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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 $$\PageIndex{1}$$

$$10 x-1=5 x ; x=\frac{1}{5}$$

Exercise $$\PageIndex{2}$$

$$w+2=\frac{5}{8} ; w=\frac{3}{8}$$

no

Exercise $$\PageIndex{3}$$

$$-12 n+5=8 n ; n=-\frac{5}{4}$$

Exercise $$\PageIndex{4}$$

$$6 a-3=-7 a, a=\frac{3}{13}$$

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 $$\PageIndex{5}$$

$$x+7=19$$

Exercise $$\PageIndex{6}$$

$$y+2=-6$$

$$y=-8$$

Exercise $$\PageIndex{7}$$

$$a+\frac{1}{3}=\frac{5}{3}$$

Exercise $$\PageIndex{8}$$

$$n+3.6=5.1$$

$$n=1.5$$

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

Exercise $$\PageIndex{9}$$

$$u-7=10$$

Exercise $$\PageIndex{10}$$

$$x-9=-4$$

$$x=5$$

Exercise $$\PageIndex{11}$$

$$c-\frac{3}{11}=\frac{9}{11}$$

Exercise $$\PageIndex{12}$$

$$p-4.8=14$$

$$p=18.8$$

In the following exercises, solve each equation.

Exercise $$\PageIndex{13}$$

$$n-12=32$$

Exercise $$\PageIndex{14}$$

$$y+16=-9$$

$$y=-25$$

Exercise $$\PageIndex{15}$$

$$f+\frac{2}{3}=4$$

Exercise $$\PageIndex{16}$$

$$d-3.9=8.2$$

$$d=12.1$$

Solve Equations That Require Simplification

In the following exercises, solve each equation.

Exercise $$\PageIndex{17}$$

$$y+8-15=-3$$

Exercise $$\PageIndex{18}$$

$$7 x+10-6 x+3=5$$

$$x=-8$$

Exercise $$\PageIndex{19}$$

$$6(n-1)-5 n=-14$$

Exercise $$\PageIndex{20}$$

$$8(3 p+5)-23(p-1)=35$$

$$p=-28$$

Translate to an Equation and Solve

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

Exercise $$\PageIndex{21}$$

The sum of $$-6$$ and $$m$$ is 25

Exercise $$\PageIndex{22}$$

Four less than $$n$$ is 13

$$n-4=13 ; n=17$$

Translate and Solve Applications

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

Exercise $$\PageIndex{23}$$

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

Exercise $$\PageIndex{24}$$

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

161 pounds

Exercise $$\PageIndex{25}$$

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 $$\PageIndex{26}$$

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

$$\ 131.19$$

### 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 $$\PageIndex{27}$$

$$8 x=72$$

Exercise $$\PageIndex{28}$$

$$13 a=-65$$

$$a=-5$$

Exercise $$\PageIndex{29}$$

$$0.25 p=5.25$$

Exercise $$\PageIndex{30}$$

$$-y=4$$

$$y=-4$$

Exercise $$\PageIndex{31}$$

$$\frac{n}{6}=18$$

Exercise $$\PageIndex{32}$$

$$\frac{y}{-10}=30$$

$$y=-300$$

Exercise $$\PageIndex{33}$$

$$36=\frac{3}{4} x$$

Exercise $$\PageIndex{34}$$

$$\frac{5}{8} u=\frac{15}{16}$$

$$u=\frac{3}{2}$$

Exercise $$\PageIndex{35}$$

$$-18 m=-72$$

Exercise $$\PageIndex{36}$$

$$\frac{c}{9}=36$$

$$c=324$$

Exercise $$\PageIndex{37}$$

$$0.45 x=6.75$$

Exercise $$\PageIndex{38}$$

$$\frac{11}{12}=\frac{2}{3} y$$

$$y=\frac{11}{8}$$

Solve Equations That Require Simplification

In the following exercises, solve each equation requiring simplification.

