# Chapter 2 Review Exercises

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- Contributed by Lynn Marecek
- Professor (Mathematics) at Santa Ana College
- Publisher: OpenStax CNX

## 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 \(\PageIndex{2}\)

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

**Answer**-
no

Exercise \(\PageIndex{3}\)

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

Exercise \(\PageIndex{4}\)

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

**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 \(\PageIndex{5}\)

\(x+7=19\)

Exercise \(\PageIndex{6}\)

\(y+2=-6\)

**Answer**-
\(y=-8\)

Exercise \(\PageIndex{7}\)

\(a+\frac{1}{3}=\frac{5}{3}\)

Exercise \(\PageIndex{8}\)

\(n+3.6=5.1\)

**Answer**-
\(n=1.5\)

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

Exercise \(\PageIndex{9}\)

\(u-7=10\)

Exercise \(\PageIndex{10}\)

\(x-9=-4\)

**Answer**-
\(x=5\)

Exercise \(\PageIndex{11}\)

\(c-\frac{3}{11}=\frac{9}{11}\)

Exercise \(\PageIndex{12}\)

\(p-4.8=14\)

**Answer**-
\(p=18.8\)

In the following exercises, solve each equation.

Exercise \(\PageIndex{13}\)

\(n-12=32\)

Exercise \(\PageIndex{14}\)

\(y+16=-9\)

**Answer**-
\(y=-25\)

Exercise \(\PageIndex{15}\)

\(f+\frac{2}{3}=4\)

Exercise \(\PageIndex{16}\)

\(d-3.9=8.2\)

**Answer**-
\(d=12.1\)

**Solve Equations That Require Simplification**

In the following exercises, solve each equation.

Exercise \(\PageIndex{17}\)

\(y+8-15=-3\)

Exercise \(\PageIndex{18}\)

\(7 x+10-6 x+3=5\)

**Answer**-
\(x=-8\)

Exercise \(\PageIndex{19}\)

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

Exercise \(\PageIndex{20}\)

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

**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 \(\PageIndex{21}\)

The sum of \(-6\) and \(m\) is 25

Exercise \(\PageIndex{22}\)

Four less than \(n\) is 13

**Answer**-
\(n-4=13 ; n=17\)

**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?

**Answer**-
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?

**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 \(\PageIndex{27}\)

\(8 x=72\)

Exercise \(\PageIndex{28}\)

\(13 a=-65\)

**Answer**-
\(a=-5\)

Exercise \(\PageIndex{29}\)

\(0.25 p=5.25\)

Exercise \(\PageIndex{30}\)

\(-y=4\)

**Answer**-
\(y=-4\)

Exercise \(\PageIndex{31}\)

\(\frac{n}{6}=18\)

Exercise \(\PageIndex{32}\)

\(\frac{y}{-10}=30\)

**Answer**-
\(y=-300\)

Exercise \(\PageIndex{33}\)

\(36=\frac{3}{4} x\)

Exercise \(\PageIndex{34}\)

\(\frac{5}{8} u=\frac{15}{16}\)

**Answer**-
\(u=\frac{3}{2}\)

Exercise \(\PageIndex{35}\)

\(-18 m=-72\)

Exercise \(\PageIndex{36}\)

\(\frac{c}{9}=36\)

**Answer**-
\(c=324\)

Exercise \(\PageIndex{37}\)

\(0.45 x=6.75\)

Exercise \(\PageIndex{38}\)

\(\frac{11}{12}=\frac{2}{3} y\)

**Answer**-
\(y=\frac{11}{8}\)

**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 \(\PageIndex{40}\)

\(24 x+8 x-11 x=-7-14\)

**Answer**-
\(x=-1\)

Exercise \(\PageIndex{41}\)

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

Exercise \(\PageIndex{42}\)

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

**Answer**-
\(d=3\)

**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 \(-27\)

**Answer**-
\(\frac{b}{9}=-27 ; b=-243\)

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.

