nEquation that represents the relationship between
    \nthe weight of the crate and the number of
    \noranges it contains is\"Envision<\/p>\n

    Explanation:
    \nGiven a\u00a0farmer ships oranges in wooden crates.
    \nSuppose each orange weighs the same amount as 0.38lb.
    \nThe total weight of a crate filled with g oranges is 24.5 pounds.
    \nSo an equation that represents the relationship between
    \nthe weight of the crate and the number of oranges
    \nit contains is 24.5 = 15lb + (0.38lb X g) and 1 pound = 1 lb.<\/p>\n

    \"Using<\/p>\n

    Question 8.
    \nJordan wrote the following description:
    \nThree fewer than one fourth of x is 12.
    \nWrite an equation to represent the description.
    \nAnswer:
    \nEquation: \\(\\frac{1}{4}\\)x – 3 = 12,<\/p>\n

    Explanation:
    \nGiven Jordan wrote the following description:
    \nThree fewer than one fourth of x is 12.
    \nSo the equation to represent the description is
    \n\\(\\frac{1}{4}\\)x – 3 = 12.<\/p>\n

    Question 9.
    \nAt a graduation dinner, an equal number of guests were seated at each of 3 large tables, and
    \n7 late-arriving guests were seated at a smaller table. There were 37 guests in all
    \n. If n represents the number of people seated at each of the large tables,
    \nwhat equation represents the situation?
    \nAnswer:
    \nThe number of people seated at each of the large tables,
    \nrepresented by an equation is \\(\\frac{1}{3}\\)n + 7 = 37,<\/p>\n

    Explanation:
    \nGiven at a graduation dinner, an equal number of
    \nguests were seated at each of 3 large tables and
    \n7 late-arriving guests were seated at a smaller table.
    \nThere were 37 guests in all. If n represents the
    \nnumber of people seated at each of the large tables,
    \nThe equation representing the situation is
    \n\\(\\frac{1}{3}\\)n + 7 = 37.<\/p>\n

    Question 10.
    \nLast night, 4 friends went out to dinner at a restaurant.
    \nThey split the bill evenly. Each friend paid $12.75 for
    \nhis or her meal and each left the same amount for a tip, t.
    \nThe total dinner bill including the tip was $61.
    \nWhat equation could you use to describe the situation?
    \nAnswer:
    \nEquation: 4 X $12.75 + 4t\u00a0 = $61,<\/p>\n

    Explanation:
    \nGiven last night, 4 friends went out to dinner at a restaurant.
    \nThey split the bill evenly. Each friend paid $12.75 for
    \nhis or her meal and each left the same amount for a tip, t.
    \nThe total dinner bill including the tip was $61.
    \nTherefore, equation to describe the situation is
    \n4 X $12.75 + 4t\u00a0 = $61.<\/p>\n

    Question 11.
    \nMia buys 4\\(\\frac{1}{5}\\) pounds of plums.
    \nThe total cost after using a coupon for 55\u00a2 off her entire
    \npurchase was $3.23. If c represents the cost of the plums
    \nin dollars per pound, what equation could represent the situation?
    \nAnswer:
    \nEquation:\u00a0 $3.23 = 4\\(\\frac{1}{5}\\) – .55,<\/p>\n

    Explanation:
    \nGiven Mia buys 4\\(\\frac{1}{5}\\) pounds of plums.
    \nThe total cost after using a coupon for 55\u00a2 off her entire
    \npurchase was $3.23. If c represents the cost of the plums
    \nin dollars per pound, the equation representing
    \nthe situation is $3.23 = 4\\(\\frac{1}{5}\\) – .55,
    \nas 1 dollar = 100 cents.<\/p>\n

    For 12 and 13, use the equation shown at the right.<\/p>\n

    Question 12.
    \nDescribe a situation that the equation could represent.
    \n\"Envision
    \nAnswer:
    \nThere are 6 groups of children each group there are 15 students
    \nand out of which 3 students are absent, so number of students
    \npresent are g represented by equation\u00a0 g + 3 \u00f7 6 = 15,<\/p>\n

    Explanation:
    \nGiven equation as g + 3 \u00f7 6 = 15, let us take the situation as
    \nthere are 6 groups of children each group there are 15 students
    \nand out of which 3 students are absent,
    \nso number of students present are g, so that g + 3 = 15 X 6,
    \ntherefore g + 3 \u00f7 6 = 15.<\/p>\n

    Given the Question 13.
    \nReasoning Would the situation you wrote for Problem 12
    \nwork if the denominator in the equation were doubled?
    \nExplain why or why not.
    \nAnswer:
    \nYes,<\/p>\n

    Explanation:
    \nReasoning is that if the situation I wrote for Problem 12
    \nwill work if the denominator in the equation were doubled
    \nbecause the value of number of students present
    \nwill change or increase.<\/p>\n

    Question 14.
    \nYou want to buy a pet iguana. You already have $12 and
    \nplan to save $9 per week.
    \n\"Envision
    \na. Model with Math If w represents the number of weeks until
    \nyou have enough money to buy the iguana, what equation
    \nrepresents your plan to afford the iguana?
    \nb. Explain how you could set up an equation to
    \nfind the amount of money you should save each week to
    \nbuy the iguana in 6 weeks.
    \nAnswer:
    \na. Equation : $48 = $12 + $9w,
    \nb. Equation to find the amount(a) of money I should save
    \neach week to buy the iguana in 6 weeks is $48 =\u00a0 $12 + 6a or $60 by 6,<\/p>\n

    Explanation:
    \nGiven I want to buy a pet iguana. I already have $12 and
    \nplan to save $9 per week and iguana costs $48,
    \na. If w represents the number of weeks until
    \nI have enough money to buy the iguana, the equation
    \nrepresenting my plan to afford the iguana is $48 = $12 + $9w,
    \nb. If the amount(a) of money I should save each week to
    \nbuy the iguana in 6 weeks is $48 =\u00a0 $12 + 6a or a = $60 by 6.<\/p>\n

    Question 15.
    \nIn a certain country, the life expectancy of a
    \nwoman born in 1995 was 80.2 years.
    \nBetween 1995 and 2005, the life expectancy
    \nincreased 0.4 year every 5 years.
    \na. If L represents the life expectancy of a woman born in 2005,
    \nwhat equation could you use to represent the situation?
    \nb. Reasoning Could two differences equations be used to
    \nfind the value of L? Explain.
    \nAnswer:
    \na. L = 80.2 + 80.2 X 2(0.4),
    \nb. Yes, Equation 1: L = 80.2 + 80.2 X 2(0.4) and
    \nEquation 2: L – 80.2 = 2 X 0.4 X 80.2,<\/p>\n

    Explanation:
    \nGiven in a certain country, the life expectancy of a
    \nwoman born in 1995 was 80.2 years.
    \nBetween 1995 and 2005, the life expectancy
    \nincreased 0.4 year every 5 years.
    \na. If L represents the life expectancy of a woman born in 2005,
    \nThe equation that could be used to represent the situation is
    \nL = 80.2 + 80.2 X 2(0.4), we multiply 80.2 with 2 of 0.4 as
    \nevery 5 years it is increased by 0.4 and it means from
    \n1995 and 2005 it will increases twice of 0.4,
    \nb. Yes two equations can be used to find the value of L as
    \nFirst we calculate L by Equation 1: L = 80.2 + 80.2 X 2(0.4) and
    \nSecond we calculate L by Equation 2: L – 80.2 = 2 X 0.4 X 80.2.<\/p>\n

    Question 16.
    \nHigher Order Thinking<\/strong> Use the equation 5x \u2013 13 = 12
    \na. Write a description that represents the equation.
    \nb. Of the numbers 1, 2, 3, 4, and 5, which are solutions to the equation?
    \nAnswer:
    \na. Description:
    \nThere are few sets of books as x in the library and
    \neach set contains 5 books in that 13 books children
    \ntook to home and rest are 12 books in the libray,
    \nSo find out how many number of books x were there\u00a0 in the library,
    \nb. Solution to the equation is 5,<\/p>\n

    Explanation:
    \nGiven the equation 5x – 13 = 12,
    \na. Wrote the description that represents the equation as
    \nthere are few sets of books as x in the library and
    \neach set contains 5 books in that 13 books children
    \ntook to home and rest are 12 books in the libray,
    \nSo find out how many number of books x were there\u00a0 in the library,
    \nb. Of the numbers 1, 2, 3, 4, and 5, the solution to the equation
    \n5x – 13 = 12 is 5x = 12 + 13,
    \n5x = 25, therefore x = 25 by 5 = 5, therefore solution to
    \nthe equation 5x – 13 = 12 is 5.<\/p>\n

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

    Question 17.
    \nA garden contains 135 flowers, each of which is either red or yellow.
    \nThere are 3 beds of yellow flowers and 3 beds of red flowers.
    \nThere are 30 yellow flowers in each yellow flower bed.
    \nPART A<\/strong>
    \nIf r represents the number of red flowers in each red flower bed,
    \nwhat equation could you use to represent the number of red and yellow flowers?<\/p>\n

    PART B<\/strong>
    \nWrite another real-world situation that your equation from Part A could represent.
    \nAnswer:
    \nPart A:
    \nEquation to represent the number of red and
    \nyellow flowers is 135 = 3 X 30 + 3r,<\/p>\n

    Part B:
    \nReal World situation :
    \nThere are 135 number of mangoes with fruit seller out of
    \nwhich 3 baskets mangoes are green in color and 3 baskets of
    \nmangoes are yellow in color and there are total 30 yellow mangoes
    \nin each yellow basket, write an equation to represent the number of
    \ngreen and yellow mangoes?,<\/p>\n

    Explanation:
    \nGiven a\u00a0garden contains 135 flowers, each of which is either red or yellow.
    \nThere are 3 beds of yellow flowers and 3 beds of red flowers.
    \nThere are 30 yellow flowers in each yellow flower bed.
    \nPART A<\/strong>
    \nIf r represents the number of red flowers in each red flower bed,
    \nthe equation I could use to represent the number of red and
    \nyellow flowers is 135 = 3 X 30 + 3r and
    \nPART B<\/strong>
    \nWrote another real-world situation that my equation
    \nfrom Part A could represent as Real World situation :
    \nThere are 135 number of mangoes with fruit seller out of
    \nwhich 3 baskets mangoes are green in color and 3 baskets of
    \nmangoes are yellow in color and there are total 30 yellow mangoes
    \nin each yellow basket, write an equation to represent the number of
    \ngreen and yellow mangoes? respectively.<\/p>\n

    Lesson 5.2 Solve Two-Step Equations<\/h3>\n

    Elizabeth wrote the following clues. What is the relationship between the shapes?
    \n\"Envision
    \nAnswer:
    \n1. 4c + t = 2s + t, 2 circles = square
    \n2. 3t + 2c = 2c + s, 3 triangles = square
    \n3. s + 2c = 6t, square = 6 triangles – 2 circles,<\/p>\n

    Explanation:
    \nElizabeth wrote the clues as
    \n1. 4 circles + 1 triangle = 2 squares and 1 triangle which means
    \n4 circles = 2 squares, so 1 square is equal to 2 circles.
    \n2. 3 triangles + 2 circles = 2 circles + 1 square,
    \nso 1 square is equal to 3 triangles,
    \n3. 1 square + 2 circles = 6 triangles,
    \nso, 1 square = 6 triangles – 2 circles.<\/p>\n

    Use Structure<\/strong>
    \nHow can you use properties of equality to reason about these equations?
    \nAnswer:
    \n1. Addition Property,
    \n2. Addition property,
    \n3. Substitution property,<\/p>\n

    Explanation:
    \nAs we know
    \n1. 4c + t = 2s + t,
    \nAddition Property says for all real numbers x, y, and z ,
    \nif x=y , then x + z = y + z, So 4c + t = 2s + t means 4c = 2s,
    \n2. 3t + 2c = 2c + s means as per Addition Property
    \n3t = s,
    \n3. 1 square + 2 circles = 6 triangles we use
    \nsubstitution property for circles and triangles as
    \n1 square = 2 circles and 1 square = 3 triangles,
    \nSo 1 square + 1 square = 2 X 3 triangles =\u00a0 2 squares,
    \n2 squares = 2 squares.<\/p>\n

    Focus on math practices<\/strong>
    \nLook for Relationships Complete the equation with only triangles
    \nusing the relationships from the clues shown above.
    \n\"Envision
    \nAnswer:
    \nEquation : 4c + 1t = 7t,<\/p>\n

    Explanation:
    \nAs we know 4c = 2s and 1s = 3t means 1s = 2c,
    \nso 3t = 2c therefore 4c = 6t as given 4c +1t = 6t + 1t = 7t
    \ntherefore the equation with only triangles
    \nusing the relationships from the clues shown above is 4c + 1t = 7t.<\/p>\n

    Essential Question<\/strong>
    \nHow is solving a two-step equation similar to solving a one-step equation?
    \nAnswer:
    \nYes, solving a two-step equation is similar to solving a one-step equation,<\/p>\n

    Explanation:
    \nSolving a one-step or two-step equation:
    \nIn solving an equation is to have only variables on
    \none side of the equal sign and numbers on the
    \nother side of the equal sign.
    \nThe other alike is to have the number in front of the variable
    \nequal to one the variable does not always have to be x.
    \nThese equations can use any letter as a variable.<\/p>\n

    Try It!<\/strong>
    \nAndrew rents bowling shoes for $4. He bowls 2 games.
    \nAndrew spent a total of $22. How much was the cost of each game, b?
    \nComplete the bar diagrams, and then solve the problem.
    \n\"Envision
    \nAnswer:
    \n\"\"<\/p>\n

    Explanation:
    \nGiven Andrew rents bowling shoes for $4. He bowls 2 games.
    \nAndrew spent a total of $22. How much was the cost of each game, b,
    \nCompleted the bar diagrams as shown above,
    \nTotal spent = shoe rental + 2 X cost of each game,
    \n$22 = $4 + 2 X b,
    \n$22 – $4 = $4 + 2b – $4,
    \n$18 = 2b,
    \nNow each game cost is
    \n$18 by 2 = 2b by 2
    \nb = $9, therefore cost of each game is $9.<\/p>\n

    Convince Me!<\/strong>
    \nWhat were the two steps you used to solve this equation?
    \nAnswer:
    \nTwo step equations can be solved in two steps using two different
    \nproperties of equality, I used subtraction property and Division property,<\/p>\n

    Explanation:
    \nTotal spent = shoe rental + 2 X cost of each game is given
    \n$22 = $4 + 2 X b,
    \nFirst we use the subtraction property of equality to
    \nisolate the term containing the variable,
    \n$22 – $4 = $4 + 2b – $4,
    \n$18 = 2b,
    \nSecondly we use the division property of equality to
    \nisolate the variable or get the variable by itself on
    \none side of the equation as
    \n$18 by 2 = 2b by 2
    \nb = $9, therefore cost of each game is $9.<\/p>\n

    Try It!<\/strong>
    \nKirsty ran 24 laps in a charity run and then walked
    \n0.2 kilometer to the presentation table.
    \nThe total distance Kirsty traveled was 29.6 kilometers.
    \nWhat was the distance of each lap? Explain how you solved the problem.
    \nAnswer:
    \n1.225km is equal to each laps,<\/p>\n

    Explanation:
    \nGiven Kristy total distance traveled = 29.6km
    \nand Kristy ran 24 laps and later walk 0.2km.
    \nIt simply implies that
    \nTotal distance traveled = distance covered running + distance covered walking,
    \nSince we know that Total distance traveled = 29.6km,
    \ndistance covered running = 24 laps,
    \ndistance covered walking = 0.2km,
    \nDistance covered running in km = total distance traveled – distance covered walking,
    \nDistance covered running in km = 29.6km – 0.2km = 29.4km.
    \nTo now find the distance for each lap.
    \nSince we have: Distance covered running in km = 29.4km.
    \nDistance covered running in lap = 24 laps i.e 24 laps = 29.4km,
    \n1 lap = x Use cross-multiple 24 laps X x =
    \n29.4km \u00d7 1 lap x = 29.4km \/ 24 x = 1.225km,
    \nTherefore 1.225km is equal to each laps.<\/p>\n

    KEY CONCEPT<\/strong>
    \nThe properties of equality can be applied the same way when
    \nsolving two-step equations as when solving one-step equations.
    \n\"Envision
    \nAnswer:
    \nYes, The properties of equality can be applied the same way when
    \nsolving two-step equations as when solving one-step equations.
    \nTwo step equations can be solved in two steps using two different
    \nproperties of equality, We used subtraction property and division property,<\/p>\n

    Explanation:
    \nWe have 5x + 27 = 122,
    \nFirst we use the subtraction property of equality to
    \nisolate the term containing the variable,
    \n5x + 27 – 27 = 122 – 27,
    \n5x = 95,
    \nSecondly we use the division property of equality to
    \nisolate the variable or get the variable by itself on
    \none side of the equation as 5x by 5 = 95 by 5
    \nx = 19, therefore the properties of equality can be applied
    \nthe same way when solving two-step equations as
    \nwhen solving one-step equations.<\/p>\n

    Do You Understand?<\/strong><\/p>\n

    Question 1.
    \nEssential Question<\/strong> How is solving a two step equation similar to
    \nsolving a one-step equation?
    \nAnswer:
    \nYes, solving a two-step equation is similar to solving a one-step equation,<\/p>\n

    Explanation:
    \nSolving a one-step or two-step equation:
    \nIn solving an equation is to have only variables on
    \none side of the equal sign and numbers on the
    \nother side of the equal sign.
    \nThe other alike is to have the number in front of the variable
    \nequal to one the variable does not always have to be x.
    \nThese equations can use any letter as a variable.<\/p>\n

    Question 2.
    \nUse Structure<\/strong> Preston uses the bar diagram below to represent 4x \u2013 3 = 13.
    \nHow would you use the bar diagram to solve for x?
    \n\"Envision
    \nAnswer:
    \nBy using bar diagram we solve x = 4,<\/p>\n

    Explanation:
    \nA helpful way to\u00a0solve problem\u00a0is to use a\u00a0bar diagram.
    \nA\u00a0bar\u00a0is used to represent the whole. So\u00a0we\u00a0label it
    \nwith the amount of the whole. Finally, we\u00a0indicate the required
    \nnumber of parts with a question mark and determine the value,
    \nBy seeing bar diagram we have the total + 3 is divided into
    \n4 parts of x now we solve x as 4x – 3 = 13 by adding 3 both sides,
    \n4x – 3 + 3 = 13 + 3,
    \n4x = 16, Now dividing both sides by 4 we get
    \n4x \u00f7 4 = 16 \u00f7 4,
    \nx = 4.<\/p>\n

    Question 3.
    \nClara has solved the problem 6p – 12 = 72 and says that p = 14.
    \nHow can you check to see if Clara is correct?
    \nAnswer:
    \nBy solving using two-step equation we get p = 14,
    \nwe check Clara is correct,<\/p>\n

    Explanation:
    \nGiven Clara equation as 6p – 12 = 72 we solve
    \nFirst by the addition property of equality to
    \nisolate the term containing the variable,
    \n6p – 12 + 12 = 72 + 12,
    \n6p = 84,
    \nSecondly use the division property of equality to
    \nisolate the variable or get the variable by itself on
    \none side of the equation as
    \n6p by 6 = 84 by 6, we get p = 14, so checked Clara is correct.<\/p>\n

    Do You Know How?<\/strong><\/p>\n

    Question 4.
    \nClyde is baking, and the recipe requires 1\\(\\frac{1}{3}\\) cups of flour.
    \nClyde has 2 cups of flour, but he is doubling the recipe to make twice as much.
    \nHow much more flour does Clyde need?
    \n\"Envision
    \na. Write an equation to represent the problem.
    \nLet c represent the amount of flour Clyde needs.
    \nb. Solve the equation.
    \nAnswer:
    \na. Equation: c = 2 X 1\\(\\frac{1}{3}\\) – 2,
    \nb. Clyde needs \\(\\frac{2}{3}\\) cup more,<\/p>\n

    Explanation:
    \nGiven Clyde is baking, and the recipe requires 1\\(\\frac{1}{3}\\) cups of flour.
    \nClyde has 2 cups of flour, but he is doubling the recipe to make twice as much.
    \nIf c represent the amount of flour Clyde needs the equation is
    \nc = 2 X 1\\(\\frac{1}{3}\\) – 2,
    \nb. Now solving c = 2 X \\(\\frac{1 X 3 + 1}{3}\\) – 2,
    \nc = 2 X \\(\\frac{4}{3}\\) – 2,
    \nc = \\(\\frac{2 X 4}{3}\\) – 2,
    \nc = \\(\\frac{8}{3}\\) – 2,
    \nc = \\(\\frac{8 – 6}{3}\\),
    \nc = \\(\\frac{2}{3}\\) .<\/p>\n

    Question 5.
    \nFour times a number, n, added to 3 is 47.
    \na. Write an equation that you can use to find the number.
    \nb. What is the number represented by n?
    \nAnswer:
    \na. Equation: 4n + 3 = 47,
    \nb. The number represented by n is 11,<\/p>\n

    Explanation:
    \nGiven four times a number n added to 3 is 47 means
    \na. 4 X n + 3 is equal to 47,
    \ntherefore the equation is 4n + 3 = 47,
    \nb. Upon solving 4n + 3 = 47 we get value of n, So
    \nfirst we subtract 3 both sides as
    \n4n + 3\u00a0 – 3 = 47 – 3,
    \n4n = 44, now we divide both sides by 4 we get
    \n4n \u00f7 4 = 44 \u00f7 4, so n = 11, therefore the number
    \nrepresented by n is 11.<\/p>\n

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

    Question 6.
    \nUse the bar diagram to help you solve the equation 4x \u2013 12 = 16.
    \n\"Envision
    \nAnswer:
    \nBy using bar diagram we solve x = 7,<\/p>\n

    Explanation:
    \nA helpful way to\u00a0solve problem\u00a0is to use a\u00a0bar diagram.
    \nA\u00a0bar\u00a0is used to represent the whole. So\u00a0we\u00a0label it
    \nwith the amount of the whole. Finally, we\u00a0indicate the required
    \nnumber of parts with a question mark and determine the value,
    \nBy seeing bar diagram we have the total + 12 is divided into
    \n4 parts of x now we solve x as 4x – 12 = 16 by adding 12 both sides,
    \n4x – 12 + 12 = 16 + 12,
    \n4x = 28, Now dividing both sides by 4 we get
    \n4x \u00f7 4 = 28 \u00f7 4,
    \nx = 7.<\/p>\n