Exercise $$\PageIndex{39}$$

$$5 r-3 r+9 r=35-2$$

Exercise $$\PageIndex{40}$$

$$24 x+8 x-11 x=-7-14$$

$$x=-1$$

Exercise $$\PageIndex{41}$$

$$\frac{11}{12} n-\frac{5}{6} n=9-5$$

Exercise $$\PageIndex{42}$$

$$-9(d-2)-15=-24$$

$$d=3$$

Translate to an Equation and Solve

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

Exercise $$\PageIndex{43}$$

143 is the product of $$-11$$ and $$y$$

Exercise $$\PageIndex{44}$$

The quotient of $$b$$ and and 9 is $$-27$$

$$\frac{b}{9}=-27 ; b=-243$$

Exercise $$\PageIndex{45}$$

The sum of q and one-fourth is one.

Exercise $$\PageIndex{46}$$

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

$$s-\frac{1}{12}=\frac{1}{4} ; s=\frac{1}{3}$$

Translate and Solve Applications

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

Exercise $$\PageIndex{47}$$

Ray paid $21 for 12 tickets at the county fair. What was the price of each ticket? Exercise $$\PageIndex{48}$$ Janet gets paid $$\ 24$$ per hour. She heard that this is $$\frac{3}{4}$$ 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 $$\PageIndex{49}$$