**Answer**-
\(s-\frac{1}{12}=\frac{1}{4} ; s=\frac{1}{3}\)

**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 \(\PageIndex{50}\)

\(10 w-5=65\)

**Answer**-
\(w=7\)

Exercise \(\PageIndex{51}\)

\(3 x+19=-47\)

Exercise \(\PageIndex{52}\)

\(32=-4-9 n\)

**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 \(\PageIndex{53}\)

\(7 y=6 y-13\)

Exercise \(\PageIndex{54}\)

\(5 a+21=2 a\)

**Answer**-
\(a=-7\)

Exercise \(\PageIndex{55}\)

\(k=-6 k-35\)

Exercise \(\PageIndex{56}\)

\(4 x-\frac{3}{8}=3 x\)

**Answer**-
\(x=\frac{3}{8}\)

**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 \(\PageIndex{58}\)

\(5 n-20=-7 n-80\)

**Answer**-
\(n=-5\)

Exercise \(\PageIndex{59}\)

\(4 u+16=-19-u\)

Exercise \(\PageIndex{60}\)

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

**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 \(\PageIndex{61}\)

\(6(x+6)=24\)

Exercise \(\PageIndex{62}\)

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

**Answer**-
\(p=\frac{13}{2}\)

Exercise \(\PageIndex{63}\)

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

Exercise \(\PageIndex{64}\)

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

**Answer**-
\(n=12\)

Exercise \(\PageIndex{65}\)

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

Exercise \(\PageIndex{66}\)

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

**Answer**-
\(m=-14\)

Exercise \(\PageIndex{67}\)

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

Exercise \(\PageIndex{68}\)

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

**Answer**-
\(q=18\)

Exercise \(\PageIndex{69}\)

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

Exercise \(\PageIndex{70}\)

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

**Answer**-
\(x=-1\)

Exercise \(\PageIndex{71}\)

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

Exercise \(\PageIndex{72}\)

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

**Answer**-
\(k=\frac{3}{4}\)

**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 \(\PageIndex{74}\)

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

**Answer**-
contradiction; no solution

Exercise \(\PageIndex{75}\)

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

Exercise \(\PageIndex{76}\)

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

**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 \(\PageIndex{77}\)

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

Exercise \(\PageIndex{78}\)

\(\frac{1}{3} x+\frac{1}{5} x=8\)

**Answer**-
\(x=15\)

Exercise \(\PageIndex{79}\)

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

Exercise \(\PageIndex{80}\)

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

**Answer**-
\(k=41\)

Exercise \(\PageIndex{81}\)

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

Exercise \(\PageIndex{82}\)

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

**Answer**-
\(y=-1\)

**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 \(\PageIndex{84}\)

\(0.36 u+2.55=0.41 u+6.8\)

**Answer**-
\(u=-85\)

Exercise \(\PageIndex{85}\)

\(0.6 p-1.9=0.78 p+1.7\)

Exercise \(\PageIndex{86}\)

\(0.6 p-1.9=0.78 p+1.7\)

**Answer**-
\(d=-20\)

__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?

**Answer**-
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?

**Answer**-
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*

- when d=510 and r=60
- in general

Exercise \(\PageIndex{92}\)

Use the formula. d=rt to solve for r

- when when d=451 and t=5.5
- in general

**Answer**-
- r=82mph
- \(r=\frac{D}{t}\)

Exercise \(\PageIndex{93}\)

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

- when A=390 and h=26
- in general

Exercise \(\PageIndex{94}\)

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

- when A=153 and b=18
- in general

**Answer**-
- \(h=17\)
- \( h=\frac{2 A}{b}\)

Exercise \(\PageIndex{95}\)

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

- I=$2,501,r=4.1%, t=5 years
- in general

Exercise \(\PageIndex{96}\)

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

- when x=−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 \(\PageIndex{99}\)

- \(x\leq 4\)
- x>−2
- x<1

Exercise \(\PageIndex{100}\)

- x>0
- x<−3
- \(x\geq −1\)

**Answer**-

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

Exercise \(\PageIndex{101}\)

- \(x<-1\)
- \(x \geq-2.5\)
- \(x \leq \frac{5}{4}\)

Exercise \(\PageIndex{102}\)

- \(x>2\)
- \(x \leq-1.5\)
- \(x \geq \frac{5}{3}\)

**Answer**-

**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 \(6 x-3=x+20\)

- 5
- \(\frac{23}{5}\)

**Answer**-
- no
- 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*

- when x=−3
- in general

**Answer**-
- y=4
- \(y=\frac{5-x}{2}\)

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?

**Answer**-
\(120=\frac{2}{3} p ;\) 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?