    Question 7.
    \nComplete the steps to solve the equation.
    \n\"Envision
    \nAnswer:
    \n\"Envision
    \nExplanation:
    \nCompleted the steps to solve the equation as shown above
    \nfirst we subtract 2 both sides as
    \n1\/5t + 2 – 2 = 17 – 2,
    \n1\/5t = 15, now we multiply both sides by 5 we get
    \n1\/5t = 15 X 5,\u00a0 we get t = 75.<\/p>\n

    Question 8.
    \nUse the bar diagram to write an equation. Then solve for x.
    \n\"Envision
    \nAnswer:
    \nEquation is 3x – 5 = 7 and x = 4,<\/p>\n

    Explanation:
    \nUsing the bar diagram we have 3 times x and subtracting 5
    \nwe are getting total as 7, so the equation will be
    \n3x – 5 = 7, now solving to get x first we add 5 both sides as
    \n3x – 5 + 5 = 7 + 5,
    \n3x = 12, now we divide both sides by 3,
    \n3x \u00f7 3 = 12 \u00f7 3, we get x = 4.<\/p>\n

    Question 9.
    \nWhile shopping for clothes, Tracy spent $38 less than 3 times what Daniel spent.
    \nWrite and solve an equation to find how much Daniel spent.
    \nLet x represent how much Daniel spent.
    \n\"Envision
    \nAnswer:
    \nEquation for Daniel spent is : 3x – 38 = 10,
    \nDaniel spent is $16,<\/p>\n

    Explanation:
    \nGiven while shopping for clothes, Tracy spent $38
    \nless than 3 times what Daniel spent.
    \nLet x represent how much Daniel spent.
    \nSo an equation to find how much Daniel spent is
    \n3x – 38 = 10, now we solve by adding 38 both sides as
    \n3x – 38 + 38 = 10 + 38,
    \n3x = 48, now we divide both sides by 3,
    \n3x \u00f7 3 = 48 \u00f7 3, we get x = 16,
    \ntherefore, Daniel spent is $16.<\/p>\n

    Question 10.
    \nSolve the equation 0.5p – 3.45 = -1.2.
    \nAnswer:
    \np = 4.5,<\/p>\n

    Explanation:
    \nGiven to solve the equation 0.5p – 3.45 = -1.2, first
    \nwe add +3.45 both side as
    \n0.5p – 3.45 + 3.45 = -1.2 + 3.45,
    \n0.5p = 2.25, now we divide both sides by 0.5 as
    \n0.5p\u00a0 \u00f7 0.5 = 2.25 \u00f7 0.5, we get p = 4.5.<\/p>\n

    Question 11.
    \nSolve the equation \\(\\frac{n}{10}\\) + 7 = 10.
    \nAnswer:
    \nn = 30,<\/p>\n

    Explanation:
    \nGiven to solve the equation \\(\\frac{n}{10}\\) + 7 = 10,
    \nfirst we subtract 7 both sides as
    \n\\(\\frac{n}{10}\\) + 7 – 7\u00a0 = 10 – 7,
    \n\\(\\frac{n}{10}\\) = 3, now we multiply both sides by 10 as
    \n\\(\\frac{n}{10}\\) X 10 = 3 X 10,
    \nwe get n = 30.<\/p>\n

    Question 12.
    \nA group of 4 friends went to the movies. In addition to their tickets,
    \nthey bought a large bag of popcorn to share for $6.25. The total was $44.25.
    \na. Write and solve an equation to find the cost of one movie ticket, m.
    \nb. Draw a model to represent the equation.
    \nAnswer:
    \na. 4m + 6.25 = 44.25,
    \nThe cost of one movie ticket m is $9.5,
    \nb.
    \n\"Envision<\/p>\n

    Explanation:
    \nGiven a\u00a0group of 4 friends went to the movies.
    \nIn addition to their tickets, they bought a large bag
    \nof popcorn to share for $6.25. The total was $44.25.
    \na. Wrote and solved an equation to find the cost of
    \none movie ticket, m as 4m + 6.25 = 44.25, first we subtract
    \n6.25 both sides as 4m + 6.25 – 6.25 = 44.25 – 6.25,
    \n4m = 38, now we divide both sides by 4 as
    \n4m \u00f7 4 = 38 \u00f7 4, we get m = $9.5.
    \nb. Drawn a bar diagram model to represent the equation
    \nas shown above.<\/p>\n

    Question 13.
    \nOliver incorrectly solved the equation 2x + 4 = 10.
    \nHe says the solution is x = 7.
    \na. What is the correct solution?
    \nb. What mistake might Oliver have made?
    \nAnswer:
    \na. The correct solution is x = 3,
    \nb. Instead of subtracting 4 both sides Oliver added
    \n4 both sides,<\/p>\n

    Explanation:
    \nGiven Oliver incorrectly solved the equation 2x + 4 = 10 as
    \nhe says the solution is x = 7,
    \na. The correct solution is first subtract 4 both sides as
    \n2x + 4 – 4 = 10 – 4,
    \n2x = 6 now divide both sides by 2 as
    \n2x \u00f7 2 = 6 \u00f7 2, we get x = 3,
    \nb. Mistake Oliver had made is if he says
    \nsolution is x = 7 and after solving we are getting
    \nx as 3 means he added more 4 both sides.<\/p>\n

    Question 14.
    \nUse the equation 4.9x \u2013 1.9 = 27.5.
    \na. Make Sense and Persevere What two properties of equality
    \ndo you need to use to solve the equation?
    \nb. The solution is x = \"Envision.
    \nAnswer:
    \na. We need two properties of equality as
    \n1. Addition property of equality and
    \n2. Division property of equality,
    \nb. The solution is x =\u00a0 6,<\/p>\n

    Explanation:
    \nGiven the equation 4.9x \u2013 1.9 = 27.5,
    \ntwo properties of equality we need to use to solve the equation is
    \nfirst addition property of equality by adding 1.9 both sides as
    \n4.9x – 1.9 + 1.9 = 27.5 + 1.9,
    \n4.9x = 29.4 now we use division property of equality by
    \ndividing 4.9 both sides as
    \n4.9x \u00f7 4.9 =29.4 \u00f7 4.9, we get x = 6,
    \nb. As solved in bit a. we get the solution for x as 6.<\/p>\n

    Question 15.
    \nHigher Order Thinking<\/strong>
    \nAt a party, the number of people who ate meatballs was
    \n11 fewer than \\(\\frac{1}{3}\\) of the total number of people.
    \nFive people ate meatballs
    \n\"Envision
    \na. Write and solve an equation to find the number of people at the party.
    \nLet x represent the number of people at the party.
    \nb. Write a one-step equation that has the same solution.
    \nAnswer:
    \na. Equation: 5 = \\(\\frac{1}{3}\\) x – 11,
    \nThe number of people at the party are x = 48,
    \nb. One-step equation that has the same solution is
    \n\\(\\frac{1}{3}\\) x\u00a0 \u00a0= 16,<\/p>\n

    Explanation:
    \nAt a party, the number of people who ate meatballs was
    \n11 fewer than \\(\\frac{1}{3}\\) of the total number of people.
    \nFive people ate meatballs, So equation to find the number of
    \npeople at the party.
    \na. Let x represent the number of people at the party. Therefore
    \nEquation is \\(\\frac{1}{3}\\) x – 11 = 5,
    \nb. As we have 5 people ate meatballs and 11 are fewer than \\(\\frac{1}{3}\\)
    \nof the total number of people. Total number of people a are equal to 16,
    \ntherefore one-step equation that has the same solution is
    \n\\(\\frac{1}{3}\\) x\u00a0 \u00a0= 16.<\/p>\n

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

    Question 16.
    \nIn a week, Tracy earns $12.45 less than twice the amount Kayla earns.
    \nTracy earns $102.45. How much does Kayla earn?
    \nAnswer:
    \nKayla earns = $57.45,<\/p>\n

    Explanation:
    \nGiven in a week, Tracy earns $12.45 less than twice the amount Kayla earns.
    \nLet Kayla earns x and Tracy earns\u00a0Earns in week= $102.45,
    \nTracy = 2x – 12.45,
    \n102.45 = 2x – 12.45,
    \n2x = 102.45 + 12.45,
    \n2x = 114.90,
    \nx= 114.90\/2,
    \nx = $57.45, therefore Kayla earns 57.45 dollars.<\/p>\n

    Question 17.
    \nSolve the equation 2x + 4\\(\\frac{1}{5}\\) = 9.
    \nExplain the steps and properties you used.
    \nAnswer:
    \nx = \\(\\frac{12}{5}\\),
    \nProperties used are 1. Subtraction property of equation and
    \n2. Division property of equation,<\/p>\n

    Explanation:
    \nGiven the equation as 2x + 4\\(\\frac{1}{5}\\) = 9,
    \n2x + \\(\\frac{4 X 5 + 1}{5}\\) = 9,
    \n2x + \\(\\frac{21}{5}\\) = 9, Now we use subtraction property of equation
    \nboth sides we subtract \\(\\frac{21}{5}\\) as
    \n2x + \\(\\frac{21}{5}\\) – \\(\\frac{21}{5}\\) = 9 – \\(\\frac{21}{5}\\),
    \n2x = \\(\\frac{45 – 21}{5}\\),
    \n2x = \\(\\frac{24}{5}\\), Now we use division property of equation so we
    \ndivide both sides by 2 as 2x\/2 = \\(\\frac{24}{5}\\)\/2, we get
    \nx = \\(\\frac{12}{5}\\).<\/p>\n

    Lesson 5.3 Solve Equations Using the Distributive Property<\/h3>\n

    Explain It!
    \n<\/strong>Six friends go jet skiing. The total cost for the adventure is $683.88,
    \nincluding a $12 fee per person to rent flotation vests.
    \nMarcella says they can use the equation 6r + 12 = 683.88
    \nto find the jet ski rental cost, r, per person.
    \nJulia says they need to use the equation 6(r + 12) = 683.88.
    \n\"Envision
    \nA. Construct Arguments Whose equation accurately represents the situation?
    \nConstruct an argument to support your response.
    \nB. What error in thinking might explain the inaccurate equation?
    \nAnswer:
    \nA. Argument : Julia is right the equation 6(r +12) = 683.33 is true,
    \nB. Error in thinking may be instead of 6 persons X $12 fee
    \nper person to rent flotation vests, Marcella says they
    \ncan use the equation 6r + 12 = 683.88 not 6r + 6 X 12 = 683.88,<\/p>\n

    Explanation: Given Six friends go jet skiing. The total cost
    \nfor the adventure is $683.88,
    \nincluding a $12 fee per person to rent flotation vests.
    \nMarcella says they can use the equation 6r + 12 = 683.88
    \nto find the jet ski rental cost, r, per person.
    \nJulia says they need to use the equation 6(r + 12) = 683.88.
    \nA. Argument : Julia is right the equation 6(r +12) = 683.33 is true,
    \nbecause the total cost for the adventure is $683.88,
    \nincluding a $12 fee per person to rent flotation vests
    \nmeans 6 multiply by r in addition to 6 multiply by $12 is
    \nequal to total cost, but not Marcella as she says they
    \ncan use the equation 6r + 12 = 683.88,
    \nB. Error in thinking may be instead of 6 persons X $12 fee per
    \nperson to rent flotation vests,\u00a0 Marcella says they can use
    \nthe equation 6r + 12 = 683.88 means she is not considering 6
    \npersons rent flotation vests only for one person she is considering,
    \ni.e Marcella is saying in the equation for 1 person rent flotation vests
    \ninstead of for six persons which makes inaccurate equation.<\/p>\n

    Focus on math practices<\/strong>
    \nUse Structure How can you use the correct equation to
    \ndetermine the jet ski rental cost per person?
    \nAnswer:
    \nThe correct equation is 6(r + $12) = $683.88 and
    \nthe jet ski rental cost per person is $101.98,<\/p>\n

    Explanation:
    \nGiven six friends go jet skiing. The total cost for the adventure is $683.88,
    \nincluding a $12 fee per person to rent flotation vests.
    \nWe have correct equation as 6(r + $12) = $683.88,
    \nr + 12 = 683.88 \u00f7 6,
    \nr = (683.88 \u00f7 6) – 12,
    \nSo the correct equation to determine the jet ski rental cost per person is
    \nr =\u00a0 $113.98 – $12 = $101.98, therefore The correct equation is 6(r + $12) = $683.88 and
    \nthe jet ski rental cost per person is $101.98.<\/p>\n

    Essential Question<\/strong>
    \nHow does the Distributive Property help you solve equations?
    \nAnswer:
    \nDistributive property helps in simplifying the problems by
    \nbreaking the expressions into addition or subtraction.
    \nThe distributive property states that when a factor is multiplied
    \nby the sum or subtract of two numbers, we can multiply each of the two numbers
    \nby that factor and then add or subtract them.<\/p>\n

    Explanation:
    \nDistributive property is defined as the algebraic property used
    \nto multiply two or more numbers within the parenthesis.
    \nWe can say that the distributive property helps in simplifying
    \nthe problems by breaking the expressions into addition or subtraction.
    \nIt multiplies the number outside parentheses which is equal
    \nto the addition or subtraction of product.
    \nApart from distributive property, there are two types of properties
    \nknown as Commutative and Associative.
    \nThis property of multiplication is basically used in addition or subtraction.
    \nTo solve equation of distributive property, we need to expand the equation,
    \nfind the products and add or subtract.<\/p>\n

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

    A collector has a box of 32 figurines. The value of each figurine
    \nincreased by $2.32 over the past year. The box of figurines is now
    \nworth $114.24. What was the original cost, x, of one figurine?
    \nThe original cost of one figurine was \"Envision.
    \n\"Envision
    \nAnswer:
    \nThe original cost of one figurine was $1.25<\/p>\n

    \"Envision
    \nExplanation:
    \nGiven a collector has a box of 32 figurines.
    \nThe value of each figurine increased by $2.32 over the past year.
    \nThe box of figurines is now worth $114.24.
    \nThe original cost of one figurine was
    \n32(x + 2.32) = 114.24, Applying distributive property as
    \n32 X x + 32 X 2.32 = 114.24,
    \n32x + 74.24 = 114.24,
    \n32x = 114.24 – 74.24,
    \n32x = 40,
    \nx = 40 \u00f7 32,
    \nx= 1.25.<\/p>\n

    Convince Me!<\/strong>
    \nCan the equation 32x + 2.32 = 114.24 be used to
    \nfind the original cost of each figurine in the problem above? Explain.
    \nAnswer:
    \nNo, the equation 32x + 2.32 = 114.24 cannot be used to
    \nfind the original cost of each figurine in the problem above,<\/p>\n

    Explanation:
    \nAs the given equation is 32x + 2.32 = 114.24 cannot be used to
    \nfind the original cost of each figurine in the problem above
    \nbecause here in the equation it is not using the value of
    \neach figurine increased by $2.32 over the past year,
    \ninstead of using for 32 figurines it is using only for 1 figurine
    \nit has to use as 32(x + 2.32) = 114.24 not 32x + 2.32 = 114.24,
    \ntherefor, the equation 32x + 2.32 = 114.24 cannot be used to
    \nfind the original cost of each figurine in the problem above.<\/p>\n

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

    Use the Distributive Property to solve each equation.
    \na. –\\(\\frac{1}{2}\\)(b \u2013 6) = 5,
    \nAnswer:
    \nb = -2,<\/p>\n

    Explanation:
    \nGiven equation as –\\(\\frac{1}{2}\\)(b \u2013 6) = 5 using
    \ndistributive property as –\\(\\frac{1}{2}\\) X b + (-\\(\\frac{1}{2}\\)) X (-6) = 5,
    \n–\\(\\frac{b}{2}\\) + \\(\\frac{6}{2}\\) = 5,
    \n–\\(\\frac{b}{2}\\) + 3 = 5,
    \n–\\(\\frac{b}{2}\\) = 5 – 3,
    \n–\\(\\frac{b}{2}\\) = 2,
    \n-b = 2 X 2,
    \n– b =\u00a0 4, therefore b = -4.<\/p>\n

    b. 0.4(x \u2013 0.45) = 9.2,
    \nAnswer:
    \nx = 23.45,<\/p>\n

    Explanation:
    \nGiven equation as 0.4(x-0.45) = 9.2 using distributive property as
    \n0.4 X x – 0.4 X 0.45 = 9.2,
    \n0.4x – 0.18 = 9.2,
    \n0.4x = 9.38,
    \nx = 9.38 \u00f7 0.4,
    \nx = 23.45.<\/p>\n

    c. -4(p – 212) = 44,
    \nAnswer:
    \np = 201,<\/p>\n

    Explanation:
    \nGiven equation as -4(p-212) = 44 using distributive property as
    \n-4 X p – 4 X -212 = 44,
    \n-4p + 848 = 44,
    \n-4p = 44 – 848,
    \np = -804 \u00f7 -4,
    \np = 201.<\/p>\n

    KEY CONCEPT<\/strong><\/p>\n

    When solving equations written in the form p(x + 9) = r,
    \nyou can use the Distributive Property to multiply the
    \ntwo terms in the parentheses by the term outside the parentheses.
    \n\"Envision<\/p>\n

    Do You Understand?<\/strong><\/p>\n

    Question 1.
    \nEssential Question<\/strong> How does the Distributive Property help you solve equations?
    \nAnswer:
    \nDistributive property helps in simplifying the problems by
    \nbreaking the expressions into addition or subtraction.
    \nThe distributive property states that when a factor is multiplied
    \nby the sum or subtract of two numbers,
    \nwe can multiply each of the two numbers
    \nby that factor and then add or subtract them,
    \nfor equation 6(x + 8.5) = 123 we use distributive property
    \nwe get x = 12,<\/p>\n

    Explanation:
    \nDistributive property is defined as the algebraic property used
    \nto multiply two or more numbers within the parenthesis.
    \nWe can say that the distributive property helps in simplifying
    \nthe problems by breaking the expressions into addition or subtraction.
    \nIt multiplies the number outside parentheses which is equal
    \nto the addition or subtraction of product.
    \nApart from distributive property, there are two types of properties
    \nknown as Commutative and Associative.
    \nThis property of multiplication is basically used in addition or subtraction.
    \nTo solve equation of distributive property, we need to expand the equation,
    \nfind the products and add or subtract for equation 6(x + 8.5) = 123
    \nwe use distributive property as 6x + 6 X 8.5 = 123,
    \n6x + 51 = 123,
    \n6x = 123 -51,
    \n6x = 72,
    \nx = 72 \u00f7 6 = 12.<\/p>\n

    Question 2.
    \nMake Sense<\/strong> and Persevere<\/strong> How are the terms in parentheses
    \naffected when multiplied by a negative coefficient when the
    \nDistributive Property is applied?
    \nAnswer:
    \nIf a number outside the\u00a0parentheses\u00a0has a\u00a0negative\u00a0sign then
    \nthe first and simplest way is to change each\u00a0positive\u00a0or\u00a0negative\u00a0sign
    \nof the\u00a0terms\u00a0that were inside the\u00a0parentheses.
    \nNegative\u00a0or minus signs become\u00a0positive\u00a0or plus signs.
    \nSimilarly,\u00a0positive\u00a0or plus signs become\u00a0negative or minus signs,<\/p>\n

    Explanation:
    \nThe terms in parentheses affected when multiplied by a
    \nnegative coefficient when the distributive property is applied is the first and
    \nsimplest way is to change each\u00a0positive\u00a0or\u00a0negative\u00a0sign
    \nof the\u00a0terms\u00a0that were inside the\u00a0parentheses.
    \nNegative\u00a0or minus signs become\u00a0positive\u00a0or plus signs.
    \nSimilarly,\u00a0positive\u00a0or plus signs become\u00a0negative or minus signs,
    \nExample : If equation is -6(x – 4) if we apply distributive property
    \n-6 X x\u00a0 -6 X -4,
    \n-6x + 24, means positive x becomes negative 6x and negative 4
    \nhas become positive 24 when multiplied by -6.<\/p>\n

    Question 3.
    \nReasoning How can an area model help you set up an equation
    \nfor a problem situation?
    \nAnswer:
    \nArea of a shape is the space occupied by the shape.
    \nThe area of the given shape is the shaded part which means
    \nwe can set up an equation for a problem situation given,<\/p>\n

    Explanation:
    \n\"Envision
    \nAs shown above the area model for the equation is
    \n(2x + 5)(x+3) as shown.<\/p>\n

    Do You Know How?<\/strong><\/p>\n

    Question 4.
    \nA family of 7 bought tickets to the circus. Each family member also
    \nbought a souvenir that cost $6. The total amount they spent was $147.
    \nHow much did one ticket cost?
    \nAnswer:
    \nEquation : 7(t + 6) = $147,
    \nThe cost of one ticket is $15,<\/p>\n

    Explanation:
    \nGiven a\u00a0family of 7 bought tickets to the circus. Each family member also
    \nbought a souvenir that cost $6. The total amount they spent was $147.
    \nSo the equation is 7(t + 6) = 147 on solving we get cost of one ticket as
    \n7t + 42 = 147,
    \n7t = 147 – 42,
    \n7t = 105,
    \nt = 105\/7 = 15,
    \ntherefore cost of one ticket is $15.<\/p>\n

    Question 5.
    \nDavid reads the problem:
    \nAlly bought a T-shirt and a pair of shorts on sale,
    \nwhich reduced prices by \\(\\frac{1}{4}\\).
    \nThe total savings on the two garments was $10.25.
    \nFind the original price for the pair of shorts.
    \n\"Envision
    \nDavid says that the original price of the shorts was $41.
    \nDoes his answer seem reasonable?
    \nDefend your answer by writing and solving an
    \nequation that represents the situation.
    \nAnswer:
    \nNo, David answer is not reasonable,
    \nthe correct equation is \\(\\frac{1}{4}\\) (18 + s) = 10.25,
    \nwhere s is price of the shorts,<\/p>\n