$$8 p+7=47$$

Exercise $$\PageIndex{50}$$

$$10 w-5=65$$

$$w=7$$

Exercise $$\PageIndex{51}$$

$$3 x+19=-47$$

Exercise $$\PageIndex{52}$$

$$32=-4-9 n$$

$$n=-4$$

Solve an Equation with Variables on Both Sides

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

Exercise $$\PageIndex{53}$$

$$7 y=6 y-13$$

Exercise $$\PageIndex{54}$$

$$5 a+21=2 a$$

$$a=-7$$

Exercise $$\PageIndex{55}$$

$$k=-6 k-35$$

Exercise $$\PageIndex{56}$$

$$4 x-\frac{3}{8}=3 x$$

$$x=\frac{3}{8}$$

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 $$\PageIndex{57}$$

$$12 x-9=3 x+45$$

Exercise $$\PageIndex{58}$$

$$5 n-20=-7 n-80$$

$$n=-5$$

Exercise $$\PageIndex{59}$$

$$4 u+16=-19-u$$

Exercise $$\PageIndex{60}$$

$$\frac{5}{8} c-4=\frac{3}{8} c+4$$

$$c=32$$

### 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 $$\PageIndex{61}$$

$$6(x+6)=24$$

Exercise $$\PageIndex{62}$$

$$9(2 p-5)=72$$

$$p=\frac{13}{2}$$

Exercise $$\PageIndex{63}$$

$$-(s+4)=18$$

Exercise $$\PageIndex{64}$$

$$8+3(n-9)=17$$

$$n=12$$

Exercise $$\PageIndex{65}$$

$$23-3(y-7)=8$$

Exercise $$\PageIndex{66}$$

$$\frac{1}{3}(6 m+21)=m-7$$

$$m=-14$$

Exercise $$\PageIndex{67}$$

$$4(3.5 y+0.25)=365$$

Exercise $$\PageIndex{68}$$

$$0.25(q-8)=0.1(q+7)$$

$$q=18$$

Exercise $$\PageIndex{69}$$

$$8(r-2)=6(r+10)$$

Exercise $$\PageIndex{70}$$

$$\begin{array}{l}{5+7(2-5 x)=2(9 x+1)} \\ {-(13 x-57)}\end{array}$$

$$x=-1$$

Exercise $$\PageIndex{71}$$

$$\begin{array}{l}{(9 n+5)-(3 n-7)} \\ {=20-(4 n-2)}\end{array}$$

Exercise $$\PageIndex{72}$$

$$\begin{array}{l}{2[-16+5(8 k-6)]} \\ {=8(3-4 k)-32}\end{array}$$

$$k=\frac{3}{4}$$

Classify Equations

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

Exercise $$\PageIndex{73}$$

$$\begin{array}{l}{17 y-3(4-2 y)=11(y-1)} \\ {+12 y-1}\end{array}$$

Exercise $$\PageIndex{74}$$

$$\begin{array}{l}{9 u+32=15(u-4)} \\ {-3(2 u+21)}\end{array}$$

Exercise $$\PageIndex{75}$$

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

Exercise $$\PageIndex{76}$$

$$\begin{array}{l}{21(c-1)-19(c+1)} \\ {=2(c-20)}\end{array}$$

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 $$\PageIndex{77}$$

$$\frac{2}{5} n-\frac{1}{10}=\frac{7}{10}$$

Exercise $$\PageIndex{78}$$

$$\frac{1}{3} x+\frac{1}{5} x=8$$

$$x=15$$

Exercise $$\PageIndex{79}$$

$$\frac{3}{4} a-\frac{1}{3}=\frac{1}{2} a-\frac{5}{6}$$

Exercise $$\PageIndex{80}$$

$$\frac{1}{2}(k-3)=\frac{1}{3}(k+16)$$

$$k=41$$

Exercise $$\PageIndex{81}$$

$$\frac{3 x-2}{5}=\frac{3 x+4}{8}$$

Exercise $$\PageIndex{82}$$

$$\frac{5 y-1}{3}+4=\frac{-8 y+4}{6}$$

$$y=-1$$

Solve Equations with Decimal Coefficients

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

Exercise $$\PageIndex{83}$$

$$0.8 x-0.3=0.7 x+0.2$$

Exercise $$\PageIndex{84}$$

$$0.36 u+2.55=0.41 u+6.8$$

$$u=-85$$

Exercise $$\PageIndex{85}$$

$$0.6 p-1.9=0.78 p+1.7$$

Exercise $$\PageIndex{86}$$

$$0.6 p-1.9=0.78 p+1.7$$

$$d=-20$$

### Solve a Formula for a Specific Variable

Use the Distance, Rate, and Time Formula

In the following exercises, solve.

Exercise $$\PageIndex{87}$$

Natalie drove for 7$$\frac{1}{2}$$ hours at 60 miles per hour. How much distance did she travel?

Exercise $$\PageIndex{88}$$

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?

5 hours

Exercise $$\PageIndex{89}$$

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 $$\PageIndex{90}$$

Link rode his bike at a steady rate of 15 miles per hour for 2$$\frac{1}{2}$$ hours. How much distance did he travel?

37.5 miles

Solve a Formula for a Specific Variable

In the following exercises, solve.

