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Elementary Algebra

Review Exercises

Elementary AlgebraReview Exercises
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Table of contents
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Use the Language of Algebra
    4. 1.3 Add and Subtract Integers
    5. 1.4 Multiply and Divide Integers
    6. 1.5 Visualize Fractions
    7. 1.6 Add and Subtract Fractions
    8. 1.7 Decimals
    9. 1.8 The Real Numbers
    10. 1.9 Properties of Real Numbers
    11. 1.10 Systems of Measurement
    12. Chapter Review
      1. Key Terms
      2. Key Concepts
    13. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations and Inequalities
    1. Introduction
    2. 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 2.2 Solve Equations using the Division and Multiplication Properties of Equality
    4. 2.3 Solve Equations with Variables and Constants on Both Sides
    5. 2.4 Use a General Strategy to Solve Linear Equations
    6. 2.5 Solve Equations with Fractions or Decimals
    7. 2.6 Solve a Formula for a Specific Variable
    8. 2.7 Solve Linear Inequalities
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Math Models
    1. Introduction
    2. 3.1 Use a Problem-Solving Strategy
    3. 3.2 Solve Percent Applications
    4. 3.3 Solve Mixture Applications
    5. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
    6. 3.5 Solve Uniform Motion Applications
    7. 3.6 Solve Applications with Linear Inequalities
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Graphs
    1. Introduction
    2. 4.1 Use the Rectangular Coordinate System
    3. 4.2 Graph Linear Equations in Two Variables
    4. 4.3 Graph with Intercepts
    5. 4.4 Understand Slope of a Line
    6. 4.5 Use the Slope–Intercept Form of an Equation of a Line
    7. 4.6 Find the Equation of a Line
    8. 4.7 Graphs of Linear Inequalities
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Systems of Linear Equations
    1. Introduction
    2. 5.1 Solve Systems of Equations by Graphing
    3. 5.2 Solve Systems of Equations by Substitution
    4. 5.3 Solve Systems of Equations by Elimination
    5. 5.4 Solve Applications with Systems of Equations
    6. 5.5 Solve Mixture Applications with Systems of Equations
    7. 5.6 Graphing Systems of Linear Inequalities
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Polynomials
    1. Introduction
    2. 6.1 Add and Subtract Polynomials
    3. 6.2 Use Multiplication Properties of Exponents
    4. 6.3 Multiply Polynomials
    5. 6.4 Special Products
    6. 6.5 Divide Monomials
    7. 6.6 Divide Polynomials
    8. 6.7 Integer Exponents and Scientific Notation
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Factoring
    1. Introduction
    2. 7.1 Greatest Common Factor and Factor by Grouping
    3. 7.2 Factor Quadratic Trinomials with Leading Coefficient 1
    4. 7.3 Factor Quadratic Trinomials with Leading Coefficient Other than 1
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Rational Expressions and Equations
    1. Introduction
    2. 8.1 Simplify Rational Expressions
    3. 8.2 Multiply and Divide Rational Expressions
    4. 8.3 Add and Subtract Rational Expressions with a Common Denominator
    5. 8.4 Add and Subtract Rational Expressions with Unlike Denominators
    6. 8.5 Simplify Complex Rational Expressions
    7. 8.6 Solve Rational Equations
    8. 8.7 Solve Proportion and Similar Figure Applications
    9. 8.8 Solve Uniform Motion and Work Applications
    10. 8.9 Use Direct and Inverse Variation
    11. Chapter Review
      1. Key Terms
      2. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Roots and Radicals
    1. Introduction
    2. 9.1 Simplify and Use Square Roots
    3. 9.2 Simplify Square Roots
    4. 9.3 Add and Subtract Square Roots
    5. 9.4 Multiply Square Roots
    6. 9.5 Divide Square Roots
    7. 9.6 Solve Equations with Square Roots
    8. 9.7 Higher Roots
    9. 9.8 Rational Exponents
    10. Chapter Review
      1. Key Terms
      2. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Quadratic Equations
    1. Introduction
    2. 10.1 Solve Quadratic Equations Using the Square Root Property
    3. 10.2 Solve Quadratic Equations by Completing the Square
    4. 10.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 10.4 Solve Applications Modeled by Quadratic Equations
    6. 10.5 Graphing Quadratic Equations
    7. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
  13. Index

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.

512.

10 x − 1 = 5 x ; x = 1 5 10 x − 1 = 5 x ; x = 1 5

513.

w + 2 = 5 8 ; w = 3 8 w + 2 = 5 8 ; w = 3 8

514.

−12 n + 5 = 8 n ; n = − 5 4 −12 n + 5 = 8 n ; n = − 5 4

515.

6 a − 3 = −7 a , a = 3 13 6 a − 3 = −7 a , a = 3 13

Solve Equations using the Subtraction and Addition Properties of Equality

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

516.

x + 7 = 19 x + 7 = 19

517.

y + 2 = −6 y + 2 = −6

518.

a + 1 3 = 5 3 a + 1 3 = 5 3

519.

n + 3.6 = 5.1 n + 3.6 = 5.1

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

520.

u − 7 = 10 u − 7 = 10

521.

x − 9 = −4 x − 9 = −4

522.

c − 3 11 = 9 11 c − 3 11 = 9 11

523.

p − 4.8 = 14 p − 4.8 = 14

In the following exercises, solve each equation.