    Explanation:
    \nGiven David reads the problem:
    \nAlly bought a T-shirt and a pair of shorts on sale,
    \nwhich reduced prices by \\(\\frac{1}{4}\\).
    \nThe total savings on the two garments was $10.25.
    \nFind the original price for the pair of shorts.
    \nDavid says that the original price of the shorts was $41.
    \nNow we will check let us take s for price of the shorts, and
    \nprice of T-shirt is $18,
    \nSo the equation is \\(\\frac{1}{4}\\) (18 + s) = 10.25,
    \n\\(\\frac{1}{4}\\) X 18 + \\(\\frac{1}{4}\\) X s = 10.25,
    \n18 + s = 10.25 X 4,
    \n18 + s = 41,
    \ntherefore s = 41 – 18 = 23,
    \nSo No, David answer is not reasonable the price of shorts is $23 not $41.<\/p>\n

    Question 6.
    \nWhich of the following shows the correct use of the
    \nDistributive Property when solving (33 – x) = 135.2?
    \nA. (33 \u2013 x) = 1_-3 \u2022 135.2
    \nB. \\(\\frac{1}{3}\\) \u2022 33 – \\(\\frac{1}{3}\\)x = \\(\\frac{1}{3}\\) \u2022 135.2
    \nC. \\(\\frac{1}{3}\\) \u2022 33 + \\(\\frac{1}{3}\\)x = 135.2
    \nD. \\(\\frac{1}{3}\\) \u2022 33 – \\(\\frac{1}{3}\\)x = 135.2
    \nAnswer:
    \nB. \\(\\frac{1}{3}\\) \u2022 33 – \\(\\frac{1}{3}\\)x = \\(\\frac{1}{3}\\) \u2022 135.2,<\/p>\n

    Explanation:
    \nGiven to show the correct use of the
    \nDistributive Property when solving (33 – x) = 135.2? as
    \nthe correct answer is B. \\(\\frac{1}{3}\\) \u2022 33 – \\(\\frac{1}{3}\\)x = \\(\\frac{1}{3}\\) \u2022 135.2, if we multiply both sides with \\(\\frac{1}{3}\\)
    \nwe get the correct equation as (33 – x) = 135.2 because
    \n\\(\\frac{1}{3}\\)(33 – x) = \\(\\frac{1}{3}\\) X135.2 means,
    \n\\(\\frac{1}{3}\\) \u2022 33 – \\(\\frac{1}{3}\\)x = \\(\\frac{1}{3}\\) \u2022 135.2,
    \ntherefore the correct answer is B. \\(\\frac{1}{3}\\) \u2022 33 – \\(\\frac{1}{3}\\)x = \\(\\frac{1}{3}\\) \u2022 135.2.<\/p>\n

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

    Leveled Practice For 7-10, use the Distributive Property to solve the equations.<\/p>\n

    Question 7.
    \n-2(x + 5) = 4
    \n\"Envision
    \nAnswer:
    \n\"Envision
    \nExplanation:
    \nGiven equation as -2(x + 5) = 4, using Distributive Property
    \nwe get -2 X x + (-2 X 5) = 4,
    \n-2x – 10 = 4,
    \n-2x = 14, So x = -14\/2 = -7,
    \ntherefore x = -7.<\/p>\n

    Question 8.
    \n3.2 = \\(\\frac{4}{5}\\)(b – 5)
    \n\"Envision
    \nAnswer:
    \n\"Envision
    \nExplanation:
    \nGiven equation as 3.2 = \\(\\frac{4}{5}\\)(b – 5) using Distributive Property,
    \nwe get 3.2 = \\(\\frac{4}{5}\\)\u00a0 X b + \\(\\frac{4}{5}\\)(-5),
    \n3.2 = \\(\\frac{4}{5}\\)b – 4,
    \n3.2 + 4 = \\(\\frac{4}{5}\\)b,
    \n7.2 = \\(\\frac{4}{5}\\)b,
    \nb = \\(\\frac{5}{4}\\) X 7.2,
    \nb = 5 X 1.8,
    \ntherefore b = 9.<\/p>\n

    Question 9.
    \n\\(\\frac{1}{8}\\)(p + 24) = 9
    \n\"Envision
    \nAnswer:
    \n\"Envision
    \nExplanation:
    \nGiven equation as \\(\\frac{1}{8}\\)(p + 24) = 9 using Distributive Property,
    \n\\(\\frac{1}{8}\\) X p + \\(\\frac{1}{8}\\) X 24 = 9,
    \n\\(\\frac{1}{8}\\)p + 3 = 9,
    \n\\(\\frac{1}{8}\\) p = 9 – 3,
    \n\\(\\frac{1}{8}\\)p = 6,
    \np = 6 X 8 = 48 or we can solve \\(\\frac{1}{8}\\)(p + 24) = 9 as
    \np + 24 = 9 X 8,
    \np + 24 =72,
    \np = 72 – 24 = 48.<\/p>\n

    Question 10.
    \n\\(\\frac{2}{3}\\)(6a + 9) = 20.4
    \n\"Envision
    \nAnswer:
    \n\"Envision
    \nExplanation:
    \nGiven equation as \\(\\frac{2}{3}\\)(6a + 9) = 20.4 using Distributive Property,
    \n\\(\\frac{2}{3}\\) X 6a + \\(\\frac{2}{3}\\) X 9 = 20.4 9,
    \n4a + 6 = 20.4,
    \n4a = 20.4 – 6,
    \n4a = 14.4,
    \na = 14.4\/4,
    \na = 3.6.<\/p>\n

    Question 11.
    \nUse the equation at the right.
    \na. Make Sense and Persevere<\/strong> If you apply the Distributive Property first to solve the equation,
    \nwhat operation will you need to use last?
    \n\"Envision
    \nb. If instead you divide first to solve the equation,
    \nwhat operation would you need to use last?
    \nAnswer:
    \na. We use division operation last, If we apply
    \nthe Distributive Property first to solve the equation,
    \nb. We use multiplication last If instead we divide first to solve the equation,<\/p>\n

    Explanation:
    \nGiven equation as 6(\\(\\frac{d}{3}\\) – 5) = 34,
    \na. If we apply the Distributive Property first to solve the equation,
    \noperation I will need to use last is division as we know
    \n1.. Multiply the term outside of the parentheses by each term
    \nin the parentheses,
    \n2. We combine like terms,
    \n3. we divide to find the final solution,
    \nso 6 X \\(\\frac{d}{3}\\) – 6 X 5 = 34,
    \n2d – 30 = 34,
    \n2d = 34 + 30,
    \n2d = 64,
    \nd = 64\/2 = 32,
    \nb. Now If instead I divide first to solve the equation,
    \noperation I would need to use last is
    \n6 X \\(\\frac{d}{3}\\) – 6 X 5 = 34,
    \n\\(\\frac{d}{3}\\) – 5 = 34\/6,
    \n\\(\\frac{d}{3}\\) = 5.66 + 5,
    \n\\(\\frac{d}{3}\\) = 10.66
    \nd= 10.66 X 3 = 31.98 \u2248 32, we use multiplication.<\/p>\n

    Question 12.
    \nA family buys 4 airline tickets online. The family buys travel
    \ninsurance that costs $19 per ticket. The total cost is $752.
    \nLet x represent the price of one ticket.
    \na. Write an equation to represent this situation.
    \nb. What is the price of one ticket?
    \nAnswer:
    \na. Equation: 4(x + 19) = 752,
    \nb. The price of one ticket is $169,<\/p>\n

    Explanation:
    \nGiven a\u00a0family buys 4 airline tickets online. The family buys travel
    \ninsurance that costs $19 per ticket. The total cost is $752.
    \nLet x represent the price of one ticket.
    \na. As 4 tickets multiplied by each ticket and 4 tickets multiplied
    \nby travel insurance of $19 is equal to $752,
    \nSo, the equation is 4(x + 19) = 752,
    \nb. Now solving the equation 4(x + 19) = 752,
    \n4x + 76 = 752,
    \n4x = 752 – 76,
    \n4x = 676,
    \nx = 676\/4 = 169, therefore the price of one ticket is $169.<\/p>\n

    Question 13.
    \nA local charity receives \\(\\frac{1}{3}\\) of funds
    \nraised during a craft fair and a bake sale.
    \nThe total amount given to charity was $137.45.
    \nHow much did the bake sale raise?
    \n\"Envision
    \nAnswer:
    \nThe Bake Sale raised funds of $159.75,<\/p>\n

    Explanation:
    \nGiven a\u00a0local charity receives \\(\\frac{1}{3}\\) of funds
    \nraised during a craft fair and a bake sale.
    \nThe total amount given to charity was $137.45 and
    \nCraft Fair raised $252.60, lets take Bake sale raised funds of b,
    \nSo \\(\\frac{1}{3}\\)(252.60 + b) = 137.45,
    \n252.60 + b = 137.45 X 3,
    \n252.60 + b = 412.35,
    \nb= 412.35 – 252.60,
    \nb = 159.75,
    \ntherefore the Bake Sale raised funds of $159.75.<\/p>\n

    Question 14.
    \nThe solution shown for the equation is incorrect.
    \na. What is the correct solution?
    \nb. What was the likely error?
    \n\"Envision
    \nAnswer:
    \na. The correct solution is r = 8,
    \nb. Instead of +8 it is showing -8,<\/p>\n

    Explanation:
    \na. The given equation is -3(6 – r) = 6,
    \n-3 X 6 -3 X -r = 6,
    \n-18 + 3r = 6,
    \n3r = 6 + 18,
    \n3r = 24,
    \nr = 24\/3 = 8,
    \nb. The likely error while solving is -3 X -r it is
    \n+3r, but instead of 3r it is taken as -3r so it is showing
    \nr = -8 but the correct solution is r =8.<\/p>\n

    Question 15.
    \nVita wants to center a towel bar on her door that is
    \n27\\(\\frac{1}{2}\\) inches wide.
    \nShe determines that the distance from each end of
    \nthe towel bar to the end of the door is 9 inches.
    \nWrite and solve an equation to find the length of the towel bar.
    \nAnswer:
    \nEquation: x + 18 = 27\\(\\frac{1}{2}\\),
    \nThe length of the towel bar is 9\\(\\frac{1}{2}\\) inches,<\/p>\n

    Explanation:
    \nLet x inches be the length of the towel bar,
    \nThe width of the door is 9 inches on the each side of the
    \ntowel bar so the width of the door is 9 + x + 9 = x + 18 inches.
    \nIt is given that the width of the door is 27\\(\\frac{1}{2}\\) inches,
    \nS0 27\\(\\frac{1}{2}\\) inches = x +18,
    \nx = 27\\(\\frac{1}{2}\\) – 18,
    \nx =\u00a0 \\(\\frac{55}{2}\\) – 18,
    \nx = \\(\\frac{55 – 36}{2}\\),
    \nx = \\(\\frac{19}{2}\\) as numerator is greater than denominator,
    \nwe write mixed fraction as x = 9\\(\\frac{1}{2}\\) inches.
    \ntherefore, the length of the towel bar is 9\\(\\frac{1}{2}\\) inches.<\/p>\n

    Question 16.
    \nHigher Order Thinking<\/strong> A cell phone plan is shown at the right.
    \nThe rates, which include an unlimited data plan, are the same
    \neach month for 7 months. The total cost for all 7 months is $180.39.
    \nLet m represent the average number of minutes that exceeds
    \n700 minutes each month.
    \na. Write an equation to represent the given situation.
    \n\"Envision
    \nb. Solve the equation to determine how many additional minutes,
    \non average, you use each month.
    \nAnswer:
    \na. Equation : 7(19.70 + 1.97 + m X 0.05) = 180.39,
    \nb. 82 additional minutes, on average, can be used each month,<\/p>\n

    Explanation:
    \nGiven a cell phone plan is shown at the right.
    \nThe rates, which include an unlimited data plan, are the same
    \neach month for 7 months. The total cost for all 7 months is $180.39.
    \nLet m represent the average number of minutes that exceeds
    \n700 minutes each month.
    \na. Equation is 7(19.70 + 1.97 + m X 0.05) = 180.39,
    \nb. Solving 7(19.70 + 1.97 + m X 0.05) = 180.39,
    \n21.67 + 0.05m = 180.39\/ 7,
    \n0.05m = 25.77 – 21.67,
    \n0.05m = 4.1,
    \nm = 4.1\/0.05 = 82,
    \ntherefore, 82 additional minutes, on average, can be used each month.<\/p>\n

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

    Question 17.
    \nFidel earns a fixed amount, m, for each television he sells, and
    \nan additional $15 if the buyer gets an extended warranty.
    \nFidel sells 12 televisions with extended warranties, earning $900.
    \nWrite an equation to represent the situation.
    \nThen solve the equation to find the amount earned for each television sold.
    \nAnswer:
    \nEquation: 12(m +15) = 900,
    \nThe amount earned for each television sold is $60,<\/p>\n

    Explanation:
    \nGiven, Amount earned by Fidel for each TV is m,
    \nAdditional amount for extended warranty is $15,
    \nTelevisions sold are 12,
    \nAmount of televisions sold is $900,
    \nAmount earned by each TV * Number of TV’s +
    \nAdditional amount * Number of TV’s = Total amount,
    \nSo the equation is 12(m + 15) = 900,
    \nDividing both sides by 12, we get\u00a0 m + 15 =\u00a0 900\/12,
    \nm +15 = 75, m =75 – 15 = 60,
    \ntherefore, Fidel earns $60 for each television sold.<\/p>\n

    TOPIC 5 MID-TOPIC CHECKPOINT<\/h3>\n

    Question 1.
    \nVocabulary<\/strong> Explain how to isolate the variable in the equation
    \n–\\(\\frac{2}{3}\\)n + 7 = 15. Lesson 5-2
    \nAnswer:
    \nn = -12,<\/p>\n

    Explanation:
    \nTo isolate a variable is to perform the following steps:
    \n1. Eliminate any <\/span>fractions\u00a0<\/span>present by multiplying both sides
    \nby any denominators present.
    \n<\/span>2. Get all terms with the variable on the left <\/span>side\u00a0<\/span>of the
    \nequation and all terms without the variable on the other side.
    \n<\/span>3. Combine <\/span>the terms on the left side, if possible.
    \n<\/span>4. Factor <\/span>out the variable on the left side, if necessary.
    \n<\/span>5. Divide <\/span>both sides by the coefficient of the variable,
    \nleaving the variable isolated. Now the equation given is
    \n–\\(\\frac{2}{3}\\)n + 7 = 15, solving
    \n–\\(\\frac{2}{3}\\)n = 15 – 7,
    \nn = – \\(\\frac{3{2}\\) X 8,
    \nn = -12.<\/span><\/p>\n

    Question 2.
    \nJake paid $13.50 for admission to the county fair and
    \nbought 9 tickets to play games.
    \nIf he spent a total of $36, what is the cost, c, of one ticket?
    \nWrite and solve an equation. Lessons 5-1 and 5-2
    \nAnswer:
    \nEquation: 9c + 13.50 = 36,
    \nThe cost of one ticket is $2.5,<\/p>\n

    Explanation:
    \nGiven Jake paid $13.50 for admission to the county fair and
    \nbought 9 tickets to play games.
    \nIf he spent a total of $36 and the cost c of one ticket is
    \nEquation 9c + 13.5 = 36,
    \n9c + $13.50 = $36, subtracting 13.5 both sides
    \n9c + 13.50 – 13.5 = 36 -13.5,
    \n9c =\u00a0 22.5, dividing both sides by 9 we get
    \n9c\/9 = 22.5\/9,
    \nc = 2.5, therefore cost of one ticket is $2.5.<\/p>\n

    Question 3.
    \nSelect all the equations that are equivalent to \\(\\frac{1}{2}\\)(4 + 8x) = 17.
    \nLesson 5-3
    \n\"Envision
    \nAnswer:
    \n\"Envision
    \nThe equations that are equivalent to \\(\\frac{1}{2}\\)(4 + 8x) = 17 are
    \n4x = 15 and 4 + 8x = 34,<\/p>\n

    Explanation:
    \nGiven equation is \\(\\frac{1}{2}\\)(4 + 8x) = 17,
    \nmultiplying both side by 2 we get 4 + 8x = 34 and now dividing
    \nboth sides by 2 we get 2 + 4x = 17, 4x = 15,
    \nso the equations that are equivalent to \\(\\frac{1}{2}\\)(4 + 8x) = 17 are
    \n4x = 15 and 4 + 8x = 34.<\/p>\n

    Question 4.
    \nClara has 9 pounds of apples. She needs 1\\(\\frac{1}{4}\\) pounds
    \nto make one apple pie. If she sets aside 1.5 pounds of
    \napples to make applesauce, how many pies, p, can she make?
    \nWrite and solve an equation. Lessons 5-1 and 5-2
    \nAnswer:
    \nEquation: 1\\(\\frac{1}{4}\\)p + 1.5 = 9,
    \nNumber of apple pies Clara can make are 6,<\/p>\n

    Explanation:
    \nGiven Clara has 9 pounds of apples.
    \nShe needs 1\\(\\frac{1}{4}\\) pounds to make one apple pie.
    \nIf she sets aside 1.5 pounds of apples to make applesauce,
    \nNumber of pie’s p she can make are
    \nequation is 1\\(\\frac{1}{4}\\)p + 1.5 = 9,
    \n\\(\\frac{4\u00a0 X 1 + 1}{4}\\)p + 1.5 = 9,
    \n\\(\\frac{5}{4}\\)p + 1.5 = 9,
    \n\\(\\frac{5}{4}\\)p = 9 – 1.5,
    \n\\(\\frac{5}{4}\\)p = 7.5,
    \n5p = 7.5 X 4,
    \n5p = 30,
    \np = 30\/5 = 6, therefore number of apple pies Clara can make are 6.<\/p>\n

    Question 5.
    \nSolve the equation -4(1.75 + x) = 18. Show your work. Lesson 5-3
    \nAnswer:
    \nSolving the equation -4(1.75 + x) = 18 we get x = -6.25,<\/p>\n

    Explanation:
    \nGiven equation is -4(1.75 + x) = 18, dividing both side by -4,
    \n-4(1.75 + x)\/-4 = 18\/-4,
    \n1.75 + x = – 4.5,
    \nx = – 4.5 -1.75,
    \nx =\u00a0 – 6.25.<\/p>\n

    Question 6.
    \nFour friends attend a school play and pay $6.75 per ticket.
    \nEach also buys a Healthy Snack Bag sold by the Theater Club.
    \nIf the friends spent a total of $37.00, how much did each Healthy
    \nSnack Bag cost, b? Write and solve an equation. Lessons 5-1, 5-2,5-3
    \nAnswer:
    \nEquation:4(6.75 + b) = 37,
    \nEach Healthy Snack Bag cost is $2.5,<\/p>\n

    Explanation:
    \nGiven four friends attend a school play and pay $6.75 per ticket.
    \nEach also buys a Healthy Snack Bag sold by the Theater Club.
    \nIf the friends spent a total of $37.00, Now if each Healthy
    \nSnack Bag cost, b then the is equation 4(6.75 + b) = 37,
    \ndividing both sides by 4 we get 6.75 + b = 37\/4,
    \n6.75 + b = 9.25,
    \nb = 9.25 – 6.75,
    \nb = 2.5, therefore each Healthy Snack Bag cost is $2.5.<\/p>\n

    How well did you do on the mid-topic checkpoint? Fill in the stars. \"Envision<\/p>\n

    \"Envision<\/p>\n

    TOPIC 5 MID-TOPIC PERFORMANCE TASK<\/h3>\n

    Marven and three friends are renting a car for a trip.
    \nRental prices are shown in the table.<\/p>\n

    \"Envision<\/p>\n

    PART A<\/strong>
    \nMarven has a coupon that discounts the rental of a full-size car by $25.
    \nThey decide to buy insurance for each day. If the cost is $465,
    \nhow many days, d, will they rent the car? Write and solve an equation.
    \nAnswer:
    \nEquation : 49d + 21d – 25 = 465,
    \nNumber of days they will rent the car is 7 days,<\/p>\n

    Explanation:
    \nGiven Marven has a coupon that discounts the rental of a full-size car by $25.
    \nThey decide to buy insurance for each day. If the cost is $465,
    \nSo number of many days, d, will they rent the car is
    \n49d + 21d – 25 = 465,
    \n70d = 465 + 25,
    \n70d = 490,
    \nd = 490\/70,
    \nd= 7, therefore number of days they will rent the car is 7 days.<\/p>\n

    PART B<\/strong>
    \nIf they still use the coupon, how many days could they rent the small car
    \nwith insurance if they have $465 to spend?
    \nAnswer:
    \nIf they still use the coupon, how many days could they rent the small car
    \nwith insurance if they have $465 to spend is 8 days,<\/p>\n

    Explanation:
    \nIf they still use the coupon, Number of days could they rent the small car
    \nwith insurance if they have $465 to spend is
    \n39d + 21d – 25 = 465,
    \n60d = 465 + 25,
    \n60d = 490,
    \nd = 8.166, minimum 8 days.<\/p>\n

    PART C<\/strong>
    \nThey rent a car with insurance for 5 days but lost their coupon.
    \nIf Marven and the three friends spend $75 each, which car did they rent?
    \nWrite and solve an equation to justify your answer.
    \nAnswer:
    \nEquation:
    \n5(p + 21) = 4 X 75,
    \nThey rent the small car,<\/p>\n

    Explanation:
    \nIf they rent a car with insurance for 5 days but lost their coupon.
    \nIf Marven and the three friends spend $75 each means 4 X 75,
    \nLet price be p for the rent of the car,
    \n5(p + 21)\u00a0 = 4 X 75,
    \n5p + 105 = 300,
    \n5p = 300 – 105,
    \n5p = 195,
    \np = 195\/5,
    \np = 39, So car did they rent is small car.<\/p>\n

    Lesson 5.4 Solve Inequalities Using Addition or Subtraction<\/h3>\n

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

    Selena and Martin are waiting at the bus stop. The number lines show the
    \npossible wait times in minutes, t, for Selena and Martin.<\/p>\n

    \"Envision
    \nA. Construct Arguments Who anticipates a longer wait?
    \nJustify your response with a mathematical explanation.
    \nAnswer:
    \nMartin anticipated longer,<\/p>\n

    Explanation:
    \nBased on inequalities, Martin anticipated a longer waiting time.
    \nThis is because he thinks that the bus will arrive greater than 15 minutes.<\/p>\n

    B. If Selena and Martin both wait 10 minutes for the bus,
    \nwhose possible wait time was closer to his or her actual wait time? Explain.
    \nAnswer:
    \nSelena is closer,<\/p>\n

    Explanation:
    \nIf both Selena and Martin waited 10 minutes for the bus. Selena is closer to the actual wait time because she anticipated the waiting time for the bus less than 15 minutes before it arrives.<\/p>\n