Exercise $$\PageIndex{91}$$

Use the formula. d=rt to solve for t

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

Exercise $$\PageIndex{92}$$

Use the formula. d=rt to solve for r

1. when when d=451 and t=5.5
2. in general
1. r=82mph
2. $$r=\frac{D}{t}$$

Exercise $$\PageIndex{93}$$

Use the formula $$A=\frac{1}{2} b h$$ to solve for b

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

Exercise $$\PageIndex{94}$$

Use the formula $$A=\frac{1}{2} b h$$ to solve for b

1. when A=153 and b=18
2. in general
1. $$h=17$$
2. $$h=\frac{2 A}{b}$$

Exercise $$\PageIndex{95}$$

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 $$\PageIndex{96}$$ Solve the formula 4x+3y=6 for y 1. when x=−2 2. in general Answer ⓐ $$y=\frac{14}{3}$$ ⓑ $$y=\frac{6-4 x}{3}$$ Exercise $$\PageIndex{97}$$ Solve $$180=a+b+c$$ for $$c$$ Exercise $$\PageIndex{98}$$ Solve the formula $$V=L W H$$ for $$H$$ Answer $$H=\frac{V}{L W}$$ ### Solve Linear Inequalities Graph Inequalities on the Number Line In the following exercises, graph each inequality on the number line. Exercise $$\PageIndex{99}$$ 1. $$x\leq 4$$ 2. x>−2 3. x<1 Exercise $$\PageIndex{100}$$ 1. x>0 2. x<−3 3. $$x\geq −1$$ Answer 1. 2. 3. In the following exercises, graph each inequality on the number line and write in interval notation. Exercise $$\PageIndex{101}$$ 1. $$x<-1$$ 2. $$x \geq-2.5$$ 3. $$x \leq \frac{5}{4}$$ Exercise $$\PageIndex{102}$$ 1. $$x>2$$ 2. $$x \leq-1.5$$ 3. $$x \geq \frac{5}{3}$$ Answer 1. 2. 3. 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 $$\PageIndex{103}$$ $$n-12 \leq 23$$ Exercise $$\PageIndex{104}$$ $$m+14 \leq 56$$ Answer Exercise $$\PageIndex{105}$$ $$a+\frac{2}{3} \geq \frac{7}{12}$$ Exercise $$\PageIndex{106}$$ $$b-\frac{7}{8} \geq-\frac{1}{2}$$ Answer 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 $$\PageIndex{107}$$ $$9 x>54$$ Exercise $$\PageIndex{108}$$ $$-12 d \leq 108$$ Answer Exercise $$\PageIndex{109}$$ $$\frac{5}{2} j<-60$$ Exercise $$\PageIndex{110}$$ $$\frac{q}{-2} \geq-24$$ Answer 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 $$\PageIndex{111}$$ $$6 p>15 p-30$$ Exercise $$\PageIndex{112}$$ $$9 h-7(h-1) \leq 4 h-23$$ Answer Exercise $$\PageIndex{113}$$ $$5 n-15(4-n)<10(n-6)+10 n$$ Exercise $$\PageIndex{114}$$ $$\frac{3}{8} a-\frac{1}{12} a>\frac{5}{12} a+\frac{3}{4}$$ Answer 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 $$\PageIndex{115}$$ Five more than z is at most 19. Exercise $$\PageIndex{116}$$ Three less than c is at least 360. Answer Exercise $$\PageIndex{117}$$ Nine times n exceeds 42. Exercise $$\PageIndex{118}$$ Negative two times a is no more than 8. Answer ### Everyday Math Exercise $$\PageIndex{119}$$ 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 $$\PageIndex{1}$$ Determine whether each number is a solution to the equation $$6 x-3=x+20$$ 1. 5 2. $$\frac{23}{5}$$ Answer 1. no 2. yes In the following exercises, solve each equation. Exercise $$\PageIndex{2}$$ $$n-\frac{2}{3}=\frac{1}{4}$$ Exercise $$\PageIndex{3}$$ $$\frac{9}{2} c=144$$ Answer c=32 Exercise $$\PageIndex{4}$$ $$4 y-8=16$$ Exercise $$\PageIndex{5}$$ $$-8 x-15+9 x-1=-21$$ Answer $$x=-5$$ Exercise $$\PageIndex{6}$$ $$-15 a=120$$ Exercise $$\PageIndex{7}$$ $$\frac{2}{3} x=6$$ Answer $$x=9$$ Exercise $$\PageIndex{8}$$ $$x-3.8=8.2$$ Exercise $$\PageIndex{9}$$ $$10 y=-5 y-60$$ Answer $$y=-4$$ Exercise $$\PageIndex{10}$$ $$8 n-2=6 n-12$$ Exercise $$\PageIndex{11}$$ $$9 m-2-4 m-m=42-8$$ Answer $$m=9$$ Exercise $$\PageIndex{12}$$ $$-5(2 x-1)=45$$ Exercise $$\PageIndex{13}$$ $$-(d-9)=23$$ Answer $$d=-14$$ Exercise $$\PageIndex{14}$$ $$\frac{1}{4}(12 m-28)=6-2(3 m-1)$$ Exercise $$\PageIndex{15}$$ $$2(6 x-5)-8=-22$$ Answer $$x=-\frac{1}{3}$$ Exercise $$\PageIndex{16}$$ $$8(3 a-5)-7(4 a-3)=20-3 a$$ Exercise $$\PageIndex{17}$$ $$\frac{1}{4} p-\frac{1}{3}=\frac{1}{2}$$ Answer $$p=\frac{10}{3}$$ Exercise $$\PageIndex{18}$$ $$0.1 d+0.25(d+8)=4.1$$ Exercise $$\PageIndex{19}$$ $$14 n-3(4 n+5)=-9+2(n-8)$$ Answer contradiction; no solution Exercise $$\PageIndex{20}$$ $$9(3 u-2)-4[6-8(u-1)]=3(u-2)$$ Exercise $$\PageIndex{21}$$ Solve the formula x−2y=5 for y 1. when x=−3 2. in general Answer 1. y=4 2. $$y=\frac{5-x}{2}$$ In the following exercises, graph on the number line and write in interval notation. Exercise $$\PageIndex{22}$$ $$x \geq-3.5$$ Exercise $$\PageIndex{23}$$ $$x<\frac{11}{4}$$ Answer In the following exercises,, solve each inequality, graph the solution on the number line, and write the solution in interval notation. Exercise $$\PageIndex{24}$$ $$8 k \geq 5 k-120$$ Exercise $$\PageIndex{25}$$ $$3 c-10(c-2)<5 c+16$$ Answer In the following exercises, translate to an equation or inequality and solve. Exercise $$\PageIndex{26}$$ 4 less than twice x is 16. Exercise $$\PageIndex{27}$$ Fifteen more than n is at least 48. Answer $$n+15 \geq 48 ; n \geq 33$$ Exercise $$\PageIndex{28}$$ 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 $$\PageIndex{29}$$

