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Chapter 2 Review Exercises

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Chapter 2 Review Exercises

Solve Equations using the Subtraction and Addition Properties of Equality

Verify a Solution of an Equation

In the following exercises, determine whether each number is a solution to the equation.

Exercise 1

10x1=5x;x=15

Exercise 2

w+2=58;w=38

Answer

no

Exercise 3

12n+5=8n;n=54

Exercise 4

6a3=7a,a=313

Answer

yes

Solve Equations using the Subtraction and Addition Properties of Equality

In the following exercises, solve each equation using the Subtraction Property of Equality.

Exercise 5

x+7=19

Exercise 6

y+2=6

Answer

y=8

Exercise 7

a+13=53

Exercise 8

n+3.6=5.1

Answer

n=1.5

In the following exercises, solve each equation using the Addition Property of Equality.

Exercise 9

u7=10

Exercise 10

x9=4

Answer

x=5

Exercise 11

c311=911

Exercise 12

p4.8=14

Answer

p=18.8

In the following exercises, solve each equation.

Exercise 13

n12=32

Exercise 14

y+16=9

Answer

y=25

Exercise 15

f+23=4

Exercise 16

d3.9=8.2

Answer

d=12.1

Solve Equations That Require Simplification

In the following exercises, solve each equation.

Exercise 17

y+815=3

Exercise 18

7x+106x+3=5

Answer

x=8

Exercise 19

6(n1)5n=14

Exercise 20

8(3p+5)23(p1)=35

Answer

p=28

Translate to an Equation and Solve

In the following exercises, translate each English sentence into an algebraic equation and then solve it.

Exercise 21

The sum of 6 and m is 25

Exercise 22

Four less than n is 13

Answer

n4=13;n=17

Translate and Solve Applications

In the following exercises, translate into an algebraic equation and solve.

Exercise 23

Rochelle’s daughter is 11 years old. Her son is 3 years younger. How old is her son?

Exercise 24

Tan weighs 146 pounds. Minh weighs 15 pounds more than Tan. How much does Minh weigh?

Answer

161 pounds

Exercise 25

Peter paid $9.75 to go to the movies, which was $46.25 less than he paid to go to a concert. How much did he pay for the concert?

Exercise 26

Elissa earned $152.84 this week, which was $2..65 more than she earned last week. How much did she earn last week?

Answer

$131.19

Solve Equations using the Division and Multiplication Properties of Equality

Solve Equations Using the Division and Multiplication Properties of Equality

In the following exercises, solve each equation using the division and multiplication properties of equality and check the solution.

Exercise 27

8x=72

Exercise 28

13a=65

Answer

a=5

Exercise 29

0.25p=5.25

Exercise 30

y=4

Answer

y=4

Exercise 31

n6=18

Exercise 32

y10=30

Answer

y=300

Exercise 33

36=34x

Exercise 34

58u=1516

Answer

u=32

Exercise 35

18m=72

Exercise 36

c9=36

Answer

c=324

Exercise 37

0.45x=6.75

Exercise 38

1112=23y

Answer

y=118

Solve Equations That Require Simplification

In the following exercises, solve each equation requiring simplification.

Exercise 39

5r3r+9r=352

Exercise 40

24x+8x11x=714

Answer

x=1

Exercise 41

1112n56n=95

Exercise 42

9(d2)15=24

Answer

d=3

Translate to an Equation and Solve

In the following exercises, translate to an equation and then solve.

Exercise 43

143 is the product of 11 and y

Exercise 44

The quotient of b and and 9 is 27

Answer

b9=27;b=243

Exercise 45

The sum of q and one-fourth is one.

Exercise 46

The difference of s and one-twelfth is one fourth.

Answer

s112=14;s=13

Translate and Solve Applications

In the following exercises, translate into an equation and solve.

Exercise 47

Ray paid $21 for 12 tickets at the county fair. What was the price of each ticket?

Exercise 48

Janet gets paid $24 per hour. She heard that this is 34 of what Adam is paid. How much is Adam paid per hour?

Answer

$32

Solve Equations with Variables and Constants on Both Sides

Solve an Equation with Constants on Both Sides

In the following exercises, solve the following equations with constants on both sides.