    Focus on math practices<\/strong>
    \nBe Precise If Selena and Martin both wait exactly 15 minutes for the bus,
    \nwhose possible wait time was closer to his or her actual wait time? Explain.
    \nAnswer:
    \nSelena is closer to actual wait time,<\/p>\n

    Explanation:
    \nIf Selena and Martin both wait exactly 15 minutes for the bus.
    \nSelena is closer to the actual wait time.
    \nThis is because her possible wait time included 15 minutes.
    \nHer anticipation for the waiting time is exactly the same as the actual waiting time.
    \nUnlike Martin that he anticipated waiting time is greater than 15 mins,
    \nwhich does not include 15 minutes.<\/p>\n

    Essential Question<\/strong>
    \nHow is solving inequalities with addition and subtraction similar to and
    \ndifferent from solving equations with addition and subtraction?
    \nAnswer:
    \nSimilarity: Both uses properties of equality or inequality,
    \nDifferent: An equation has one solution while an inequality has more than one solution,<\/p>\n

    Explanation:
    \nThere is a similarity when solving inequalities with addition and subtraction
    \nfrom solving equations with addition and subtraction because
    \nboth uses properties of equality or inequality.
    \nInverse relationship between addition and subtraction to
    \nisolate the variable in an equation or inequality. The difference
    \nbetween the two is the solution for inequality and equation,
    \nThere is only one solution for an equation while in an inequality
    \nthere can be more than one solution.<\/p>\n

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

    Kyoko has completed 26 hours of community service.
    \nHer goal is to complete at least 90 hours this semester.
    \nWrite and solve an inequality to show how many more hours, h,
    \nKyoko needs to complete to meet her goal.
    \nUse the number line to graph the solutions.
    \n\"Envision
    \nAnswer:
    \n\"Envision
    \nExplanation:
    \nGiven Kyoko has completed 26 hours of community service.
    \nHer goal is to complete at least 90 hours this semester.
    \nWrote and solved an inequality to show how many more hours, h,
    \nKyoko needs to complete to meet her goal as h + 26 \u2264 90,
    \nUsed the number line to graph the solutions as shown above.<\/p>\n

    Convince Me!<\/strong>
    \nIs there more than one solution to the problem about Kyoko?
    \nExplain. Give one value that is a solution and one value that is not a solution.
    \nAnswer:
    \nYes , there more than one solution to the problem about Kyoko,<\/p>\n

    Explanation:
    \nYes , there more than one solution to the problem about Kyoko
    \nbecause of the indicated range for time.
    \nKyoko must complete at least 64 hours more to achieve her
    \ngoal in the community service.
    \nShe cannot go beyond 64 hours because it will be more than
    \nthe required number of hours. 65 hours is not a solution to the problem.<\/p>\n

    \n
    \n
    \n
    Try It!<\/strong><\/div>\n<\/div>\n<\/div>\n<\/section>\n

    The speed limit on a road drops down to 15 miles per hour around a curve.
    \nMr. Gerard slows down by 10 miles per hour as he drives around the curve.
    \nHe never drives above the speed limit.
    \nAt what speed was Mr. Gerard driving before the curve? Graph the solution.
    \nAnswer:
    \nMr. Gerard was driving less than or equal to 25 miles per hour before the curve,
    \n\"Envision
    \nExplanation:
    \nGiven the speed limit on a road drops down to
    \n15 miles per hour around a curve.
    \nMr. Gerard slows down by 10 miles per hour
    \nas he drives around the curve.
    \nHe never drives above the speed limit.
    \nThe speed Mr. Gerard was driving before the curve is
    \nx – 10 \u2264 15, inequality for the situation,
    \nx – 10 + 10 \u2264 15 + 10, addition property of inequality,
    \nx \u2264 25, his driving speed before the curve,
    \ntherefore, Mr. Gerard was driving less than or equal to
    \n25 miles per hour before the curve.<\/p>\n

    Try It!<\/strong>
    \nSolve the inequality n – 1\\(\\frac{3}{4}\\) \u2264 –\\(\\frac{5}{8}\\) Then graph the solution.
    \nAnswer:
    \nn \u2264 1\\(\\frac{1}{8}\\)\u00a0 or n \u2264 1.125,<\/p>\n

    Explanation:
    \nn – 1\\(\\frac{3}{4}\\) \u2264 –\\(\\frac{5}{8}\\),
    \nn – 1\\(\\frac{3}{4}\\) + 1\\(\\frac{3}{4}\\) \u2264 –\\(\\frac{5}{8}\\) + 1\\(\\frac{3}{4}\\), addition property of inequality,
    \nn \u2264 –\\(\\frac{5}{8}\\) + \\(\\frac{7}{4}\\), simplifying
    \nn \u2264 \\(\\frac{-5 + 14}{8}\\),
    \nn \u2264 \\(\\frac{9}{8}\\), as numerator is greater we write in mixed fraction as
    \nn \u2264 1\\(\\frac{1}{8}\\)\u00a0 or n \u2264 1.125.<\/p>\n

    KEY CONCEPT<\/strong>
    \nSolving inequalities with addition and subtraction is the same as solving equations with addition and subtraction.
    \nUse the inverse relationship between addition and subtraction to isolate the variable.
    \n\"Envision<\/p>\n

    Do You Understand?<\/strong><\/p>\n

    Question 1.
    \nEssential Question How is solving inequalities with addition and subtraction
    \nsimilar to and different from solving equations with addition and subtraction?
    \nAnswer:
    \nSimilarity: Both uses properties of equality or inequality,
    \nDifferent: An equation has one solution while an inequality has more than one solution,<\/p>\n

    Explanation:
    \nThere is a similarity when solving inequalities with addition and subtraction
    \nfrom solving equations with addition and subtraction because
    \nboth uses properties of equality or inequality.
    \nInverse relationship between addition and subtraction to
    \nisolate the variable in an equation or inequality. The difference
    \nbetween the two is the solution for inequality and equation,
    \nThere is only one solution for an equation while in an inequality
    \nthere can be more than one solution.<\/p>\n

    Question 2.
    \nBe Precise How do the solutions of the two inequalities differ?
    \nAre any of the solutions the same? Explain.
    \na. x+ 5 < 8 and x + 5 > 8
    \nb. x + 5 \u2264 8 and x + 5 \u2265 8<\/p>\n

    Answer:
    \na. The number 3 is not part of the solution for both inequalities,
    \n\"Envision
    \nb. The number 3 is one the solutions for both inequalities,
    \n\"Envision<\/p>\n

    Explanation:
    \na. x + 5 – 5 < 8 – 5, (subtraction property of inequality)
    \nx < 3, (solution to the equation) and
    \nx + 5 – 5 > 8 – 5, (subtraction property of inequality)
    \nx > 3, (solution to the equation),
    \nBoth solution uses subtraction property of equality,
    \nHowever the value of the solutions are different,
    \nOne is less than 3 while the other is greater than 3.<\/p>\n

    b. x + 5 – 5 \u2264 8 – 5, (subtraction property of inequality)
    \nx \u2264 3, (solution to the equation) and
    \nx + 5 – 5 \u2265 8 – 5, (subtraction property of inequality)
    \nx \u2265 3, (solution to the equation),
    \nBoth solution uses subtraction property of equality,
    \nBoth inequalities have 3 as part of the solution.<\/p>\n

    Question 3.
    \nReasoning Write two different inequalities in which one of the solutions is
    \nthe same as the solution to x – 23 = 191.
    \nAnswer:
    \nThe inequalities are x- 23 \u2264 191 and x – 23 < 191,
    \nthe first inequality has a similar solution to the indicated equation
    \nbecause of the inequality symbol,<\/p>\n

    Explanation:
    \n1. x -23 \u2264 191 (first inequality)
    \nx -23 + 23 \u2264 191 + 23 (addition property)
    \nx \u2264 214 (range of solution to the inequality),
    \nx -23 < 191 (second inequality)
    \nx -23 + 23 < 191 + 23 (addition property)
    \nx < 214 (range of solution to the inequality),<\/p>\n

    2. The solution to the equation is
    \nx – 23 + 23 = 191 + 23,(addition property of equality)
    \nx = 214 (solution to the equation).<\/p>\n

    Do You Know How?<\/strong><\/p>\n

    Question 4.
    \nSolve each inequality. Then graph the solution.
    \na. x + 5 > 3
    \n\"Envision
    \nAnswer:
    \nx > -2,
    \n\"Envision<\/p>\n

    Explanation:
    \nx + 5 > 3( inequality),
    \nx + 5 – 5 > 3 – 5, subtract 5 from both sides,
    \nx > -2,
    \nThe graph of the solution x > -2 is shown above.<\/p>\n

    b. x + 5 \u2264 3
    \n\"Envision
    \nAnswer:
    \nx \u2264 -2,
    \n\"Envision<\/p>\n

    Explanation:
    \nx + 5 \u2264 3 (inequality),
    \nx + 5 – 5 \u2264 3 – 5, subtract 5 from both sides,
    \nx \u2264 -2,
    \nThe graph of the solution x \u2264 -2 is shown above.<\/p>\n

    c. x – \\(\\frac{3}{2}\\) \u2264 -3
    \n\"Envision
    \nAnswer:
    \nx \u2264 –\\(\\frac{3}{2}\\),
    \n\"Envision<\/p>\n

    Explanation:
    \nx – \\(\\frac{3}{2}\\) \u2264 -3 (inequality),
    \nx – \\(\\frac{3}{2}\\) + \\(\\frac{3}{2}\\) \u2264 -3 +<\/span>\u00a0<\/span>\\(\\frac{3}{2}\\)<\/span>(adding <\/span> \\(\\frac{3}{2}\\) both sides),
    \n<\/span>x\u00a0\u2264 –\\(\\frac{3}{2}\\),
    \nThe graph of the solution x > -2 is shown above.<\/span><\/p>\n

    Question 5.
    \n<\/span>Elanor is driving below the speed limit on a highway.
    \na. Write the inequality to show how much faster Elanor can drive without going over the speed limit.
    \n\"Envision
    \nb. Solve the inequality you wrote. By how much can Elanor increase her speed?
    \nAnswer:
    \na. x + 43.5 < 55,
    \nb. Elanor increase her speed by x < 11.5,<\/p>\n

    Explanation:
    \na. Let x be the increase in the speed limit,
    \nThe inequality for the given situation where Elanor can
    \ndrive without going over the speed limit is x + 43.5 < 55,<\/p>\n

    b. Now we find the value of x in the given inequality is
    \nx + 43.5 < 55, subtracting 43.5 both sides as
    \nx + 43.5 – 43.5 < 55 – 43.5, we get
    \nx< 11.5, So Elanor increase her speed by x < 11.5.<\/p>\n

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

    Leveled Practice<\/strong> In 6 and 7, fill in the boxes to solve each inequality. Then graph the solutions.<\/p>\n

    Leveled Practice In 6 and 7, fill in the boxes to solve each inequality. Then graph the solutions.<\/p>\n

    Question 6.
    \nx + 5 < 7
    \n\"Envision
    \nAnswer:
    \nx < 2,
    \n\"Envision<\/p>\n

    Explanation:
    \nGiven inequality as x + 5 < 7, now we subtract 5 from both sides,
    \nx + 5 – 5 < 7 – 5, we get
    \nx < 2.<\/p>\n

    Question 7.
    \nx – 4 \u2265 12
    \n\"Envision
    \nAnswer:
    \nx\u00a0 \u2265 16,
    \n\"Envision<\/p>\n

    Explanation:
    \nGiven inequality as x – 4 \u2265 12, now we add 4 on both sides,
    \nx – 4 + 4 \u2265 12 + 4, we get
    \nx \u2265 16.<\/p>\n

    Question 8.
    \nSolve x + 10 \u2265 14. Then graph the solution.
    \n\"Envision
    \nAnswer:
    \nx \u2265 4,
    \n\"Envision<\/p>\n

    Explanation:
    \nGiven inequality as x + 10 \u2265 14, now we subtract 10 from both sides,
    \nx + 10 – 10 \u2265 14 – 10, we get x \u2265 4.<\/p>\n

    Question 9.
    \nSolve x – 20 \u2264 -11. Then graph the solution.
    \n\"Envision
    \nAnswer:
    \nx \u2264 9,
    \n\"Envision<\/p>\n

    Explanation:
    \nGiven inequality as x – 20 \u2264 – 11, now we add 20 on both sides,
    \nx – 20 + 20 \u2265 -11 + 20, we get
    \nx \u2264 9.<\/p>\n

    Question 10.
    \nThe maximum number of students in a classroom is 26.
    \nIf there are 16 students signed up for the art class, how many more
    \nstudents can join the class without exceeding the maximum?
    \nAnswer:
    \nThere are at most 10 students who can join the art class,<\/p>\n

    Explanation:
    \nLet x be the number of students that can join the art class,
    \nx + 16 \u2264 26,
    \nx + 16 – 16 \u2264 26 – 16, subtract 16 from both sides,
    \nx \u2264 10, Therefore there are at most 10 students who can join the art class.<\/p>\n

    Question 11.
    \nHigher Order Thinking<\/strong> The inequality x + c > -2.55 has the
    \nsolution x > 4.85 What is the value of c? How do you know?
    \nAnswer:
    \nc = -7.4,<\/p>\n

    Explanation:
    \nx + c = -2.55,
    \n4.85 + c = -2.55, Substitute the value of x,
    \n4.85 + c – 4.85 = -2.55 – 4.85, subtract -4.85 from both sides,
    \nc = -7.4,
    \nThe value of c = -7.4, substitute this to the original inequality,
    \nso that it will make the inequality true,
    \nx + c > -2.55,
    \nx + (-7.4) > -2.55, substitute the value of c,
    \nx + (-7.4) + 7.4 > -2.55 + 7.4, add 7.4 on both sides,
    \nx > 4.85 which is true.<\/p>\n

    Question 12.
    \nRina is climbing a mountain. She has not yet reached base camp.
    \nWrite an inequality to show the remaining distance, d, in feet
    \nshe must climb to reach the peak.
    \n\"Envision
    \nAnswer:
    \nInequality: d + 9,695 > 12,358,
    \nRina must climb d > 2,663 feet to reach the peak,<\/p>\n

    Explanation:
    \nThe inequality to determine the distance is
    \nd + 9,695 > 12,358,
    \nd + 9,695 – 9,695 > 12,358 – 9,695, subtract 9,695 from both sides,
    \nd > 2,663, therefore, Rina must climb d > 2,663 feet to reach the peak.<\/p>\n

    Question 13.
    \nOn a math test, students must solve the inequality x – 5< 11 and then graph the solution. Mason said the solution is x < 6 and graphed the solution as shown below.
    \n\"Envision
    \na. What error did Mason make?
    \nb. Show the correct solution on the number line.
    \n\"Envision
    \nAnswer:
    \na. Mason did not use the inverse operations to determine the solution,
    \nb. \"Envision<\/p>\n

    Explanation:
    \na. Mason did not use the inverse operations to determine the solution of x,
    \nMason subtracted 5 to the right side when it should have been added this
    \nis the correct way of using inverse operation,
    \nb. x – 5 < 11,
    \nx – 5 + 5 < 11 + 6, adding 5 on both sides,
    \nx < 16, the graph of solution is shown on number line above.<\/p>\n

    Question 14.
    \nModel with Math<\/strong> Dani’s neighbors paid her to take care of their bird during their vacation.
    \nDani spent $4 of her earnings on an afternoon snack and $16 on a new book.
    \nAfterward, she had at least $8 left. Write an inequality to represent how much Dani’s neighbors paid her.
    \nAnswer:
    \nInequality : x – (4 + 16) \u2265 8,
    \nDani’s neighbors paid her x \u2265 28,<\/p>\n

    Explanation:
    \nGiven Dani’s neighbors paid her to take care of their bird
    \nduring their vacation.
    \nDani spent $4 of her earnings on an afternoon snack and
    \n$16 on a new book. Afterward, she had at least $8 left,
    \nLet x be the amount Dani’s neighbors paid her, So the inequality is
    \nx – (4 + 16) \u2265 8, now solving the inequality
    \nx – 20 + 20 \u2265 8 + 20, adding 20 on both sides,
    \nx \u2265 28, therefore, Dani’s neighbors paid her x \u2265 28.<\/p>\n

    Question 15.
    \nReasoning<\/strong> The temperature in a greenhouse should be 67\u00b0F or higher.
    \nOne morning, the heater stopped working.
    \nThe temperature dropped 4 degrees before someone fixed the heater.
    \nThe temperature was still at least 67\u00b0F when the heater started working again.
    \nHow can you best describe the temperature in the greenhouse before
    \nthe heater stopped working?
    \n\"Envision
    \nAnswer:
    \nThe temperature in the greenhouse before
    \nthe heater stopped working is at least 71\u00b0F,<\/p>\n

    Explanation:
    \nThe temperature in a greenhouse should be 67\u00b0F or higher.
    \nOne morning, the heater stopped working.
    \nThe temperature dropped 4 degrees before someone fixed the heater.
    \nThe temperature was still at least 67\u00b0F when the heater started working again.
    \nLet x be the initial temperature of the heater, so
    \nx – 4 \u2265 67,
    \nx + 4 + 4 \u2265 67 + 4, adding 4 on both sides,
    \nx \u2265 71, therefore, the temperature in the greenhouse before
    \nthe heater stopped working is at least 71\u00b0F.<\/p>\n

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

    Question 16.
    \nRamiro has $21. He wants to buy a skateboard that costs $47.
    \nHow much more money does he need to have at least $47?
    \nWrite an inequality that represents the situation.
    \nSolve the inequality and graph your solution.
    \n\"Envision
    \nAnswer:
    \nInequality : 47 \u2264 x + 21,
    \n26 \u2264 x or x \u2265 26,
    \n\"Envision<\/p>\n

    Explanation:
    \nGiven Ramiro has $21. He wants to buy a skateboard that costs $47.
    \nLet x be more money does he need to have at least $47,
    \nSo the inequality is 47 \u2264 x+21,
    \n47 – 21\u2264\u00a0 x + 21 – 21, subtract 21 from both sides,
    \n26 \u2264 x, The graph 26 \u2264 x or x \u2265 26 is shown above.<\/p>\n

    Question 17.
    \nKendra has $7.35 in her purse. She needs at least $2.87 more to buy a special bead.
    \nWhat is the total amount, x, she needs for the bead?
    \nWhich inequalities can be used to represent the situation?
    \nSelect all that apply.
    \n\"Envision
    \nAnswer:
    \nInequalities that can be used to represent the
    \nsituation x = 10.22 are
    \n1. x – 7.35 \u2264 2.87,
    \n2.\u00a0 x + 7.35\u00a0 \u2265 2.87,
    \n3. x – 7.35 \u2265 2.87,
    \n4. x \u2265 10.22,
    \n5. x \u2264 10.22,<\/p>\n

    Explanation:
    \nGiven Kendra has $7.35 in her purse. She needs at least $2.87
    \nmore to buy a special bead.
    \nThe total amount, x, she needs for the bead is x = 7.35 + 2.87,
    \nSo x = 10.22, Given
    \n1. x + 7.35 \u2264 2.87, on solving
    \nx + 7.35 – 7.35 \u2264 2.87 – 7.35, subtracting 7.35 from both sides,
    \nx \u2264 4.48 but we got x = 10.22, So
    \nx + 7.35 \u2264 2.87 is incorrect as x is not equal to x = 10.22 and
    \nalso not less than 4.48.<\/span><\/p>\n

    2. x – 7.35 \u2264 2.87, on solving
    \nx – 7.35 + 7.35 \u2264 2.87 + 7.35, adding 7.35 from both sides,
    \nx \u2264 10.22 we got x = 10.22, So
    \nx – 7.35 \u2264 2.87 is correct as x is equal to x = 10.22.<\/p>\n

    3. x + 7.35 \u2265 2.87,on solving
    \nx + 7.35 – 7.35 \u2265 2.87 – 7.35, subtracting 7.35 from both sides,
    \nx \u2265 4.48 but we got x = 10.22, So
    \nx + 7.35 \u2265 2.87 is correct as x = 10.22 whose value is
    \ngreater than 4.48.<\/p>\n

    4. x – 7.35 \u2265 2.87,on solving
    \nx – 7.35 + 7.35 \u2265 2.87 + 7.35, adding 7.35 from both sides,
    \nx \u2265 10.22 but we got x = 10.22, So
    \nx – 7.35 \u2265 2.87 is correct as x value is equal to 10.22.<\/span><\/p>\n

    5. x \u2265 10.22 is correct as x is equal to x = 10.22.<\/p>\n

    6. x \u2264 2.87 is incorrect as x is not equal to 10.22 and is not
    \nalso less than 2.87.<\/p>\n

    7. x \u2264 10.22 is correct as x is equal to 10.22.<\/p>\n

    8. x \u2264 4.48 is incorrect as x value is not equal to 10.22 and
    \nalso not less than 4.48.
    \nTherefore,<\/p>\n

    Inequalities that can be used to represent the
    \nsituation x = 10.22 are
    \n1. x – 7.35 \u2264 2.87,
    \n2.\u00a0 x + 7.35\u00a0 \u2265 2.87,
    \n3. x – 7.35 \u2265 2.87,
    \n4. x \u2265 10.22,
    \n5. x \u2264 10.22,<\/p>\n

    Lesson 5.5 Solve Inequalities Using Multiplication or Division<\/h3>\n

    Solve & Discuss It!<\/strong><\/p>\n

    Alex and Hope were trying to solve -6x > 24.
    \nWhose inequality shows the solution? Show your work.
    \n\"Envision<\/p>\n

    Answer:
    \nHope’s work x < -4 shows the inequality to the solution,<\/p>\n

    Explanation:
    \nGiven Alex and Hope were trying to solve -6x > 24.
    \nHope shows the solution as x < -4 means x values are less than
    \n-4 which will be -5,-6, -7 and so on, if we substitute x as -5,
    \nwe get -6(-5) = 30 which is greater than 24,
    \ntherefore, Hope’s work x < -4 shows the inequality to the solution,<\/p>\n

    Construct Arguments<\/strong> Why does more than one value of x
    \nmake the inequality true?
    \nAnswer:
    \nYes, because there may be more than one solutions to make
    \nmake the inequality true,<\/p>\n