Jenna bought a coat on sale for $$\ 120,$$ which was $$\frac{2}{3}$$ of the original price. What was the original price of the coat?

$$120=\frac{2}{3} p ;$$ The original price was $$\ 180$$

Exercise $$\PageIndex{30}$$

Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 7$$\frac{2}{3}$$ 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 $$\PageIndex{74}$$

$$\dfrac{x}{4}=\dfrac{3}{5}$$

$$\dfrac{12}{5}$$

Exercise $$\PageIndex{75}$$

$$\dfrac{3}{y}=\dfrac{9}{5}$$

Exercise $$\PageIndex{76}$$

$$\dfrac{s}{s+20}=\dfrac{3}{7}$$

$$15$$

Exercise $$\PageIndex{77}$$

$$\dfrac{t−3}{5}=\dfrac{t+2}{9}$$

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

Exercise $$\PageIndex{78}$$

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

$$1161$$ calories

Exercise $$\PageIndex{79}$$

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 $$\PageIndex{80}$$

$$∆ABC$$ is similar to $$∆XYZ$$. The lengths of two sides of each triangle are given in the figure. Find the lengths of the third sides. $$b=9$$; $$x=2\dfrac{1}{3}$$

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 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.

$$23$$ 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 $$x>−3$$ 99. $$4x−2\leq 4$$ and $$7x−1>−8$$

100. $$5(3x−2)\leq 5$$ and $$4(x+2)<3$$ 101. $$34(x−8)\leq 3$$ and $$15(x−5)\leq 3$$

102. $$34x−5\geq −2$$ and $$−3(x+1)\geq 6$$ 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 $$6+3x\leq 4$$ 105. $$3(2x−3)<−5$$ or $$4x−1>3$$

106. $$34x−2>4$$ or $$4(2−x)>0$$ 107. $$2(x+3)\geq 0$$ or $$3(x+4)\leq 6$$

108. $$12x−3\leq 4$$ or $$13(x−6)\geq −2$$ 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.

$$6\leq w\leq 12$$