Exercise 49

8p+7=47

Exercise 50

10w5=65

Answer

w=7

Exercise 51

3x+19=47

Exercise 52

32=49n

Answer

n=4

Solve an Equation with Variables on Both Sides

In the following exercises, solve the following equations with variables on both sides.

Exercise 53

7y=6y13

Exercise 54

5a+21=2a

Answer

a=7

Exercise 55

k=6k35

Exercise 56

4x38=3x

Answer

x=38

Solve an Equation with Variables and Constants on Both Sides

In the following exercises, solve the following equations with variables and constants on both sides.

Exercise 57

12x9=3x+45

Exercise 58

5n20=7n80

Answer

n=5

Exercise 59

4u+16=19u

Exercise 60

58c4=38c+4

Answer

c=32

Use a General Strategy for Solving Linear Equations

Solve Equations Using the General Strategy for Solving Linear Equations

In the following exercises, solve each linear equation.

Exercise 61

6(x+6)=24

Exercise 62

9(2p5)=72

Answer

p=132

Exercise 63

(s+4)=18

Exercise 64

8+3(n9)=17

Answer

n=12

Exercise 65

233(y7)=8

Exercise 66

13(6m+21)=m7

Answer

m=14

Exercise 67

4(3.5y+0.25)=365

Exercise 68

0.25(q8)=0.1(q+7)

Answer

q=18

Exercise 69

8(r2)=6(r+10)

Exercise 70

5+7(25x)=2(9x+1)(13x57)

Answer

x=1

Exercise 71

(9n+5)(3n7)=20(4n2)

Exercise 72

2[16+5(8k6)]=8(34k)32

Answer

k=34

Classify Equations

In the following exercises, classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

Exercise 73

17y3(42y)=11(y1)+12y1

Exercise 74

9u+32=15(u4)3(2u+21)

Answer

contradiction; no solution

Exercise 75

8(7m+4)=6(8m+9)

Exercise 76

21(c1)19(c+1)=2(c20)

Answer

identity; all real numbers

Solve Equations with Fractions and Decimals

Solve Equations with Fraction Coefficients

In the following exercises, solve each equation with fraction coefficients.

Exercise 77

25n110=710

Exercise 78

13x+15x=8

Answer

x=15

Exercise 79

34a13=12a56

Exercise 80

12(k3)=13(k+16)

Answer

k=41

Exercise 81

3x25=3x+48

Exercise 82

5y13+4=8y+46

Answer

y=1

Solve Equations with Decimal Coefficients

In the following exercises, solve each equation with decimal coefficients.

Exercise 83

0.8x0.3=0.7x+0.2

Exercise 84

0.36u+2.55=0.41u+6.8

Answer

u=85

Exercise 85

0.6p1.9=0.78p+1.7

Exercise 86

0.6p1.9=0.78p+1.7

Answer

d=20

Solve a Formula for a Specific Variable

Use the Distance, Rate, and Time Formula

In the following exercises, solve.

Exercise 87

Natalie drove for 712 hours at 60 miles per hour. How much distance did she travel?

Exercise 88

Mallory is taking the bus from St. Louis to Chicago. The distance is 300 miles and the bus travels at a steady rate of 60 miles per hour. How long will the bus ride be?

Answer

5 hours

Exercise 89

Aaron’s friend drove him from Buffalo to Cleveland. The distance is 187 miles and the trip took 2.75 hours. How fast was Aaron’s friend driving?

Exercise 90

Link rode his bike at a steady rate of 15 miles per hour for 212 hours. How much distance did he travel?

Answer

37.5 miles

Solve a Formula for a Specific Variable

In the following exercises, solve.

Exercise 91

Use the formula. d=rt to solve for t

  1. when d=510 and r=60
  2. in general
Exercise 92

Use the formula. d=rt to solve for r

  1. when when d=451 and t=5.5
  2. in general
Answer
  1. r=82mph
  2. r=Dt
Exercise 93

Use the formula A=12bh to solve for b

  1. when A=390 and h=26
  2. in general
Exercise 94

Use the formula A=12bh to solve for b

  1. when A=153 and b=18
  2. in general
Answer
  1. h=17
  2. h=2Ab
Exercise 95

Use the formula I=Prt to solve for the principal, P for

  1. I=$2,501,r=4.1%, t=5 years
  2. in general
Exercise 96

Solve the formula 4x+3y=6 for y

  1. when x=−2
  2. in general
Answer

y=143y=64x3

Exercise 97

Solve 180=a+b+c for c

Exercise 98

Solve the formula V=LWH for H

Answer

H=VLW

Solve Linear Inequalities

Graph Inequalities on the Number Line

In the following exercises, graph each inequality on the number line.