    Explanation:
    \nMost\u00a0of the time, an\u00a0inequality has more than one\u00a0or even infinity\u00a0solutions.
    \nFor example the\u00a0inequality: x>3 . The\u00a0solutions\u00a0of this\u00a0inequality\u00a0are
    \n“all numbers strictly\u00a0greater than 3”, therefore more than one value of x
    \nmake the inequality true.<\/p>\n

    Focus on math practices<\/strong>
    \nBe Precise What do you notice about the inequality symbols used
    \nin the original inequality and in the correct solution?
    \nAnswer:
    \nThe inequality symbol in the solution changed,<\/p>\n

    Explanation:
    \nThe inequality symbol (>) changed in the solution,
    \nThe inequality symbol became <, This is because of the
    \nnegative integer divided to both sides of the inequality.<\/p>\n

    Essential Question<\/strong>
    \nHow is solving inequalities with multiplication and division similar to and
    \ndifferent from solving equations with multiplication and division?
    \nAnswer:
    \nSolving inequalities is very similar to solving equations,
    \nbut sometimes we have to reverse the symbol.
    \nwe must reverse the inequality symbol when we multiply or
    \ndivide both sides of the equation by a negative number.
    \n1) It must be multiplication or division (not addition or subtraction)
    \n2) The number being multiplied or divided must be negative.<\/p>\n

    Explanation:
    \nSolving inequalities is very similar to solving equations,
    \nbut sometimes we have to reverse the symbol.<\/p>\n

    For example:
    \nMultiplication Properties of Inequality,
    \nc is positive,
    \na<b, then ac<bc \u2190 -2<3,
    \na>b, then ac>bc \u2190-2(4)<3(4),
    \nc is negative,
    \na<b, then ac>bc \u2190 -2<3,
    \na>b, then ac<bc \u2190-2(-4)>3(-4),
    \nDivision property of inequality,
    \nc is positive,
    \na<b, then a\/c<b\/c \u2190 2<8,
    \na>b, than a\/c>b\/c \u21902\/4<8\/4,
    \nc is negative,
    \na<b, then a\/c>b\/c \u21902<8,
    \na>b, then a\/c<b\/c \u2190 2\/-4>8\/-4, or<\/p>\n

    Solving Inequalities is very similar to solving Equations.
    \nThe same general technique applies.
    \nThat technique for solving equations is:
    \nWhatever we do to one side of the equation,
    \nwe have to do to the other side to preserve the equality,
    \nThe technique for solving inequalities is:
    \nWhatever we do to one side of the inequality,
    \nwe have to do to the other side to preserve the inequality
    \nthe techniques are the same.
    \nThe technique means:
    \nIf we multiply or divide one side of an equation or inequality
    \nby the same number, we have to multiply or divide the other side
    \nof the equation or inequality by the same number.
    \nThe difference between solving equations and solving inequalities is:
    \nIf you multiply or divide an inequality by a negative number,
    \nthen the inequality reverses.
    \nThe following examples will show this rule in action.
    \nThe first example will be multiplication.
    \nThe second example will be division.
    \nThe first example starts with:
    \n7 is greater than 5.
    \nMultiply both sides of that inequality by -5 and we get:
    \n7*-5 = -35 on the left side of the inequality.
    \n5*-5 = -25 on the right side of the inequality.
    \nThe result is -35 is smaller than -25.
    \nThe inequality started as greater than and
    \nbecame smaller than because we were multiplying both sides
    \nof the inequality by a negative number.
    \nIt’s clear to see that 7 really is greater than 5.
    \nIt is also clear to see that -35 is less than -25.
    \nThe second example is simply the reverse of the first example and starts with:
    \n-35 is smaller than -25.
    \nDivide both sides of this inequality by -5 and we get:
    \n-35 \/ -5 = 7 on the left side of the inequality.
    \n-25 \/ -5 = 5 on the right side of the inequality.
    \nThe result is 7 is greater than 5.
    \nThe inequality started as less than and became greater than
    \nbecause you were dividing both sides of the inequality by a negative number.
    \nWe needed to do that to preserve the inequality.
    \nIt’s clear with these numbers that reversing the inequality is
    \nessential when we are multiplying both sides of the inequality or
    \ndividing both sides of the inequality by a negative number.<\/p>\n

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

    Solve the inequality \\(\\frac{d}{7}\\) > 15. Then graph the solution.
    \n\"Envision
    \nAnswer:
    \nd> 105,
    \n\"Envision
    \nExplanation:
    \nSolved the inequality \\(\\frac{d}{7}\\) > 15 as
    \n\\(\\frac{d}{7}\\) > 15, multiplying both sides by 7,
    \nwe get d > 105, Shown the graph of the solution d > 105 above.<\/p>\n

    Convince Me!<\/strong>
    \nFrances solved the inequality 5g \u2265 35. She says that 7 is
    \na solution to the inequality. Is Frances correct? Explain.
    \nAnswer:
    \nYes, Frances is correct,<\/p>\n

    Explanation:
    \nGiven Frances solved the inequality 5g \u2265 35.
    \nShe says that 7 is a solution to the inequality on solving
    \n5g \u2265 35, dividing both sides by 5 we get
    \ng \u2265 7, means g is greater than 7 and we get g is
    \nequal to 7, therefore, Yes, Frances is correct.<\/p>\n

    Try It!<\/strong>
    \nSolve each inequality. Then graph the solution.<\/p>\n

    a. 149.76 > -19.2x
    \n\"Envision
    \nAnswer:
    \n-7.8 < x or x > -7.8,
    \n\"Envision
    \nExplanation:
    \nGiven to solve inequality 149.76 > -19.2x,
    \n149.76\/- 19.2 > -19.2x\/-19.2, dividing both sides by -19.2 and
    \nreverse the inequality symbol,
    \n-7.8 < x, shown the value of x as -7.8 < x or x > -7.8 on the graph above.<\/p>\n

    b. -3.25y < -61.75
    \n\"Envision
    \nAnswer:
    \n-3.25y < -61.75,
    \n\"Envision
    \nExplanation:
    \nGiven to solve inequality -3.25y < -61.75,
    \n-3.25y\/- 3.25 > -61.75\/-3.25, dividing both sides by -3.25 and
    \nreverse the inequality symbol,
    \ny > 19, shown the value of y as y > 19 on the graph above.<\/p>\n

    Try It!<\/strong>
    \nSolve each inequality. Then graph the solution.<\/p>\n

    a. \\(\\frac{k}{-0.5}\\) < 12
    \n\"Envision
    \nAnswer:
    \nk> -6,
    \n\"Envision
    \nExplanation:
    \nGiven to solve \\(\\frac{k}{-0.5}\\) < 12 inequality,
    \n– 0.5\\(\\frac{k}{-0.5}\\) > – 0.5(12), multiplying both sides by -0.5 and
    \nreverse the inequality symbol,
    \nk > -6, The graph of the solution k > -6 is as shown above.<\/p>\n

    b. –\\(\\frac{5}{4}\\)h \u2265 25
    \n\"Envision
    \nAnswer:
    \nh \u2264 -20,
    \n\"Envision
    \nExplanation:
    \nGiven to solve –\\(\\frac{5}{4}\\)h \u2265 25 inequality,
    \n-4 (-\\(\\frac{5}{4}\\)h) \u2264 -4(25) multiplying both sides by -4 and
    \nreverse the inequality symbol,
    \n5h \u2264 -100,
    \n5h\/5 \u2264 -100\/5, divide both sides by 5,
    \n5h \u2264 -20,The graph of the solution 5h \u2264 -20 is as shown above.<\/p>\n

    KEY CONCEPT<\/strong><\/p>\n

    Solving inequalities with multiplication and division is the
    \nsame as solving equations with multiplication and division
    \nwhen the values are positive. Use the inverse relationship
    \nbetween multiplication and division to isolate the variable.
    \n\"Envision
    \nWhen multiplying or dividing by negative values, the inequality symbol is reversed.
    \n\"Envision<\/p>\n

    Do You Understand?<\/strong><\/p>\n

    Question 1.
    \nEssential Question<\/strong>
    \nHow is solving inequalities with multiplication and division
    \nsimilar to and different from solving equations with multiplication and division?
    \nAnswer:
    \nSolving inequalities is very similar to solving equations,
    \nbut sometimes you have to reverse the symbol.
    \nwe must reverse the inequality symbol when we multiply or
    \ndivide both sides of the equation by a negative number.
    \n1) It must be multiplication or division (not addition or subtraction)
    \n2) The number being multiplied or divided must be negative.<\/p>\n

    Explanation:
    \nSolving inequalities is very similar to solving equations,
    \nbut sometimes you have to reverse the symbol.<\/p>\n

    For example:
    \nMultiplication Properties of Inequality,
    \nc is positive,
    \na<b, then ac<bc \u2190 -2<3,
    \na>b, then ac>bc \u2190-2(4)<3(4),
    \nc is negative,
    \na<b, then ac>bc \u2190 -2<3,
    \na>b, then ac<bc \u2190-2(-4)>3(-4),
    \nDivision property of inequality,
    \nc is positive,
    \na<b, then a\/c<b\/c \u2190 2<8,
    \na>b, than a\/c>b\/c \u21902\/4<8\/4,
    \nc is negative,
    \na<b, then a\/c>b\/c \u21902<8,
    \na>b, then a\/c<b\/c \u2190 2\/-4>8\/-4, or<\/p>\n

    Solving Inequalities is very similar to solving Equations.
    \nThe same general technique applies.
    \nThat technique for solving equations is:
    \nWhatever we do to one side of the equation,
    \nwe have to do to the other side to preserve the equality,
    \nThe technique for solving inequalities is:
    \nWhatever we do to one side of the inequality,
    \nwe have to do to the other side to preserve the inequality
    \nthe techniques are the same.
    \nThe technique means:
    \nIf you multiply or divide one side of an equation or inequality
    \nby the same number, we have to multiply or divide the other side
    \nof the equation or inequality by the same number.
    \nThe difference between solving equations and solving inequalities is:
    \nIf we multiply or divide an inequality by a negative number,
    \nthen the inequality reverses.
    \nThe following examples will show this rule in action.
    \nThe first example will be multiplication.
    \nThe second example will be division.
    \nThe first example starts with:
    \n7 is greater than 5.
    \nMultiply both sides of that inequality by -5 and we get:
    \n7*-5 = -35 on the left side of the inequality.
    \n5*-5 = -25 on the right side of the inequality.
    \nThe result is -35 is smaller than -25.
    \nThe inequality started as greater than and
    \nbecame smaller than because we were multiplying both sides
    \nof the inequality by a negative number.
    \nIt’s clear to see that 7 really is greater than 5.
    \nIt is also clear to see that -35 is less than -25.
    \nThe second example is simply the reverse of the first example and starts with:
    \n-35 is smaller than -25.
    \nDivide both sides of this inequality by -5 and we get:
    \n-35 \/ -5 = 7 on the left side of the inequality.
    \n-25 \/ -5 = 5 on the right side of the inequality.
    \nThe result is 7 is greater than 5.
    \nThe inequality started as less than and became greater than
    \nbecause you were dividing both sides of the inequality by a negative number.
    \nWe needed to do that to preserve the inequality.
    \nIt’s clear with these numbers that reversing the inequality is
    \nessential when we are multiplying both sides of the inequality or
    \ndividing both sides of the inequality by a negative number.
    \nor
    \nThe similarity is the method of solving it when the
    \nvalues are positive. The difference is the method of
    \nsolving it when the values are negative.<\/p>\n

    Explanation:
    \nThe similarities of solving equation and solving inequalities
    \nBoth are used in determining the values of variable
    \nBoth can be solved either by one step or two steps
    \nBoth have same process when the values are positive
    \nThe difference between solving equation and solving inequalities
    \nThe properties used when solving equations and solving inequalities
    \nThe method of solving equation and inequalities differ
    \nwhen the values are negative.<\/p>\n

    Question 2.
    \nConstruct Arguments Why is -x < 3 equivalent to x > -3?
    \nProvide a convincing argument.
    \nAnswer:
    \nThe two inequalities are equivalent,<\/p>\n

    Explanation:
    \nThe variable cannot be negative, therefore it is need
    \nto show the variable as positive.
    \n-x < 3,
    \n-1x\/-1 > 3\/-1, divide both sides by -1 and
    \nreverse the inequality symbol,
    \nx > -3,
    \nThe two inequalities are equivalent, this is because
    \nthe first inequality has negative variable.
    \nit needs to be changed into a positive variable.
    \nAfter solving the first inequality, the result is the same
    \nwith the second inequality.<\/p>\n

    Question 3.
    \nIf a, b, and care rational numbers and a > b,
    \nis ac > bc always true? Justify your answer.
    \nAnswer:
    \nThe given inequality is not always true,<\/p>\n

    Explanation:
    \nThe inequality ac > bc is not always true,
    \nThis is because when c is a negative integer,
    \nthe inequality becomes false.
    \nExample for this is a = 3, b = 2, c = -1,
    \nSubstituting to the inequality ac > bc =
    \n3(-1) > 2(-1) = -3 < -2.<\/p>\n

    Do You Know How?<\/strong><\/p>\n

    Question 4.
    \nSolve each inequality. Then graph the solution.
    \na. 4x > 12
    \n\"Envision
    \nAnswer:
    \nx > 3,
    \n\"Envision
    \nExplanation:
    \nGiven to solve inequality 4x > 12,
    \n4x\/4 > 12\/4, divide both sides by 4
    \nx > 3, shown the value of x as x > 3 on the graph above.<\/p>\n

    b. \\(\\frac{x}{4}\\) \u2264 -12
    \n\"Envision
    \nAnswer:
    \nx \u2264 -48,
    \n\"Envision<\/p>\n

    Given to solve inequality \\(\\frac{x}{4}\\) \u2264 -12
    \n4 X \\(\\frac{x}{4}\\) \u2264 4 X -12, multiplying both sides by 4
    \nx \u2264 -48, shown the value of x as x \u2264 -48 on the graph above.<\/p>\n

    c. -4x > 12
    \n\"Envision
    \nAnswer:
    \nx < -3,
    \n\"Envision
    \nExplanation:
    \nGiven to solve -4x > 12 inequality,
    \n-4x\/-4 < 12\/-4 dividing both sides by -4 and
    \nreverse the inequality symbol,
    \nx < -3,
    \nx < -3,The graph of the solution x <-3 is as shown above.<\/p>\n

    Question 5.
    \nVanna is saving for a trip. The hotel room will be $298.17 for 3 nights, and
    \nthere will be additional fees. What is her daily cost?
    \n\"Envision
    \na. Write an inequality for the situation.
    \nb. Solve the inequality. Then provide a statement that
    \nrepresents the solution of the problem.
    \nAnswer:
    \na. 3x \u2265 298.17,
    \nb. x\u2265 99.39 , The stay of Vanna in the hotel plus additional cost fees
    \ncost at least $99.39,<\/p>\n

    Explanation:
    \nGiven Vanna is saving for a trip. The hotel room will be $298.17 for 3 nights, and
    \nthere will be additional fees, Let x be the daily cost of the stay in the hotel room, So
    \na. The inequality for the situation is 3x \u2265 298.17,
    \nb. Solving 3x \u2265 298.17 we get daily cost
    \n3x\/3 \u2265 298.17\/3, dividing both sides by 3 we get
    \nx \u2265 99.39, therefore the stay of Vanna in the hotel plus additional cost fees
    \ncost at least $99.39.<\/p>\n

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

    Leveled Practice<\/strong> In 6-9, fill in the boxes to solve the inequality.
    \nThen graph the solution.<\/p>\n

    Question 6.
    \n\"Envision
    \nAnswer:
    \nm \u2264 7,
    \n\"Envision<\/p>\n

    Explanation:
    \nGiven to fill the boxes and to solve the 8m \u2264 56 inequality,
    \n8m\/8 \u2264 56\/8 dividing both sides by 8,
    \nm \u2264 7,The graph of the solution m \u2264 7 is as shown above.<\/p>\n

    Question 7.
    \n\"Envision
    \nAnswer:
    \nx> 6,
    \n\"Envision
    \nExplanation:
    \nGiven to fill the boxes and to solve the –\\(\\frac{4}{3}\\)x < -8 inequality,
    \n-3 X \\(\\frac{4}{3}\\) x > -3(-8), multiplying both sides by -3 and
    \nreverse the inequality symbol,
    \n4x > 24,
    \n4x\/4 > 24\/4, dividing both sides by 4 we get
    \nx > 6, so the graph of the solution x > 6 is as shown above.<\/p>\n

    Question 8.
    \n\"Envision
    \nAnswer:
    \nx < -8,<\/p>\n

    \"Envision
    \nExplanation:
    \nGiven to fill the boxes and to solve the -7x > 56 inequality,
    \n– 7x\/(-7) < 56\/(-8), dividing both sides by -7 and
    \nreverse the inequality symbol,
    \nx < -8, so the graph of the solution x < -8 is as shown above.<\/p>\n

    Question 9.
    \n\"Envision
    \nAnswer:
    \nm \u2264 -10,
    \n\"Envision
    \nExplanation:
    \nGiven to fill the boxes and to solve the \\(\\frac{m}{-5}\\) \u2265 2 inequality,
    \n-5 X \\(\\frac{m}{-5}\\) \u2264 -5(2), multiplying both sides by -5 and
    \nreverse the inequality symbol,
    \nm \u2264 -10, so the graph of the solution m \u2264 -10 is as shown above.<\/p>\n

    Question 10.
    \nKyra and five friends shared a bag of fruit snacks.
    \nEach person got no more than 3 fruit snacks.
    \nThe inequality x \u00f7 6 \u2264 3 represents this situation.
    \nSolve the inequality to find the possible numbers of fruit
    \nsnacks that were in the bag.
    \nAnswer:
    \nThere are at most 18 fruit snacks that were in the bag,<\/p>\n

    Explanation:
    \nGiven Kyra and five friends shared a bag of fruit snacks.
    \nEach person got no more than 3 fruit snacks.
    \nThe inequality x \u00f7 6 \u2264 3 represents this situation the
    \npossible numbers of fruit snacks that were in the bag are
    \nx \u00f7 6 \u2264 3,
    \n6 X (x \u00f7 6) \u2264 6 X 3, multiplying both sides by 6,
    \nx \u2264 18, therefore, there are at most 18 fruit snacks that were in the bag.<\/p>\n

    Question 11.
    \nOver the next 17 months, Eli needs to read more than 102 e-books.
    \nThe inequality 17x > 102 represents the number of e-books he
    \nneeds to read per month.
    \nSolve the inequality to find the number of e-books Eli needs to read per month.
    \nAnswer:
    \nEli needs to read at least 6 e -books per month,<\/p>\n

    Explanation:
    \nGiven over the next 17 months, Eli needs to read more than 102 e-books.
    \nThe inequality 17x > 102 represents the number of e-books he
    \nneeds to read per month. Now solving the inequality to
    \nfind the number of e-books Eli needs to read per month as
    \n17x > 102,
    \n17x \u00f717 > 102 \u00f717, dividing both sides by 17,
    \nx > 6, therefore, Eli needs to read at least 6 e -books per month.<\/p>\n

    Question 12.
    \nBrittney can spend no more than $15 for new fish in her aquarium.
    \na. Let f be the number of fish she can buy. What inequality represents the problem?
    \n\"Envision
    \nb. How many fish can Brittney buy?
    \nAnswer:
    \na. 3f \u2264 15,
    \nb. Brittney can buy no more than 5 fishes,<\/p>\n

    Explanation:
    \nGiven Brittney can spend no more than $15 for new fish in her aquarium.
    \na. Let f be the number of fish she can buy.
    \nSo inequality that represents the problem is 3f \u2264 15,
    \nb. Solving the inequality equation 3f \u2264 15,
    \n3f \u00f7 3 \u2264 15 \u00f7 3, dividing both sides by 3 we get,
    \nf \u2264 5, So Brittney can buy no more than 5 fishes.<\/p>\n

    Question 13.
    \nIsaac has a bag of n peanuts. He shares the peanuts with 5 of his friends.
    \nEach person gets at least 18 peanuts. The inequality 18 \u2264 n \u00f7 6
    \nrepresents this situation. Graph the solution of this inequality.
    \nAnswer:
    \nThe number of peanuts Isaac has a bag of at least 108,
    \n\"Envision<\/p>\n

    Explanation:
    \nGiven Isaac has a bag of n peanuts. He shares the peanuts with 5 of his friends.
    \nEach person gets at least 18 peanuts. The inequality 18 \u2264 n \u00f7 6
    \nrepresents this situation. So the number of peanuts Isaac have are
    \n6 X 18 \u2264 6 X n \u00f7 6, multiplying the inequality both sides by 6,
    \n108 \u2264 n or n \u2265 108, so the graph of the solution
    \n108 \u2264 n or n \u2265 108 is as shown above.<\/p>\n

    Question 14.
    \na. Solve the inequality – 3x < 12.
    \nb. Reasoning<\/strong>
    \nDescribe how you know the direction of the inequality sign
    \nwithout solving the inequality.
    \nAnswer:
    \na. x > -4,
    \nb. The inequality sign changes because of the
    \nnegative integer in the variable,<\/p>\n

    Explanation:
    \na. Given to solve the inequality -3x < 12,
    \n-3x \u00f7 -3< 12 \u00f7 -3, dividing\u00a0both sides by -3 and
    \nreverse the inequality symbol,
    \nx > -4.
    \nb. The inequality sign can be determined without solving
    \nbecause there is negative integer in the variable.
    \nWith the negative integer, automatically the inequality sign
    \nchanges.<\/p>\n

    Question 15.
    \nHigher Order Thinking<\/strong> Renata and her family go through an
    \naverage of more than 15 cans of sparkling water each day.
    \nThey buy cases of 24 cans at $3.50 a case.
    \n\"Envision
    \na. Write an inequality for the number of cases they go through in 30 days.
    \nb. Solve the inequality in part a. If they buy only full cases,
    \nhow much do they spend on sparkling water in 30 days?
    \nAnswer:
    \na. x \u2265 18.75 is the inequality for the number of cases
    \nthey go through in 30 days,
    \nb. $66.50 they spend on sparkling water in 30 days,<\/p>\n