Exercise 99
  1. x4
  2. x>−2
  3. x<1
Exercise 100
  1. x>0
  2. x<−3
  3. x1
Answer
  1. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 0 is graphed on the number line, with an open parenthesis at x equals 0, and a dark line extending to the right of the parenthesis.
  2. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than negative 3 is graphed on the number line, with an open parenthesis at x equals negative 3, and a dark line extending to the left of the parenthesis.
  3. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to 1 is graphed on the number line, with an open bracket at x equals 1, and a dark line extending to the right of the bracket.

In the following exercises, graph each inequality on the number line and write in interval notation.

Exercise 101
  1. x<1
  2. x2.5
  3. x54
Exercise 102
  1. x>2
  2. x1.5
  3. x53
Answer
  1. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 2 is graphed on the number line, with an open parenthesis at x equals 2, and a dark line extending to the right of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, 2 comma infinity, parenthesis.
  2. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to negative 1.5 is graphed on the number line, with an open bracket at x equals negative 1.5, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma negative 1.5, bracket.
  3. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to 5/3 is graphed on the number line, with an open bracket at x equals 5/3, and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, 5/3 comma infinity, parenthesis.

Solve Inequalities using the Subtraction and Addition Properties of Inequality

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

Exercise 103

n1223

Exercise 104

m+1456

Answer

At the top of this figure is the solution to the inequality: m is less than or equal to 42. Below this is a number line ranging from 40 to 44 with tick marks for each integer. The inequality m is less than or equal to 42 is graphed on the number line, with an open bracket at m equals 42, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 42, bracket

Exercise 105

a+23712

Exercise 106

b7812

Answer

At the top of this figure is the solution to the inequality: b is greater than or equal to 3/8. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. The inequality b is greater than or equal to 3/8 is graphed on the number line, with an open bracket at b equals 3/8 (written in), and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, 3/8 comma infinity, bracket

Solve Inequalities using the Division and Multiplication Properties of Inequality

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

Exercise 107

9x>54

Exercise 108

12d108

Answer

At the top of this figure is the solution to the inequality: d is greater than or equal to negative 9. Below this is a number line ranging from negative 11 to negative 7 with tick marks for each integer. The inequality d is greater than or equal to negative 9 is graphed on the number line, with an open bracket at d equals negative 9, and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, negative 9 comma infinity, parenthesis.

Exercise 109

52j<60

Exercise 110

q224

Answer

At the top of this figure is the solution to the inequality: q is less than or equal to 48. Below this is a number line ranging from 46 to 50 with tick marks for each integer. The inequality q is less than or equal to 48 is graphed on the number line, with an open bracket at q equals 48, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 48, bracket.

Solve Inequalities That Require Simplification

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

Exercise 111

6p>15p30

Exercise 112

9h7(h1)4h23

Answer

At the top of this figure is the solution to the inequality: h is greater than or equal to 15. Below this is a number line ranging from 13 to 17 with tick marks for each integer. The inequality h is greater than or equal to 15 is graphed on the number line, with an open bracket at h equals 15, and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, 15 comma infinity, parenthesis.

Exercise 113

5n15(4n)<10(n6)+10n

Exercise 114

38a112a>512a+34

Answer

At the top of this figure is the solution to the inequality: a is less than negative 6. Below this is a number line ranging from negative 8 to negative 4 with tick marks for each integer. The inequality a is less than negative 6 is graphed on the number line, with an open parenthesis at a equals negative 6, and a dark line extending to the left of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma negative 6, parenthesis.

Translate to an Inequality and Solve

In the following exercises, translate and solve. Then write the solution in interval notation and graph on the number line.

Exercise 115

Five more than z is at most 19.

Exercise 116

Three less than c is at least 360.

Answer

At the top of this figure is the inequality c minus 3 is greater than or equal to 360. To the right of this is the solution to the inequality: c is greater than or equal to 363. To the right of the solution is the solution written in interval notation: bracket, 363 comma infinity, parenthesis. Below all of this is a number line ranging from 361 to 365 with tick marks for each integer. The inequality c is greater than or equal to 363 is graphed on the number line, with an open bracket at c equals 363, and a dark line extending to the right of the bracket.

Exercise 117

Nine times n exceeds 42.

Exercise 118

Negative two times a is no more than 8.

Answer

At the top of this figure is the inequality negative 2a is less than or equal to 8. To the right of this is the solution to the inequality: a is greater than or equal to negative 4. To the right of the solution is the solution written in interval notation: bracket, negative 4 comma infinity, parenthesis. Below all of this is a number line ranging from negative 6 to negative 2 with tick marks for each integer. The inequality a is greater than or equal to negative 4 is graphed on the number line, with an open bracket at a equals negative 4, and a dark line extending to the right of the bracket.

Everyday Math

Exercise 119

Describe how you have used two topics from this chapter in your life outside of your math class during the past month.

Chapter 2 Practice Test

Exercise 1

Determine whether each number is a solution to the equation 6x3=x+20

  1. 5
  2. 235
Answer
  1. no
  2. yes

In the following exercises, solve each equation.

Exercise 2

n23=14

Exercise 3

92c=144

Answer

c=32

Exercise 4

4y8=16

Exercise 5

8x15+9x1=21

Answer

x=5

Exercise 6

15a=120

Exercise 7

23x=6

Answer

x=9

Exercise 8

x3.8=8.2

Exercise 9

10y=5y60

Answer

y=4

Exercise 10

8n2=6n12

Exercise 11

9m24mm=428

Answer

m=9

Exercise 12

5(2x1)=45

Exercise 13

(d9)=23

Answer

d=14

Exercise 14

14(12m28)=62(3m1)

Exercise 15

2(6x5)8=22

Answer

x=13

Exercise 16

8(3a5)7(4a3)=203a

Exercise 17

14p13=12

Answer

p=103

Exercise 18

0.1d+0.25(d+8)=4.1

Exercise 19

14n3(4n+5)=9+2(n8)

Answer

contradiction; no solution

Exercise 20

9(3u2)4[68(u1)]=3(u2)

Exercise 21

Solve the formula x−2y=5 for y

  1. when x=−3
  2. in general
Answer
  1. y=4
  2. y=5x2

In the following exercises, graph on the number line and write in interval notation.

Exercise 22

x3.5

Exercise 23

x<114

Answer

This figure is a number line ranging from 1 to 5 with tick marks for each integer. The inequality x is less than 11/4 is graphed on the number line, with an open parenthesis at x equals 11/4, and a dark line extending to the left of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 11/4, parenthesis.

In the following exercises,, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

Exercise 24

8k5k120

Exercise 25

3c10(c2)<5c+16

Answer

This figure is a number line ranging from negative 2 to 3 with tick marks for each integer. The inequality c is greater than 1/3 is graphed on the number line, with an open parenthesis at c equals 1/3, and a dark line extending to the right of the parenthesis. Below the number line is the solution: c is greater than 1/3. To the right of the solution is the solution written in interval notation: parenthesis, 1/3 comma infinity, parenthesis

In the following exercises, translate to an equation or inequality and solve.

Exercise 26

4 less than twice x is 16.

Exercise 27

Fifteen more than n is at least 48.

Answer

n+1548;n33

Exercise 28

Samuel paid $25.82 for gas this week, which was $3.47 less than he paid last week. How much had he paid last week?

Exercise 29

Jenna bought a coat on sale for $120, which was 23 of the original price. What was the original price of the coat?

Answer

120=23p; The original price was $180

Exercise 30

Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 723 hours, what was the speed of the bus?

 

Review for 2.7 Ratio and Proportions and Similar Triangles

Solve Proportions

In the following exercises, solve.

Exercise 74

x4=35

Answer

125

Exercise 75

3y=95

Exercise 76

ss+20=37

Answer

15

Exercise 77

t35=t+29

​​​​​​​In the following exercises, solve using proportions.

Exercise 78

Rachael had a 21 ounce strawberry shake that has 739 calories. How many calories are there in a 32 ounce shake?