    Explanation:
    \nGiven Renata and her family go through an
    \naverage of more than 15 cans of sparkling water each day.
    \nThey buy cases of 24 cans at $3.50 a case.
    \na.
    \n1 day = 15 cans and 24 cans = $3.50,
    \nSo let x be number of cases
    \n30 multiply 15 \u00f7 24 = 18.75 almost 19,
    \ntherefore x \u2265 18.75 is the inequality for the
    \nnumber of cases they go through in 30 days,
    \nb. 19 X 3.5 = $66.50 they spend on sparkling water in 30 days.<\/p>\n

    Question 16.
    \nSolve the inequality. Graph the solution on the number line.
    \n-6.25x > -38\\(\\frac{3}{4}\\)
    \n\"Envision
    \nAnswer:
    \nx < 6.2,
    \n\"Envision
    \nExplanation:
    \nGiven to solve the inequality -6.25x > -38\\(\\frac{3}{4}\\),
    \nconverting the mixed number to decimal before solving the inequality,
    \n-38\\(\\frac{3}{4}\\) = -38.75,
    \n-6.25x > -38.75,
    \n-6.25x \u00f7 -6.25 > -38.75 \u00f7 -6.25, dividing both sides by -6.25 and
    \nreverse the inequality symbol,
    \nx < 6.2, the graph of the solution x < 6.2 is shown above.<\/p>\n

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

    Question 17.
    \nCynthia plans to build a tree house that is \\(\\frac{1}{3}\\) the
    \nsize of Andrew’s tree house. Cynthia plans to make the area of her
    \ntree house at least 13 square feet.
    \nPART A<\/strong>
    \nWrite and solve an inequality to find the area of Andrew’s tree house.
    \nLet x be the area of Andrew’s tree house.
    \nAnswer:
    \n\\(\\frac{1}{3}\\)x \u2265 13,<\/p>\n

    Explanation:
    \nGiven Cynthia plans to build a tree house that is \\(\\frac{1}{3}\\) the
    \nsize of Andrew’s tree house. Cynthia plans to make the area of her
    \ntree house at least 13 square feet.
    \nThe inequality for the given situation is \\(\\frac{1}{3}\\)x \u2265 13,
    \nNow solving 3 X \\(\\frac{1}{3}\\)x \u2265 3 X 13,
    \nmultiplying both sides by 3 we get
    \nx \u2265 39 is the area of Andrew’s tree house.<\/p>\n

    PART B<\/strong>
    \nDescribe how you know which tree house is larger without solving the inequality.
    \nAnswer:
    \nIt is because of the factor that was multiplied to the
    \narea of the Andrew’s tree house which is less than 1,<\/p>\n

    Explanation:
    \nSince the factor that was multiplied to the
    \narea of the Andrew’s tree house which is less than 1,
    \nit can be easily be determined that Cynthia’s tree house is
    \nsmaller than Andrew’s tree house.<\/p>\n

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

    \"Envision<\/p>\n

    ACT 1<\/strong><\/p>\n

    Question 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 Make a prediction to answer this Main Question. Explain your prediction.<\/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. Too small
    \n\"Envision
    \n\"Envision
    \nAnswer:<\/p>\n

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

    ACT 2<\/strong><\/p>\n

    Question 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<\/strong> What tools can you use to solve the problem?
    \nExplain how you would use them strategically
    \nAnswer:<\/p>\n

    Question 8.
    \nModel with Math<\/strong> Represent the situation using mathematics.
    \nUse 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 initial prediction? Explain why.
    \n\"Envision
    \nAnswer:<\/p>\n

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

    Question 10.
    \nWrite the answer you saw in the video.
    \nAnswer:<\/p>\n

    Question 11.
    \nReasoning<\/strong> 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<\/strong> and Persevere Would you change your model now that you know the answer? Explain.
    \nAnswer:<\/p>\n

    \"Envision<\/p>\n

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

    Extension<\/p>\n

    Reflect<\/strong><\/p>\n

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

    Question 14.
    \nReasoning<\/strong> If all single tracks were on sale for 10% off, how would your model change?
    \nHow would the answer to the Main Question change?
    \n\"Envision
    \nAnswer:<\/p>\n

    SEQUEL<\/strong><\/p>\n

    Question 15.
    \nMake Sense and Persevere<\/strong> Suppose you have a $50 gift card to the same site.
    \nYou want to buy an album with 16 tracks for $12.99 and then use the rest of the
    \ngift card for single tracks. How many songs can you buy with the gift card?
    \nAnswer:<\/p>\n

    Lesson 5.6 Solve Two-Step Inequalities<\/h3>\n

    Solve & Discuss It!<\/strong><\/p>\n

    Rico and Halima are shopping for craft sticks, glue, and electrical tape for a science project.
    \nTogether, they have $30 to spend on supplies.
    \nHow should they spend their $30 if they need at least 1,000 craft sticks?
    \n\"Envision<\/p>\n

    Focus on math practices<\/strong>
    \nMake Sense and Persevere At the store, Rico and Halima find
    \nboxes of 500 craft sticks for $7.50.
    \nWhich boxes of craft sticks should they buy?
    \nAnswer:
    \nRico and Halima should buy the box with 500 craft sticks,<\/p>\n

    Explanation:
    \nRico and Halima should buy the box with 500 craft sticks,
    \nbuying 2 boxes of 500 craft sticks will give them a total
    \nof 1000 craft sticks at a lower price which will cost them $15,
    \ninstead of buying at least 4 boxes of 275 craft sticks.<\/p>\n

    Essential Question<\/strong>
    \nHow is solving a two-step inequality similar to and different from solving a two-step equation?
    \nAnswer:
    \nEquations\u00a0and\u00a0Inequalities\u00a0–\u00a0Two-step equations\u00a0and\u00a0inequalities –
    \nIt takes two steps\u00a0to\u00a0solve\u00a0an\u00a0equation\u00a0or\u00a0inequality\u00a0that has more than one operation:
    \nSimplify using the inverse of addition or subtraction. Simplify further by using
    \nthe inverse of multiplication or division.<\/p>\n

    Explanation:
    \nIt takes two steps to solve an equation or inequality that has more than one operation:
    \n<\/span>Simplify using the inverse of addition or subtraction.
    \n<\/span>Simplify further by using the inverse of multiplication or division.
    \n<\/span>when you multiply or divide an inequality by a negative number,
    \n<\/span>we must reverse the inequality symbol.<\/span><\/p>\n

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

    Erin has $52 to spend at the florist. She wants to buy a vase for $11.75 and
    \nseveral roses for $3.50 each. What are the possible numbers of roses Erin can buy?
    \n\"Envision
    \nAnswer:
    \nErwin can buy up to 11 roses,
    \n\"Envision<\/p>\n

    Explanation:
    \nGiven Erin has $52 to spend at the florist. She wants to buy a vase for $11.75 and
    \nseveral roses for $3.50 each. The possible numbers of roses Erin can buy are
    \n11.75 + 3.50r \u2264 52,
    \n11.75 + 3.50r – 11.75 \u2264 52 – 11.75, subtracting 11.75 from both sides,
    \n3.50r \u2264 40.25,
    \n3.50r \u00a0\u00f7 3.50 \u2264 40.25 \u00a0\u00f7 3.50, dividing both sides by 3.50 we get
    \nr \u2264 11.50, therefore Erwin can buy up to 11 roses.<\/p>\n

    Convince Me!<\/strong>
    \nWhat properties did you use to solve the inequality? 300 5-6 Solve Two-Step Inequalities.
    \nAnswer:
    \nSubtraction property,
    \nDivision property.<\/p>\n

    Explanation:
    \nSolving two-step inequalities,
    \nExample:
    \nx + 8 > 5,
    \nIsolated the variable x by subtracting 8 from both sides of the inequality.
    \nx + 8 \u2013 8 > 5 \u2013 8 => x > \u22123, Therefore, x > \u22123.<\/p>\n

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

    The Jazz Band needs to raise at least $600 to travel to an upcoming competition.
    \nThe members of the band have already raised $350. If they sell calendars for $8 each,
    \nhow many calendars would they need to sell to exceed their goal?
    \nAnswer:
    \nJazz band needs to sell at least 32 calendars to exceed their goal,<\/p>\n

    Explanation:
    \nGiven the Jazz Band needs to raise at least $600 to travel to an upcoming competition.
    \nThe members of the band have already raised $350. If they sell calendars for $8 each,
    \nnumber of calendars would they need to sell to exceed their goal if x is the
    \nnumber of calendars,
    \n350 + 8x \u2265 600,
    \n350 + 8x – 350 \u2265 600 -350, subtracting 350 from both sides,
    \n8x \u2265 250,
    \n8x \u00f7 8 \u2265 250 \u00f7 8, dividing both sides by 8,
    \nx \u2265 31.25, therefore, Jazz band needs to sell at least 32 calendars to exceed their goal.<\/p>\n

    Try It!<\/strong>
    \nSolve the inequality 5 – \\(\\frac{1}{2}\\)x > 30.
    \nAnswer:
    \nx < -50,<\/p>\n

    Explanation:
    \nGiven the inequality 5 – \\(\\frac{1}{2}\\)x > 30,
    \n5 – \\(\\frac{1}{2}\\)x -5 > 30 – 5, subtract 5 from both the sides,
    \n–\\(\\frac{1}{2}\\)x\u00a0 > 25,
    \n-2 X – \\(\\frac{1}{2}\\)x > -2 X 25, multiplying both sides by -2 and reverse
    \nthe inequality symbol as x < -50.<\/p>\n

    KEY CONCEPT<\/strong><\/p>\n

    Like two-step equations, solving two-step inequalities involves carrying out
    \ntwo different operations\u2014addition or subtraction, and multiplication or division.
    \nUnlike two-step equations, which have a single solution,
    \ntwo-step inequalities have multiple solutions.<\/p>\n

    Do You Understand?<\/strong><\/p>\n

    Question 1.
    \nEssential Question<\/strong> How is solving a two-step inequality similar to and
    \ndifferent from solving a two-step equation?
    \nAnswer:
    \nThe similarity between the two is that they are carrying out two
    \ndifferent operations,
    \nThe difference is that the number of solutions they have,<\/p>\n

    Explanation:
    \nThe similarity between solving a two-step inequality and
    \ntwo-step equation is that they are carrying out two
    \ndifferent operations,
    \nThe difference\u00a0between solving a two-step inequality and
    \ntwo-step equation is that two-step equation only have
    \nsingle solution unlike two-step inequalities have multiple solutions.<\/p>\n

    Question 2.
    \nReasoning<\/strong> What is the difference between the number of solutions for
    \na two-step equation and for a two-step inequality?
    \nAnswer:
    \nTwo-step equations only have a single solution,
    \nTwo-step inequalities have multiple solutions,<\/p>\n

    Explanation:
    \nTwo-step equations there is only a single solution,
    \nTwo-step inequalities there are multiple solutions,
    \nthis is because of the inequality symbol, which makes
    \nthe inequality true for as long as the value of the variable
    \nis within the given solution.<\/p>\n

    Question 3.
    \nWhy are inverse relationships between operations used to solve two-step inequalities?
    \nAnswer:
    \nInverse relationships between operations are used to isolate the variable on the one side
    \nof the inequality,<\/p>\n

    Explanation:
    \nInverse relationships between operations are used to solve two-step inequalities
    \nin order to isolate the variable on the one side of the inequality, removing the
    \nvalues together with the variable means that the inverse operation should be used
    \non both sides.<\/p>\n

    Do You Know How?<\/strong><\/p>\n

    Question 4.
    \nJoe ran 3 miles yesterday and wants to run at least 12 miles this week.
    \nWrite an inequality that can be used to determine the additional number
    \nof days Joe must run this week if each run is 3 miles. Then solve the inequality.
    \n\"Envision
    \nAnswer:
    \nThe inequality for the given situation is 3 + 3x \u00a0\u2265 12,
    \nJoe need to run at least 3 days to meet his target,<\/p>\n

    Explanation:
    \nGiven Joe ran 3 miles yesterday and wants to run at least 12 miles this week.
    \nLet x be number of days Joe must run this week if each run is 3 miles.
    \nThe inequality for the given situation is 3 + 3x \u00a0\u2265 12,
    \n3 + 3x -3 \u2265 12 -3, subtracting 3 both sides,
    \n3x \u00a0\u2265 9, dividing both sides by 3 we get
    \nx \u00a0\u2265 3, therefore, Joe need to run at least 3 days to meet his target.<\/p>\n

    Question 5.
    \nSolve 4 + 6.5x < 36.5.
    \nAnswer:
    \nSolving 4 + 6.5x < 36.5, we get x < 5,<\/p>\n

    Explanation;
    \nGiven the inequality 4 + 6.5x < 36.5,
    \n4 + 6.5x -4 < 36.5 -4, subtracting 4 from both the sides,
    \n6.5x < 32.5,
    \n6.5x \u00f7 6.5 < 32.5 \u00f7 6.5, dividing both sides by 6.5,
    \nx < 5.<\/p>\n

    Question 6.
    \nTomas has $1,000 to spend on a vacation. His plane ticket costs $348.25.
    \nIf he stays 5.5 days at his destination, how much can he spend each day?
    \nWrite an inequality and then solve.
    \nAnswer:
    \nInequality: 348.25 + 5.5x \u2264 1000,
    \nTomas can spend no more than $118.50 per day,<\/p>\n

    Explanation:
    \nGiven Tomas has $1,000 to spend on a vacation. His plane ticket costs $348.25.
    \nIf he stays 5.5 days at his destination let x be the amount,
    \nthe inequality for the given situation is 348.25 + 5.5x \u2264 1000,
    \n348.25 + 5.5x – 348.25 \u2264 1000 – 348.25, subtracting 348.25 from both the sides,
    \n5.5x \u2264 651.75,
    \n5.5x \u00f7 5.5 \u2264 651.75 \u00f7 5.5, dividing both sides by 5.5,
    \nx \u2264 118.50, therefore, Tomas can spend no more than $118.50 per day.<\/p>\n

    Question 7.
    \nSolve 12 – \\(\\frac{3}{5}\\)x > 39.
    \nAnswer:
    \nSolving 12 – \\(\\frac{3}{5}\\)x > 39, we get x < -45,<\/p>\n

    Explanation:
    \nGiven 12 – \\(\\frac{3}{5}\\)x > 39,
    \n12 – \\(\\frac{3}{5}\\)x – 12 > 39 – 12, subtracting 12 from both sides,
    \n– \\(\\frac{3}{5}\\)x > 27,
    \n– 5 X- \\(\\frac{3}{5}\\)x < – 5 X 27, multiplying both sides by -5 and
    \nreverse the inequality symbol,<\/p>\n

    3x < -135,
    \n3x \u00f7 3 < -135 \u00f7 3, divide both sides by 3,
    \nx < -45.<\/p>\n

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

    Leveled Practice For 8 and 9, fill in the boxes to write and solve each inequality.<\/p>\n

    Question 8.
    \nEight less than the product of a number n and \\(\\frac{1}{5}\\) is no more than 95.
    \n\"Envision
    \nAnswer:
    \nn \u2264\u00a0 515,
    \n\"Envision<\/p>\n

    Explanation:
    \nThe inequality is \\(\\frac{1}{5}\\) x – 8 \u2264 95,
    \n\\(\\frac{1}{5}\\) x – 8 + 8 \u2264 95 +8, adding 8 on both\u00a0 sides,
    \n\\(\\frac{1}{5}\\) x \u2264 103,
    \n5 X \\(\\frac{1}{5}\\) x \u2264 5 X 103, multiplying both sides by 5,
    \nx \u2264 515.<\/p>\n

    Question 9.
    \nSeven more than the quotient of a number b and 45 is greater than 5.
    \n\"Envision
    \nAnswer:
    \nb > -90,
    \n\"Envision
    \nExplanation:
    \nGiven \\(\\frac{b}{45}\\) + 7 > 5,
    \n\\(\\frac{b}{45}\\) + 7\u00a0 – 7 > 5 – 7, subtracting 7 from both sides
    \n\\(\\frac{b}{45}\\)\u00a0 > -2,
    \n45 X \\(\\frac{b}{45}\\) > 45 X (-2), multiplying both sides by 45 we get,
    \nb > -90.<\/p>\n

    Question 10.
    \nSolve the inequalities and compare.
    \na. Solve 2x + 6 < 10.
    \nb. Solve -2x + 22 < 18.
    \nc. Which is the correct comparison of solutions for
    \n2x + 6 < 10 and -2x + 22 < 18?
    \nA. The inequalities have some common solutions.
    \nB. The inequalities have one common solution.
    \nC. The inequalities have no common solutions.
    \nD. The inequalities have the same solutions.
    \nAnswer:
    \na. x < 2,
    \nb. x > 2,
    \nc. C. The inequalities have no common solutions,<\/p>\n

    Explanation:
    \na. Solving 2x + 6 < 10,
    \n2x + 6 -6 < 10 -6, subtracting 6 from both sides,
    \n2x < 4,
    \n2x \u00f7 2 < 4 \u00f7 2, dividing both sides by 2,
    \nso x < 2,
    \nb. Solving -2x + 22 < 18,
    \n-2x +22 -22 < 18 -22, subtracting 22 from both sides,
    \n-2x < -4,
    \n-2x \u00f7 -2 > -4 \u00f7 -2, dividing both sides by -2 and reverse the inequality symbol,
    \nso x > 2,
    \nc. The \u00a0correct comparison of solutions for the two given
    \ninequalities 2x + 6 < 10 and -2x + 22 < 18 are
    \nthe inequalities have no common solutions, the first inequality
    \nhas a solution that the value of x should be less than 2 to make the
    \ninequality true, the second inequality has a solution that the value of
    \nx should be greater than 2.<\/p>\n

    Question 11.
    \nMake Sense and Persevere Talia has a daily budget of $94 for a car rental.
    \nWrite and solve an inequality to find the greatest distance Talia can drive
    \neach day while staying within her budget.
    \n\"Envision
    \nAnswer:
    \nTali can drive no more than 320 miles,<\/p>\n

    Explanation:
    \nLet x be the distance per mile,
    \nThe inequality for the given situation is 30 + 0.20x \u2264 94,
    \nsolving 30 + 0.20x \u2264 94,
    \n30 + 0.20x -30 \u2264 94 -30, subtracting 30 from both sides,
    \n0.20 x \u2264 64,
    \n0.20 x \u00f70.20 \u2264 64 \u00f7 0.20,dividing both sides by 0.20
    \nx \u2264 320, therefore, Tali can drive no more than 320 miles.<\/p>\n

    Question 12.
    \nModel with Math<\/strong> A manager needs to rope off a
    \nrectangular section for a private party. The length of
    \nthe section must be 7.6 meters. The manager can use
    \nno more than 28 meters of rope. What inequality could
    \nyou use to find the possible width, w, of the roped-off section?
    \nAnswer:
    \nInequality:
    \n2(7.6) + 2(w) \u2264 28,
    \nThe width of the roped – off section should be no
    \nmore than 6.4 meters,<\/p>\n

    Explanation:
    \nGiven A manager needs to rope off a
    \nrectangular section for a private party. The length of
    \nthe section must be 7.6 meters. The manager can use
    \nno more than 28 meters of rope. The inequality to find
    \nthe possible width, w, of the roped-off section is 2(7.6) + 2(w) \u2264 28,
    \n15.2 + 2w \u2264 28, multiplying the values,
    \n15.2 + 2w – 15.2 \u2264 28 – 15.2, substracting 15.2 from both sides,
    \n2w \u2264 12.8,
    \n2w \u00f7 2 \u2264 12.8 \u00f7 2, dividing both sides by 2,
    \nw \u2264 6.4 , therefore, the width of the roped – off section should be no
    \nmore than 6.4 meters.<\/p>\n

    Question 13.
    \nHigher Order Thinking<\/strong> Andrea went to the store to buy a
    \nsweater that was on sale for 40% off the original price.
    \nIt was then put on clearance at an additional 25% off the sale price.
    \nShe also used a coupon that saved her an additional $5.
    \nAndrea did not spend more than $7.60 for the sweater.
    \nWhat are the possible values for the original price of the sweater?
    \n\"Envision
    \nAnswer:
    \nThe price of the sweater is no more than $28,<\/p>\n

    Explanation:
    \nGiven Andrea went to the store to buy a
    \nsweater that was on sale for 40% off the original price.
    \nIt was then put on clearance at an additional 25% off the sale price.
    \nShe also used a coupon that saved her an additional $5.
    \nAndrea did not spend more than $7.60 for the sweater.
    \nLet x be the original price of the sweater so the inequality
    \nis 0.75(0.6x) – 5 \u2264 7.6, multiplying the values
    \n0.75(0.6x) – 5 \u2264 7.6, adding 5 both sides,
    \n0.45 x – 5 + 5\u00a0 \u2264 7.6 + 5,
    \n0.45x \u2264 12.6,
    \n0.45x \u00f7 0.45 \u2264 12.6 \u00f7 0.45, dividing both sides by 0.45,
    \nx \u2264 28, the price of the sweater is no more than $28.<\/p>\n

    Question 14.
    \nA pool can hold 850 gallons. It now has 598 gallons of
    \nwater and is being filled at the rate shown.
    \nHow many more minutes, m, can water continue to
    \nflow into the pool before it overflows? Write and solve an inequality.
    \n\"Envision
    \nAnswer:
    \nInequality :
    \n598 + 15.75m \u2264 850,
    \nThe water can\u00a0 continue to flow into the pool before it overflows
    \nno more 16 minutes,<\/p>\n

    Explanation:
    \nGiven A pool can hold 850 gallons. It now has 598 gallons of
    \nwater and is being filled at the rate shown. So\u00a0more minutes, m,
    \ncan water continue to flow into the pool before it overflows is
    \nsolving the inequality 598 + 15.75m \u2264 850,
    \n598 + 15.75m – 598 \u2264 850 – 598, subtracting 598 from both sides,
    \n15.75m \u2264 252,
    \n15.75m \u00f715.75 \u2264 252 \u00f7 15.75, dividing both sides by 15.75 we get
    \nm \u2264 16, therefore the water can\u00a0 continue to flow into the pool
    \nbefore it overflows no more 16 minutes.<\/p>\n

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

    Question 15.
    \nUse the rectangle diagram at the right.
    \nPART A<\/strong>
    \nWrite and solve an inequality to find the values of x for
    \nwhich the perimeter of the rectangle is less than 120.
    \n\"Envision
    \nAnswer:
    \nInequality:
    \n2(x + 4) + 2x < 120,
    \nx < 28,<\/p>\n