Answer

1161 calories

Exercise 79

Leo went to Mexico over Christmas break and changed $525 dollars into Mexican pesos. At that time, the exchange rate had $1 US is equal to 16.25 Mexican pesos. How many Mexican pesos did he get for his trip?

​​​​​​​Solve Similar Figure Applications

In the following exercises, solve.

Exercise 80

∆ABC is similar to ∆XYZ. The lengths of two sides of each triangle are given in the figure. Find the lengths of the third sides.

This image shows two triangles. The large triangle is labeled A B C. The length from A to B is labeled 8. The length from B to C is labeled 7. The length from C to A is labeled b. The smaller triangle is triangle x y z. The length from x to y is labeled 2 and two-thirds. The length from y to z is labeled x. The length from x to z is labeled 3.

Answer

b=9; x=2\dfrac{1}{3}

Exercise \PageIndex{81}

On a map of Europe, Paris, Rome, and Vienna form a triangle whose sides are shown in the figure below. If the actual distance from Rome to Vienna is 700 miles, find the distance from

  1. a. Paris to Rome
  2. b. Paris to Vienna

This is an image of a triangle. Clockwise beginning at the top, each vertex is labeled. The top vertex is labeled “Paris”, the next vertex is labeled “Vienna”, and the next vertex is labeled “Rome”. The distance from Paris to Vienna is 7.7 centimeters. The distance from Vienna to Rome is 7 centimeters. The distance from Rome to Paris is 8.9 centimeters.

Exercise \PageIndex{82}

Tony is 5.75 feet tall. Late one afternoon, his shadow was 8 feet long. At the same time, the shadow of a nearby tree was 32 feet long. Find the height of the tree.

Answer

23 feet

Exercise \PageIndex{83}

The height of a lighthouse in Pensacola, Florida is 150 feet. Standing next to the statue, 5.5 foot tall Natalie cast a 1.1 foot shadow How long would the shadow of the lighthouse be?

 

Review for 2.9 Compound Inequalities 

Solve Compound Inequalities with “and”

In each of the following exercises, solve each inequality, graph the solution, and write the solution in interval notation.

98. x\leq 5 and x>−3

Answer

The solution is negative 3 is less than x which is less than or equal to 5. The number line shows an open circle at negative 3 and a closed circle at 5. The interval notation is negative 3 to 5 within a parenthesis and a bracket.

99. 4x−2\leq 4 and 7x−1>−8

100. 5(3x−2)\leq 5 and 4(x+2)<3

Answer

The solution is negative x is less than negative five-fourths. The number line shows an open circle at negative five-fourths with shading to its left. The interval notation is negative infinity to negative five-fourths within parentheses.

101. 34(x−8)\leq 3 and 15(x−5)\leq 3

102. 34x−5\geq −2 and −3(x+1)\geq 6

Answer

The solution is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation

103. −5\leq 4x−1<7

Solve Compound Inequalities with “or”

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

104. 5−2x\leq −1 or 6+3x\leq 4

Answer

The solution is x is less than negative two-thirds or x is greater than or equal to 3. The number line shows a closed circle at negative two-thirds with shading to its left and a closed circle at 3 with shading to its right. The interval notation is the union of negative infinity to negative two-thirds within a parenthesis and a bracket and 3 to infinity within a bracket and a parenthesis.

105. 3(2x−3)<−5 or 4x−1>3

106. 34x−2>4 or 4(2−x)>0

Answer

The solution is x is less than 2 or x is greater than 8. The number line shows an open circle at 2 with shading to its left and an open circle at 8 with shading to its right. The interval notation is the union of negative infinity to 8 within parentheses and 8 to infinity within parentheses.

107. 2(x+3)\geq 0 or 3(x+4)\leq 6

108. 12x−3\leq 4 or 13(x−6)\geq −2

Answer

The solution is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.

Solve Applications with Compound Inequalities

In the following exercises, solve.

109. Liam is playing a number game with his sister Audry. Liam is thinking of a number and wants Audry to guess it. Five more than three times her number is between 2 and 32. Write a compound inequality that shows the range of numbers that Liam might be thinking of.

110. Elouise is creating a rectangular garden in her back yard. The length of the garden is 12 feet. The perimeter of the garden must be at least 36 feet and no more than 48 feet. Use a compound inequality to find the range of values for the width of the garden.

Answer

6\leq w\leq 12

 


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