    Explanation:
    \nGiven length of rectangle x + 4 and breadth as x and
    \nfor which the perimeter of the rectangle is less than 120,
    \nThe inequality for the given situation is 2(x + 4) + 2x < 120,
    \n2x + 8 +2x < 120, using distributive property,
    \n4x + 8 < 120, combining like terms,
    \n4x + 8 – 8 < 120 – 8, subtracting 8 from both sides,
    \n4x < 112,
    \n4x \u00f7 4 < 112 \u00f7 4, dividing both sides by 4 we get
    \nx < 24.<\/p>\n

    PART B<\/strong>
    \nBased on your answer to Part A, are there any values that
    \ncan be eliminated from the solution set? Explain.
    \nAnswer:
    \nYes, 28 can be eliminated from the solution set,<\/p>\n

    Explanation:
    \nThere is a value that can be eliminated in the solution set,
    \n28 is not part of the solution set since the solution set is
    \nless than 28, it means any value that is 27 or below.<\/p>\n

    Question 16.
    \nKari is building a rectangular garden bed. The length is 6 feet.
    \nShe has 20 feet of boards to make the sides.
    \nWrite and solve an inequality to find the possible width of her garden bed.
    \nAnswer:
    \n2(6 + w) \u2264 20,
    \nThe width of Kari garden bed is less than 4,<\/p>\n

    Explanation:
    \nGiven Kari is building a rectangular garden bed. The length is 6 feet.
    \nShe has 20 feet of boards to make the sides.
    \nLet w be the possible width of her garden bed.
    \nThe inequality is 2(6 + w) \u2264 20,solving for w,
    \n2(6 + w) \u2264 20, multiplying the values,
    \n12 + 2w \u2264 20, subtracting 12 both sides,
    \n12 + 2w -12 \u2264 20 -12, subtracting 12 both sides,
    \n2w \u2264 8,
    \n2w \u00f7 2 \u2264 8 \u00f7 2, dividing both sides by 2,
    \nw \u2264 4, therefore the width of Kari garden bed is less than 4.<\/p>\n

    Lesson 5.7 Solve Multi-Step Inequalities<\/h3>\n

    Explore It<\/strong><\/p>\n

    Charlene has 2 flash drives of the same size that
    \nshe uses to store pictures and videos. Each drive is
    \nholding the same number of GB of data, d.
    \nShe wants to move everything to a memory card that can hold up to 8 GB.
    \n\"Envision<\/p>\n

    A. Charlene is going to delete 1 GB of data from each flash drive.
    \nHow can the total amount of data left on the two flash drives be
    \nrepresented as an expression?
    \nB. How can the expression you wrote be used to
    \nwrite an inequality that shows the maximum amount of
    \ndata each flash drive can have on it in order to have all
    \nthe data transfer to the 8 GB memory card?
    \nAnswer:
    \na. 2(d – 1),
    \nb. 2(d – 1) \u2264 8,<\/p>\n

    Explanation:
    \nGiven Charlene has 2 flash drives of the same size that
    \nshe uses to store pictures and videos. Each drive is
    \nholding the same number of GB of data, d.
    \nShe wants to move everything to a memory card that can hold up to 8 GB.
    \na. Charlene is going to delete 1 GB of data from each flash drive.
    \nSo the total amount of data left on the two flash drives be
    \nrepresented as an expression as 2(d – 1).
    \nb. The expression I wrote can be used
    \nfor the inequality that shows the maximum amount of
    \ndata each flash drive can have on it in order to have all
    \nthe data transfer to the 8 GB memory card is 2(d – 1) \u2264 8.<\/p>\n

    Focus on math practices<\/strong>
    \nReasoning<\/strong> If each flash drive has 5 GB of memory, can all of
    \nthe data be transferred to the memory card? Explain.
    \nAnswer:
    \nA flash drive has 5 GB of memory, can have all of its
    \ndata be transferred to the memory card,<\/p>\n

    Explanation:
    \nUsing the inequality to determine If each flash drive
    \nhas 5 GB of memory can be transferred to the memory card so
    \n2(d – 1) \u2264 8,
    \n2d – 2 \u2264 8, applying distributive property,
    \n2d – 2 + 2 \u2264 8 + 2, adding 2 on both sides,
    \n2d \u2264 10,
    \n2d \u00f7 2 \u2264 10 \u00f7 2, dividing both sides by 2,
    \nd \u2264 5,
    \nFrom the solution set, a flash drive can hold no more than
    \n5 GB, therefore, a\u00a0flash drive has 5 GB of memory,
    \ncan have all of its data be transferred to the memory card.<\/p>\n

    Essential Question<\/strong>
    \nHow is solving a multi-step inequality similar to and different
    \nfrom solving a multi-step equation.
    \nAnswer:
    \nTwo are similar because of the methods used in solving,
    \nThe difference between the two is the number of solution,<\/p>\n

    Explanation:
    \nSolving a multi-step inequality and solving a multi-step equation is similar
    \nbecause of the methods used in solving like using the Distributive Property,
    \ncombining like terms and use inverse relationships and properties to solve them.
    \nThe difference between a multi-step inequality and solving a multi-step equation is
    \nthe number of solution. For multi-step equation there is one possible solution,
    \nfor multi-step inequality there are many possible solutions.<\/p>\n

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

    Twice the difference of Felipe’s age, f, and 4 is at least 2.
    \nWhat are possible values for Felipe’s age? Graph the solution.
    \n\"Envision<\/p>\n

    \"Envision
    \nExplanation:
    \nGiven twice the difference of Felipe’s age, f, and 4 is at least 2,
    \nthe possible values for Felipe’s age are inequality 2(f-4) \u2265 2,
    \n2f – 8 \u2265 2, used distributive property,
    \n2f – 8 + 8 \u2265 2 + 8, adding 8 on both sides,
    \n2f \u00f7 2 \u2265 10 \u00f7 2, dividing both sides by 2,
    \nf \u2265 5, so the graph of the solution f \u2265 5 is as shown above.<\/p>\n

    Convince Me!<\/strong>
    \nDescribe the similarity between the process of solving an inequality with two steps and
    \nsolving an inequality with more than two steps.<\/p>\n

    Try It!<\/strong>
    \nSolve the inequality – 1 – 6(6 + 2x) < 11. Then graph the solution.
    \n\"Envision
    \nSolve the inequality 3(4 – 6) + 2 \u2265 2(-t + 3) + 4. Then graph the solution.
    \n\"Envision
    \nAnswer:
    \n\"Envision
    \n\"Envision
    \nExplanation:
    \nSolving the inequality – 1 – 6(6 + 2x) < 11,
    \n-1 -36 -12x < 11, used distributive property,
    \n-37 -12x < 11, combining like terms,
    \n-37 -12x + 37 < 11 + 37, Adding 37 on both sides,
    \n-12x > 48,
    \n-12x \u00f7 12 > 48 \u00f7 12, dividing both sides by -12 and reverse inequality symbol,
    \nx > 4, so the graph of the solution x > 4 is as shown above.<\/p>\n

    Solving the inequality 3(4 – 6) + 2 \u2265 2(-t + 3) + 4,
    \n12 – 18 + 2 \u2265 -2t + 6 + 4, used distributive property,
    \n-4 \u2265 -2t + 10, combining like terms,
    \n-4 – 10 \u2265 -2t + 10 – 10, subtracting 10 from both sides,
    \n-14 \u2265 -2t,
    \n-14 \u00f7 -2 \u2264 -2t \u00f7 -2, dividing both sides by -2 and reverse inequality symbol,
    \n7 \u2264 t, so the graph of the solution 7 \u2264 t is as shown above.<\/p>\n

    KEY CONCEPT<\/strong><\/p>\n

    Solving multi-step inequalities is similar to solving multi-step equations.
    \nYou may need to use the Distributive Property, combine like terms, and
    \nuse inverse relationships and properties to solve them.
    \n4(y \u2013 4) + 8 \u2264 20
    \n4y – 16 + 8 \u2264 20
    \n4y – 8 \u2264 20
    \n4y – 8 + 8 \u2264 20 + 8
    \n4y \u2264 28
    \n\"Envision<\/p>\n

    Do You Understand?<\/strong><\/p>\n

    Question 1.
    \nEssential Question<\/strong> How is solving a multi step inequality similar to and
    \ndifferent from solving a multi-step equation?
    \nAnswer:
    \nTwo are similar because of the methods used in solving,
    \nThe difference between the two is the number of solution,<\/p>\n

    Explanation:
    \nSolving a multi-step inequality and solving a multi-step equation is similar
    \nbecause of the methods used in solving like using the Distributive Property,
    \ncombining like terms and use inverse relationships and properties to solve them.
    \nThe difference between a multi-step inequality and solving a multi-step equation is
    \nthe number of solution. For multi-step equation there is one possible solution,
    \nfor multi-step inequality there are many possible solutions.<\/p>\n

    Question 2.
    \nBe Precise<\/strong> Explain how you would combine like terms and use properties of
    \noperations to solve the inequality 5(2t + 3) \u2013 3t < 16.
    \nAnswer:
    \nt < 1\/7,<\/p>\n

    Explanation:
    \nGiven to solve the inequality 5(2t + 3) \u2013 3t < 16,
    \n10t + 15 – 3t <16, used distributive property,
    \n7t + 15 < 16, combining like terms,
    \n7t + 15 -15 < 16 – 15, subtracting 15 from both sides of the inequality,
    \n7t < 1,
    \n7t \u00f7 7 < 1 \u00f77, dividing both sides by 7 we get,
    \nt < 1\/7.<\/p>\n

    Question 3.
    \nCritique Reasoning Gloria’s solution to a multi-step inequality is r> 7.
    \nShe states that the graph will have an open dot at 7 and extend with
    \nan arrow to the right indefinitely. Is she correct? Explain.
    \nAnswer:
    \nGloria’s describes the solution correctly,<\/p>\n

    Explanation:
    \nGloria is correct because it will open dot as the solution does not
    \ninclude 7 as the part of the solution, the inequality symbol is greater
    \nthan which means that the arrow will extend to the right.<\/p>\n

    Do You Know How?<\/strong><\/p>\n

    Question 4.
    \nSolve the inequality 2(n + 3) \u2013 4 < 6. Then graph the solution.
    \n\"Envision
    \nAnswer:
    \n\"Envision<\/p>\n

    Explanation:
    \nGiven 2(n + 3) \u2013 4 < 6,
    \n2n + 6 – 4 < 6, using distributive property,
    \n2n + 2 < 6, combined like terms,
    \n2n + 2 – 2 < 6 – 2, subtracting 2 from both sides,
    \n2n < 4,
    \n2n \u00f7 2 < 4 \u00f7 2, dividing both sides by 2,
    \nn < 2, so the graph of the solution n < 2 is as shown above.<\/p>\n

    Question 5.
    \nSolve the inequality -2(x + 3) + 2 \u2265 6. Then graph the solution.
    \n\"Envision
    \nAnswer:
    \n\"Envision<\/p>\n

    Explanation:
    \nGiven -2(x + 3) + 2 \u2265 6,
    \n-2x – 6 + 2 \u2265 6, using distributive property,
    \n-2x – 4 \u2265 6, combined like terms,
    \n-2x – 4 + 4 \u2265 6 + 4, adding 4 on both sides,
    \n-2x \u2265 10,
    \n-2x \u00f7 -2 \u2264\u00a010 \u00f7 -2, dividing both sides by -2 and reversing inequality symbol,
    \nx \u2264 5, so the graph of the solution x \u2264 5 is as shown above.<\/p>\n

    Question 6.
    \nThree times the difference of Federico’s age and 4, increased by 7, is greater than 37.
    \nWhat are possible values of Federico’s age? Graph his possible ages on the number line.
    \n\"Envision
    \nAnswer:
    \n\"Envision<\/p>\n

    Explanation:
    \nGiven three times the difference of Federico’s age and 4, increased by 7
    \nis greater than 37. The possible values of Federico’s age are
    \n3(x – 4) + 7 > 37,
    \n3x – 12 + 7 > 37, using distributive property,
    \n3x – 5 > 37, combined like terms,
    \n3x – 5 + 5 > 37 + 5, adding 5 on both sides,
    \n3x > 42,
    \n3x \u00f7 3 > 42 \u00f7 3, dividing both sides by 3,
    \nx > 14, so there Federico’s age should be greater than 14,
    \nThe graph of the possible ages of Federico’s is as shown above.<\/p>\n

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

    Question 7.
    \nUse the inequality 18 < -3(4x \u2013 2).
    \na. Solve the inequality for x.
    \nb. Which graph shows the solution to the inequality?
    \n\"Envision
    \nAnswer:
    \n\"Envision
    \nExplanation:
    \na. Given to solve the inequality 18 < -3(4x -2),
    \n18 < -12x + 6, using distributive property,
    \n18 – 6 < -12x + 6 – 6, subtracting 6 from both sides,
    \n12 < -12x,
    \n12 \u00f7 -12 > -12x \u00f7 -12, dividing both sides by -12 and\u00a0reversing inequality symbol,
    \n-1 > x,
    \nb. Graph A shows the solution to the inequality 18 < -3(4x -2) as -1 > x which
    \nis shown above.<\/p>\n

    Question 8.
    \nMichelle says that the solution to the inequality 2(4y – 3) > -22 is y> -3.5.
    \nHer work is shown.
    \n2(4y – 3) > -22
    \n8y > -28
    \ny > -3.5
    \na. What was Michelle’s mistake?
    \nb. What is the solution to the inequality?
    \nAnswer:
    \na. Michelle did not use the inverse relationship in order to isolate the variable,
    \nb. The solution is y > -2,<\/p>\n

    Explanation:
    \na. After using distributive property, Michelle did not use the inverse relationship
    \nin order to isolate the variable, she thought that subtracting 6 on the right side of inequality
    \nwill remove the 6 as well on left side of the inequality,
    \nb. Solving the inequality 2(4y – 3) > -22,
    \n8y – 6 > -22, using distributive property,
    \n8y – 6 + 6 > -22 + 6, adding 6 on both sides,
    \n8y > -16,
    \n8y \u00f7 8 > -16 \u00f7 8, dividing both sides by 8 we get
    \ny > -2.<\/p>\n

    Question 9.
    \nModel with Math<\/strong> The length of a picture frame is 7 inches more than the width.
    \nFor what values of x is the perimeter of the picture frame greater than 154 inches?
    \n\"Envision
    \nAnswer:
    \nThe width of the picture frame is greater than 35 inches,<\/p>\n

    Explanation:
    \nGiven the length of a picture frame is 7 inches more than the width.
    \nSo the values of x for which the perimeter of the picture frame will be
    \ngreater than 154 inches is
    \n2(x + 7) + 2 (x) > 154,
    \n2x + 14 + 2x > 154, using distributive property,
    \n4x + 14 > 154, combined like terms,
    \n4x + 14 – 14 > 154 – 14, subtracting 14 on both sides,
    \n4x > 140,
    \n4x \u00f7 4 > 140 \u00f7 4, dividing both sides by 4 we get
    \nx > 35, therefore the width of the picture frame is greater than 35 inches.<\/p>\n

    Question 10.
    \nCritique Reasoning<\/strong> Sierra says that she can simplify the left side of
    \nthe inequality 2(-3 + 5) + 2 \u2265 -4(x – 2) \u2013 3 by combining the terms
    \nwithin the parentheses, but that she can’t do the same on the right side.
    \nIs Sierra correct? Explain.
    \nAnswer:
    \nSierra is correct about combining the terms,<\/p>\n

    Explanation:
    \nSierra is correct as the terms inside the parentheses are both constant,
    \nPerforming the operation inside the parentheses can combine
    \nthe two constants. She is also correct that on the right side of inequality,
    \nthe term are not the same, therefore it cannot be combined.<\/p>\n

    Question 11.
    \na. Solve the inequality 30 \u2265 6(\\(\\frac{2}{3}\\)z + \\(\\frac{1}{3}\\)).
    \nb. Solve the inequality 15.6 < 2.7(z \u2013 1) – 0.6.
    \nc. Are there any values of z that solve both inequalities?
    \nUse a number line to support your answer.
    \nAnswer:
    \na. 7 \u2265 z,
    \nb. 7< z,
    \nc. \"Envision<\/p>\n

    Explanation:
    \na. Solving the inequality 30 \u2265 6(\\(\\frac{2}{3}\\)z + \\(\\frac{1}{3}\\)),
    \n30 \u2265 4z + 2, using distributive property,
    \n30 – 2 \u2265 4z + 2 – 2, subtracting 2 on both sides,
    \n28 \u2265 4z,
    \n28 \u00f7 4 \u2265 4z \u00f7 4, dividing both sides by 4 we get 7 \u2265 z.<\/p>\n

    b. Solving the inequality 15.6 < 2.7(z \u2013 1) – 0.6,
    \n15.6 < 2.7z – 2.7 – 0.6, using distributive property,
    \n15.6 < 2.7z – 3.3, combined like terms,
    \n15.6 + 3.3 < 2.7z – 3.3 + 3.3, adding 3.3 on both sides,
    \n18.9 < 2.7z,
    \n18.9 \u00f7 2.7 < 2.7z \u00f7 2.7, dividing both sides by 2.7,
    \nwe get 7 < z.<\/p>\n

    c. There are no values that could solve both inequalities,
    \nThis is because, the first inequality has the solution set of any
    \nvalue no more than 7 while the second inequality has the solution
    \nset of any value greater than 7. The solution of first inequality, includes
    \n7 while the solution for the second inequality does not include 7.<\/p>\n

    Question 12.
    \nMr. Lin baked banana bread for a bake sale to raise money for the math team.
    \nHe said that he added a spoonful of walnuts for each of the students in
    \nhis three classes, and that he added more than 250 walnuts. He used the
    \ninequality 16W + 24W + 10w > 250 to represent the situation, where w represents
    \nthe number of walnuts in each spoonful. How many walnuts could be in each spoonful?
    \n\"Envision
    \nAnswer:
    \nEach spoonful have greater than 5 walnuts,<\/p>\n

    Explanation:
    \nGiven Mr. Lin baked banana bread for a bake sale to raise money for the math team.
    \nHe said that he added a spoonful of walnuts for each of the students in
    \nhis three classes and that he added more than 250 walnuts. He used the
    \ninequality 16W + 24W + 10w > 250 to represent the situation, where w represents
    \nthe number of walnuts in each spoonful. So number of many walnuts could be in
    \neach spoonful is writing inequality 16W + 24W + 10w > 250 and solving as
    \n50w > 250, combining like terms,
    \n50w \u00f7 50 > 250 \u00f7 50, dividing both sides by 50,
    \nwe get w >5, therefore, each spoonful have greater than 5 walnuts.<\/p>\n

    Question 13.
    \nUse both the Addition and Multiplication Properties of Inequality
    \nto solve the inequality.
    \nGraph the solutions on a number line. 2(3y \u2013 5) < -16
    \n\"Envision
    \nAnswer:
    \n\"Envision<\/p>\n

    Explanation:
    \nSolving the inequality 2(3y – 5) < -16,
    \n6y – 10 < -16, using distributive property,
    \n6y – 10 + 10 < -16 + 10, adding 10 on both sides,
    \n6y < -6,
    \n6y \u00f7 6 < -6 \u00f7 6, dividing both sides by 6,
    \nwe get y < -1.<\/p>\n

    Question 14.
    \nHigher Order Thinking<\/strong> Solve each of the given inequalities for z.
    \nWhich of the inequalities has 5 as a solution?
    \n\"Envision
    \nAnswer:
    \nThe inequality that has a solution of 5 is Inequality 1,<\/p>\n

    Explanation:
    \nGiven to solve the inequality 4(2.8z +1.75) > -26.6,
    \n11.2z + 7 > -26.6, using distributive property,
    \n11.2z + 7 – 7 > -26.6 – 7, subtracting 7 on both sides,
    \n11.2z > -33.6,
    \n11.2z \u00f7 11.2 > -33.6 \u00f7 11.2, dividing both sides by 11.2,
    \nz > -3,
    \nGiven to solve the inequality 2(1.9z +1.5) \u2264 18.2,
    \n3.8z + 3 \u2264 18.2, using distributive property,
    \n3.8z + 3 – 3 \u2264 18.2 – 3 subtracting 7 on both sides,
    \n3.8z \u2264 15.2,
    \n3.8z \u00f7 3.8 \u2264 15.2 \u00f7 3.8, dividing both sides by 3.8,
    \nz \u2264 4,
    \nSo the inequality that has a solution of 5 is Inequality 1.<\/p>\n

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

    Question 15.
    \nThe school band needs $500 to buy new hats. They already have $200.
    \nThey are selling bumper stickers for $1.50 each.
    \nHow many bumper stickers do they need to sell to have
    \nat least $500? Write and solve an inequality that represents the situation.
    \nAnswer:
    \n200 bumper stickers do they need to sell,
    \nInequality : 1.50b + 200 = 500,<\/p>\n

    Explanation:
    \nGiven the school band needs $500 to buy new hats. They already have $200.
    \nThey are selling bumper stickers for $1.50 each.
    \nLet b be the number of\u00a0 bumper stickers do they need to sell to have
    \nat least $500? The inequality that represents the situation is
    \n1.50b + 200 \u2264 500, solving,
    \n1.50b + 200 – 200 \u2264 500 – 200, subtracting 200 both sides,
    \n1.50b \u2264 300,
    \n1.50b \u00f71.50 \u2264 300 \u00f7 1.50, dividing both sides by 1.50,
    \nb \u2264 200, therefore 200 bumper stickers do they need to sell and
    \nInequality : 1.50b + 200 = 500.<\/p>\n

    Topic 5 REVIEW<\/h3>\n

    Topic Essential Question<\/strong><\/p>\n

    How can you solve real-world and mathematical problems with numerical and
    \nalgebraic equations and inequalities?
    \nAnswer:
    \nThrough the use of equations and inequalities , we will be able to determine
    \nthe possible solutions to the problem,<\/p>\n

    Explanation:
    \nReal world and mathematical problems can be solved through the use
    \nof numerical and algebraic equations and inequalities.
    \nIn a certain problem there are unknown that needs to be solved.
    \nTherefore, using equations and inequalities in order to determine
    \nthe unknown it will help to easily solve the problem.
    \nThrough the use of equations and inequalities we will be
    \nable to determine the possible solutions to the problem.<\/p>\n

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

    Complete each definition and then provide an example of each vocabulary word used.
    \n\"Envision<\/p>\n

    Question 1.
    \nYou \"Envision when you divide both sides of the equation 3n = 12 by 3.
    \nAnswer:
    \nIsolate the variable,
    \nWhen we divide both sides of the equation 3n = 12 by 3, we isolate
    \nthe variable,<\/p>\n

    Explanation:
    \n3n = 12,
    \n3n \u00f7 3 = 12 \u00f7 3, Dividing both sides by 3 we get n = 4.<\/p>\n

    Question 2.
    \nA statement that contains >, <, \u2265, \u2264, or \u2260 to compare two expressions is a(n)
    \nAnswer:
    \nInequality,<\/p>\n

    Explanation:
    \nInequality is a statement that contains >, <, \u2265, \u2264, or \u2260 to compare two expressions.<\/p>\n

    Question 3.
    \nYou can use the \"Envision to remove parentheses in
    \nthe process of solving the equation – 10(x + 5) = 40.
    \nAnswer:
    \nDistributive Property,<\/p>\n

    Explanation:
    \nIn the the process of solving the equation – 10(x + 5) = 40,
    \nwe use distributive property to remove parentheses as
    \n-10x -50 = 40.<\/p>\n

    Use Vocabulary in Writing<\/strong>
    \nWrite an equation or inequality to represent the following situation:
    \n17 is at least 5 more than 3 times x. Explain how you wrote your
    \nequation or inequality. Use vocabulary from Topic 5 in your explanation.
    \nAnswer:
    \nEquation or Inequality : 17 > 3x + 5,<\/p>\n

    Explanation:
    \nGiven the situation as 17 is at least 5 more than 3 times x,
    \nSo wrote the equation or inequality as 17 > 5 + 3x means
    \n17 is greater than 5 and 3 times of x(3x), Vocabulary used is
    \nequation or inequality.<\/p>\n

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

    LESSONS 5-1 AND 5-2<\/strong>
    \nWrite Two-Step Equations | Solve Two-Step Equations<\/strong><\/p>\n

    Quick Review<\/strong>
    \nEquations can be used to represent situations. Two-step equations have two
    \ndifferent operations.
    \nThe properties of equality can be applied the same way when solving
    \ntwo-step equations as when solving one-step equations.<\/p>\n

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

    Question 1.
    \nThe total number of students in the seventh grade is 9 more than 4 times
    \nas many students as are in the art class. There are 101 students in the seventh grade.
    \nWrite and solve an equation to find the number of students in the art class.
    \nLet x represent the number of students in the art class.
    \nAnswer:
    \nEquation : 4x + 9 = 101,
    \nThere are 23 students in the art class,<\/p>\n

    Explanation:
    \nGiven the total number of students in the seventh grade is 9 more than 4 times
    \nas many students as are in the art class. There are 101 students in the seventh grade.
    \nLet x represent the number of students in the art class.
    \nSo the equation to find the number of students in the art class is\u00a0 4x + 9 = 101, on solving
    \n4x + 9 – 9 = 101 – 9, subtracting 9 from both sides,
    \n4x \u00f7 4 = 92 \u00f7 4, Dividing both sides by 4 we get
    \nx = 23, therefore there are 23 students in the art class,<\/p>\n

    Question 2.
    \nList the steps to solve the following equation: 5x \u2013 6 = 44.
    \nThen solve for x.
    \nAnswer:
    \nx = 10,<\/p>\n

    Explanation:
    \nThe steps to solve the following equation : 5x \u2013 6 = 44 is
    \n5x – 6 + 6 = 44 + 6, adding 6 on both sides,
    \n5x = 50,
    \n5x \u00f7 5 = 50 \u00f7 5, dividing both sides by 5 we get x = 10.<\/p>\n

    Question 3.
    \nSolve for the given variable.
    \na. 4y + 3 = 19
    \nb. \\(\\frac{1}{2}\\)n – 3 = 5
    \nAnswer:
    \na. y = 4,
    \nb. n = 16,<\/p>\n

    Explanation:
    \nSolving:
    \na. 4y + 3 = 19,
    \n4y + 3 – 3 = 19 – 3, subtracting 3 on both sides,
    \n4y = 16,
    \n4y \u00f7 4 = 16 \u00f7 4, dividing both sides by 4 we get y = 4.<\/p>\n

    b. \\(\\frac{1}{2}\\)n – 3 = 5,
    \n\\(\\frac{1}{2}\\)n – 3\u00a0 + 3 = 5 + 3, adding 3 on both sides,
    \n\\(\\frac{1}{2}\\)n = 8,
    \n2 X \\(\\frac{1}{2}\\)n = 2 X 8, multiplying both sides by 2 we get
    \nn = 16.<\/p>\n

    LESSON 5.3 Solve Equations Using the Distributive Property<\/strong><\/p>\n

    Quick Review<\/strong>
    \nUse the Distributive Property to solve problems of the form p(x + q) = r.<\/p>\n

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

    Question 1.
    \nThere are 450 seats in the lower level of a concert hall with
    \nb balcony seats in the upper level. So far, 170 tickets have been sold,
    \nwhich is \\(\\frac{1}{5}\\) of the total number of seats in the concert hall.
    \nHow many tickets sold are balcony seats?
    \nAnswer:
    \nThere are 400 balcony seats in the concert hall,<\/p>\n

    Explanation:
    \nGiven there are 450 seats in the lower level of a concert hall with
    \nb balcony seats in the upper level. So far, 170 tickets have been sold,
    \nwhich is \\(\\frac{1}{5}\\) of the total number of seats in the concert hall.
    \nNumber of tickets sold which are balcony seats are
    \nEquation: \\(\\frac{1}{5}\\)(b + 450) = 170,
    \n\\(\\frac{1}{5}\\)b + 90 = 170, Using distributive property,
    \n\\(\\frac{1}{5}\\)b + 90 – 90 = 170 – 90, subtracting 90 from both sides,
    \n\\(\\frac{1}{5}\\)b = 80,
    \n5 X \\(\\frac{1}{5}\\)b = 5 X 80 , multiplying both sides by 5 we get
    \nb = 400, therefore there are 400 balcony seats in the concert hall.<\/p>\n

    Question 2.
    \nSolve the equation -4(8 + y) = 90.
    \nAnswer:
    \ny = -30.5,<\/p>\n

    Explanation:
    \nGiven to solve -4(8 + y) = 90,
    \n-32 – 4y = 90, Using distributive property,
    \n-32 – 4y + 32 = 90 + 32, adding 32 on both sides,
    \n-4y = 122,
    \n-4y \u00f7 -4 = 122 \u00f7 -4, dividing both sides by -4 we get y = -30.5.<\/p>\n

    LESSON 5-4 Solve Inequalities Using Addition or Subtraction<\/strong><\/p>\n

    Quick Review<\/strong>
    \nWhen you add or subtract the same number on both sides of an inequality,
    \nthe relationship between the sides stays the same.
    \nSolutions to inequalities can be graphed on number lines.<\/p>\n

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

    Question 1.
    \nCarson’s wheelbarrow can hold 345 pounds. If he has 121 pounds
    \nof rock in the wheelbarrow, what number of pounds, p, can he put in
    \nthe wheelbarrow without going over the weight limit?
    \nAnswer:
    \nThe weight that can put in the wheelbarrow is no more than 224 pounds,<\/p>\n

    Explanation:
    \nGiven Carson’s wheelbarrow can hold 345 pounds.
    \nIf he has 121 pounds of rock in the wheelbarrow,
    \nSo number of pounds p can he put in the wheelbarrow without
    \ngoing over the weight limit is\u00a0 121 + p \u2264 345,
    \n121 + p – 121 \u2264 345 – 121, subtracting 121 from both sides we get
    \np \u2264 224, therefore the weight that can put in the wheelbarrow is
    \nno more than 224 pounds.<\/p>\n

    Question 2.
    \nSolve x \u2013 19 < 81. Then graph the solution.
    \nAnswer:
    \n\"Envision
    \nExplanation:
    \nSolving x – 19 < 81 as
    \nx – 19 + 19 < 81 + 19, adding 19 on both sides,
    \nx < 100 and graph of the solution x < 100 is as shown above.<\/p>\n

    LESSON 5-5 Solve Inequalities Using Multiplication or Division<\/strong><\/p>\n

    Quick Review<\/strong>
    \nWhen you multiply or divide both sides of an inequality by the same positive number,
    \nthe inequality remains true. When you multiply or divide both sides of an inequality by
    \nthe same negative number, you need to reverse the inequality symbol,
    \nbut the inequality remains true.<\/p>\n

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

    Question 1.
    \nTravis has 3 months to save money for a trip. An airplane ticket costs more than $300.
    \nIf he saves the same amount of money, a, each month, how much does he need to
    \nsave each month to pay for the ticket?
    \nAnswer:
    \nInequality : 3a > 300,
    \nTravis should save more than $100 each month to pay for the ticket,<\/p>\n

    Explanation:
    \nGiven Travis has 3 months to save money for a trip.
    \nAn airplane ticket costs more than $300.
    \nIf he saves the same amount of money a each month,
    \nSo he need to save each month to pay for the ticket is 3a > 300,
    \nNow on simplifying 3a \u00f7 3 = 300 \u00f7 3, Dividing both sides by 3 we get a >100,
    \ntherefore, Travis should save more than $100 each month to pay for the ticket.<\/p>\n

    Question 2.
    \nSolve –\\(\\frac{1}{8}\\)y \u2264 34. Then graph the solution.
    \nAnswer:
    \n\"Envision
    \nExplanation:
    \nSolving –\\(\\frac{1}{8}\\)y \u2264 34,
    \n-8 X –\\(\\frac{1}{8}\\)y \u2264 -8 X 34, Multiplying both sides by -8 and
    \nreverse the inequality symbol,
    \ny \u2265 -272 and graph of the solution y \u2265 -272 is as shown above.<\/p>\n

    LESSON 5-6 Solve Two-Step Inequalities<\/strong><\/p>\n

    Quick Review<\/strong>
    \nInverse relationships and properties can be used to isolate the variable and
    \nsolve two step inequalities in the form px + q < r or px + q > r in the same way
    \nthat they are used to solve two-step equations.<\/p>\n

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

    Question 1.
    \nThe school band gets $5 for each T-shirt they sell at a fundraiser.
    \nThey have a goal of raising $150. If $45 has been raised so far,
    \nhow many more T-shirts do they have to sell to reach or exceed the goal?
    \nAnswer:
    \nThe school band needs to sell at least\u00a0 21 T-shirts,<\/p>\n

    Explanation:
    \nGiven the school band gets $5 for each T-shirt they sell at a fundraiser.
    \nThey have a goal of raising $150. If $45 has been raised so far,
    \nmore T-shirts they do have to sell to reach or exceed the goal are
    \n45 + 5x \u2265 150,
    \n45 + 5x – 45 \u2265 150 – 45, subtracting 45 from both sides,
    \n5x \u2265 105,
    \n5x \u00f7 5 \u2265 105 \u00f7 5, dividing both sides by 5 we get x \u2265 21,
    \ntherefore the school band needs to sell at least 21 T-shirts.<\/p>\n

    Question 2.
    \nSolve the inequality \u20138 –\\(\\frac{1}{3}\\)n \u2264 -25.
    \nAnswer:
    \nn \u2265 51,<\/p>\n

    Explanation:
    \nSolving the inequality \u20138 –\\(\\frac{1}{3}\\)n \u2264 -25,
    \n\u20138 –\\(\\frac{1}{3}\\)n + 8 \u2264 -25 + 8, adding 8 on both sides,
    \n–\\(\\frac{1}{3}\\)n \u2264 -17,
    \n-3 X –\\(\\frac{1}{3}\\)n \u2265 – 3 X -17, Multiplying both sides by -3 and
    \nreverse the inequality symbol we get n \u2265 51.<\/p>\n

    LESSON 5-7 Solve Multi-Step Inequalities<\/strong><\/p>\n

    Quick Review<\/strong>
    \nSolving a multi-step inequality is similar to solving a multi-step equation.
    \nAll of the rules and properties for solving one- and
    \ntwo step inequalities apply to solving multi-step inequalities.<\/p>\n

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

    Question 1.
    \nSolve 1.9(2.3n + 6) + 10.45 > 43.7. Then graph the solution.
    \n\"Envision
    \nAnswer:
    \n\"Envision<\/p>\n

    Explanation:
    \nGiven to solve 1.9(2.3n + 6) + 10.45 > 43.7,
    \n4.37n + 11.4 + 10.45 > 43.7, using distributive property,
    \n4.37n + 21.85 > 43.7, combining like terms,
    \n4.37n + 21.85 – 21.85 > 43.7 – 21.85, subtracting 21.85 from both sides,
    \n4.37n > 21.85,
    \n4.37n \u00f7 4.37 > 21.85 \u00f7 4.37, dividing both sides by 4.37 we get
    \nn > 5 and graph of the solution n > 5<\/span>\u00a0is as shown above.<\/span><\/p>\n

    Question 2.
    \nSolve 4(-2n + 2.5) \u2013 8 \u2264 50. Then graph the solution.
    \n\"Envision
    \nAnswer:
    \n\"Envision
    \nExplanation:
    \nGiven to solve 4(-2n + 2.5) \u2013 8 \u2264 50,
    \n-8n + 10 – 8 \u2264 50, using distributive property,<\/span>
    \n-8n + 2 \u2264 50, combining like terms,
    \n-8n +2 -2 \u2264 50 – 2, subtracting -2 from both sides,
    \n-8n \u2264 48,
    \n-8n \u00f7 -8<\/span> \u2265 48 \u00f7 -8, dividing both sides by -8 and reverse
    \nthe inequality symbol we get n \u2265 -6 and
    \ngraph of the solution n \u2265 -6<\/span>\u00a0is as shown above.<\/span><\/p>\n

    Topic 5 Fluency Practice<\/h3>\n

    Crisscrossed<\/strong><\/p>\n

    Solve each problem. Write your answers in the cross-number puzzle below.
    \nEach digit, decimal point, dollar sign, and percent symbol of your answer goes in its own box.
    \nRound money amounts to the nearest cent as needed.
    \n\"Envision<\/p>\n

    ACROSS<\/strong>
    \nA. Antonia buys 0.75 yard of fabric at $12.00 per yard.
    \nIf she pays 5% sales tax, what is the total cost of the fabric?
    \nB. Five friends plan to split a restaurant bill evenly.
    \nThe total cost of the meal is $89.75, and they want to leave a 20% tip.
    \nWhat amount should each friend pay?
    \nE. Kaylie buys a sweater on sale for $40.11. If the discount is 20% off and
    \nshe pays $1.91 in sales tax, what is the original price of the sweater?
    \nF. Randy buys a pair of shoes that were originally priced at $147.
    \nHe receives a 35% discount and pays 8.5% sales tax. How much does Randy pay?
    \nG. A basketball player makes 8 of 22 shots in Game 1, 6 of 15 shots in
    \nGame 2, and 10 of 23 shots in Game 3.
    \nWhat percent of the shots did the player make in the three games?
    \nAnswer:
    \n\"Envision
    \nAcross:
    \nA. The total cost of the fabric is $9.45,
    \nB. The amount should each friend pay is $21.54,
    \nE. The original price of the sweater is $47.75,
    \nF. Randy paid $103.67,
    \nG. Percent of the shots did the player make in the three games is 40%,<\/p>\n

    Explanation:
    \nAcross:
    \nA. Given Antonia buys 0.75 yard of fabric at $12.00 per yard.
    \nIf she pays 5% sales tax, so 0.75 X $12 = $9,
    \nNow 5% on $9 is 5 X 9 \u00f7100 = $0.45,
    \n<\/span>therefore the total cost of the fabric is $9 + $0.45 = $9.45.
    \nB. Given five friends plan to split a restaurant bill evenly.
    \nThe total cost of the meal is $89.75 and they want to leave a 20% tip.
    \nSo amount should each friend pay must be
    \n89.75 X 20 \u00f7100 = $17.95 is the\u00a0 tip,
    \nthe total amount is $89.75 + $17.95 = $107.7 which is paid by
    \n5 friends, therefore the amount each friend must pay is
    \n$107.7 \u00f7 5 = $21.54.
    \nE. Given Kaylie buys a sweater on sale for $40.11.
    \nIf the discount is 20% off and she pays $1.91 in sales tax,
    \nLet x be the original price of the sweater so
    \nOriginal Price = (Paid Price – Tax) + 20% discount,
    \nx = (40.11 -1.91) + 20% discount,
    \nx\u00a0 = $38.20 +( (20 X x) \u00f7 100),
    \nx – (20x \u00f7 100)= $38.20,
    \n100x – 20x = $38.20 X 100,
    \n80x = $3,820.
    \nx\u00a0 = $3,820 \u00f7 80,
    \nx = $47.75.
    \nF. Given Randy buys a pair of shoes that were originally priced at $147.
    \nHe receives a 35% discount and pays 8.5% sales tax.
    \nLet Randy paid p,
    \nRandy receives 35% discount means
    \n($147 – ($147 X 35% discount)) = ($147 – $51.45) =\u00a0 $95.55,
    \non $95.55 Randy pays 8.5% sales tax so 95.55 X 8.5 \u00f7 100 = $8.12175,
    \nTherefore Randy paid p = $95.55 + $8.12175 = $103.67175 \u2248 $103.67.
    \nG. Given a\u00a0basketball player makes 8 of 22 shots in Game 1,
    \n6 of 15 shots in Game 2 and 10 of 23 shots in Game 3,
    \nIn Game 1 percent of the shots did the player make is 8 \u00f7 22 = 0.36 = 36%,
    \nIn Game 2 percent of the shots did the player make is 6 \u00f7 15 = 0.40 = 40%,
    \nIn Game 3 percent of the shots did the player make is 10 \u00f7 23 = 0.43 = 43%,
    \nTherefore percent of the shots did the player make in the three games is
    \n(36% + 40% + 43%) \u00f7 3 = 39.33% \u2248 40%.<\/p>\n

    DOWN<\/strong>
    \nA. Jack buys a tablet that costs $99 and a memory card that costs $15.
    \nHe has a coupon for a 15% discount. What is the amount of the discount on the two items?
    \nB. Tara buys two pairs of socks for $4.99 each and three T-shirts for $11.45 each.
    \nIf she pays 6% sales tax, what is the total amount of her purchase?
    \nC. Sunil receives a 20% discount on a concert ticket that costs $75.
    \nIf Sunil pays $3.30 in sales tax on the discounted ticket, what is the sales tax rate?
    \nD. Dylan works for 4 hours and is paid $17.50 per hour. He must pay 15% in income taxes.
    \nWhat amount does he earn after taxes?
    \nE. Miles earns a 6% commission on each vehicle he sells.
    \nToday he sold a truck for $18,500 and a car for $9,600.
    \nWhat is the total amount of his commission for these vehicles?
    \nAnswer:
    \nA. The amount of the discount on the two items is $17.10,
    \nB. The total amount of Tara’s purchase is $47.00,
    \nC. The sales tax rate is 5.5%,
    \nD. Dylan earn after taxes is $59.50,
    \nE. The total amount of Miles commission for these vehicles is $1,686,<\/p>\n

    Explanation:
    \nA. Given Jack buys a tablet that costs $99 and a memory card that costs $15.
    \nHe has a coupon for a 15% discount. So tablet costs $99 – ($99 X 15 \u00f7 100) =
    \n$99 –\u00a0 $14.85 = $84.15 and Memory card costs $15 – ($15 X 15 \u00f7 100) =
    \n$15 – $2.25 = $12. 75, Therefore the amount of the discount on the two items is
    \ndiscount amount on tablet + discount amount on memory card = $ 14.85 +\u00a0 $ 2.25 = $17.10.
    \nB. Given Tara buys two pairs of socks for $4.99 each and three T-shirts for $11.45 each.
    \nIf she pays 6% sales tax, To find the total amount of her purchase first we will calculate
    \ncost for socks + cost for T-shirts and then add sales tax too, so
    \n2($4.99) + 3(11.45) = $9.98 + $34.35 = $44.33,
    \nNow sales tax = $44.33 X 6 \u00f7 100 = $2.6598, therefore $44.33 + $2.6598 = $46.9898 \u2248 $47.00.
    \nC. Given Sunil receives a 20% discount on a concert ticket that costs $75.
    \nIf Sunil pays $3.30 in sales tax on the discounted ticket,
    \nThe discounted ticket price is ($75 – $75 X 20 \u00f7 100) = $75 – $15 = $60,
    \nlet the sales rate tax is r, so $3.30 = r% of 60, therefore r = (3.30 \u00f7 60) X 100= 0.55 X 100 = 5.5%.
    \nD. Given Dylan works for 4 hours and is paid $17.50 per hour.
    \nHe must pay 15% in income taxes. So amount does he earn after taxes would be
    \nfirst he gets $17.50 X 4 = $70, So amount paid to income tax will be\u00a0 $70 X 15 \u00f7 100 = $10.5,
    \nTherefore Dylan earn after taxes is $70 – $10.5 = $59.50.
    \nE. Given Miles earns a 6% commission on each vehicle he sells.
    \nToday he sold a truck for $18,500 and a car for $9,600.
    \nSo first commission on truck sold is $18,500 X 6 \u00f7 100 = $1,110 and
    \ncommission on car sold is $9,600 X 6 \u00f7 100 = $576,
    \ntherefore the total amount of his commission for these vehicles is
    \n$1,110 + $576 = $1,686.<\/p>\n","protected":false},"excerpt":{"rendered":"

    Go through the\u00a0enVision Math Common Core Grade 7 Answer Key\u00a0Topic 5 Solve Problems Using Equations and Inequalities\u00a0regularly and improve your accuracy in solving questions. enVision Math Common Core 7th Grade Answers Key Topic 5 Solve Problems Using Equations and Inequalities Review What You Know! Vocabulary Choose the best term from the box to complete each …<\/p>\n","protected":false},"author":1,"featured_media":25565,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"https:\/\/envisionmathanswerkey.com\/wp-content\/uploads\/2021\/09\/Envision-Math-Common-Core-Grade-7-Answer-Key-Topic-5-Solve-Problems-Using-Equations-and-Inequalities.png","_links":{"self":[{"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts\/1920"}],"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=1920"}],"version-history":[{"count":4,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts\/1920\/revisions"}],"predecessor-version":[{"id":28114,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts\/1920\/revisions\/28114"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/media\/25565"}],"wp:attachment":[{"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/media?parent=1920"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/categories?post=1920"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/tags?post=1920"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}