524.

n − 12 = 32 n − 12 = 32

525.

y + 16 = −9 y + 16 = −9

526.

f + 2 3 = 4 f + 2 3 = 4

527.

d − 3.9 = 8.2 d − 3.9 = 8.2

Solve Equations That Require Simplification

In the following exercises, solve each equation.

528.

y + 8 − 15 = −3 y + 8 − 15 = −3

529.

7 x + 10 − 6 x + 3 = 5 7 x + 10 − 6 x + 3 = 5

530.

6 ( n − 1 ) − 5 n = −14 6 ( n − 1 ) − 5 n = −14

531.

8 ( 3 p + 5 ) − 23 ( p − 1 ) = 35 8 ( 3 p + 5 ) − 23 ( p − 1 ) = 35

Translate to an Equation and Solve

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

532.

The sum of −6−6 and mm is 25.

533.

Four less than nn is 13.

Translate and Solve Applications

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

534.

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

535.

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

536.

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?

537.

Elissa earned $152.84 this week, which was $21.65 more than she earned last week. How much did she earn last week?

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.

538.

8 x = 72 8 x = 72

539.

13 a = −65 13 a = −65

540.

0.25 p = 5.25 0.25 p = 5.25

541.

− y = 4 − y = 4

542.

n 6 = 18 n 6 = 18

543.

y −10 = 30 y −10 = 30

544.

36 = 3 4 x 36 = 3 4 x

545.

5 8 u = 15 16 5 8 u = 15 16

546.

−18 m = −72 −18 m = −72

547.

c 9 = 36 c 9 = 36

548.

0.45 x = 6.75 0.45 x = 6.75

549.

11 12 = 2 3 y 11 12 = 2 3 y

Solve Equations That Require Simplification

In the following exercises, solve each equation requiring simplification.

550.

5 r − 3 r + 9 r = 35 − 2 5 r − 3 r + 9 r = 35 − 2

551.

24 x + 8 x − 11 x = −7 − 14 24 x + 8 x − 11 x = −7 − 14

552.

11 12 n − 5 6 n = 9 − 5 11 12 n − 5 6 n = 9 − 5

553.

−9 ( d − 2 ) − 15 = −24 −9 ( d − 2 ) − 15 = −24

Translate to an Equation and Solve

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

554.

143 is the product of −11−11 and y.

555.

The quotient of b and and 9 is −27−27.

556.

The sum of q and one-fourth is one.

557.

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

Translate and Solve Applications

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

558.

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

559.

Janet gets paid $24 per hour. She heard that this is 3434 of what Adam is paid. How much is Adam paid per hour?

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.

560.

8 p + 7 = 47 8 p + 7 = 47

561.

10 w − 5 = 65 10 w − 5 = 65

562.

3 x + 19 = −47 3 x + 19 = −47

563.

32 = −4 − 9 n 32 = −4 − 9 n

Solve an Equation with Variables on Both Sides

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

564.

7 y = 6 y − 13 7 y = 6 y − 13

565.

5 a + 21 = 2 a 5 a + 21 = 2 a

566.

k = −6 k − 35 k = −6 k − 35

567.

4 x − 3 8 = 3 x 4 x − 3 8 = 3 x

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.

568.

12 x − 9 = 3 x + 45 12 x − 9 = 3 x + 45

569.

5 n − 20 = −7 n − 80 5 n − 20 = −7 n − 80

570.

4 u + 16 = −19 − u 4 u + 16 = −19 − u

571.

5 8 c − 4 = 3 8 c + 4 5 8 c − 4 = 3 8 c + 4

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.

572.

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

573.

9 ( 2 p − 5 ) = 72 9 ( 2 p − 5 ) = 72

574.

− ( s + 4 ) = 18 − ( s + 4 ) = 18

575.

8 + 3 ( n − 9 ) = 17 8 + 3 ( n − 9 ) = 17

576.

23 − 3 ( y − 7 ) = 8 23 − 3 ( y − 7 ) = 8

577.

1 3 ( 6 m + 21 ) = m − 7 1 3 ( 6 m + 21 ) = m − 7

578.

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

579.

0.25 ( q − 8 ) = 0.1 ( q + 7 ) 0.25 ( q − 8 ) = 0.1 ( q + 7 )

580.

8 ( r − 2 ) = 6 ( r + 10 ) 8 ( r − 2 ) = 6 ( r + 10 )

581.

5 + 7 ( 2 − 5 x ) = 2 ( 9 x + 1 ) 5 + 7 ( 2 − 5 x ) = 2 ( 9 x + 1 )
− ( 13 x − 57 ) − ( 13 x − 57 )

582.

( 9 n + 5 ) − ( 3 n − 7 ) ( 9 n + 5 ) − ( 3 n − 7 )
= 20 − ( 4 n − 2 ) = 20 − ( 4 n − 2 )

583.

2 [ −16 + 5 ( 8 k − 6 ) ] 2 [ −16 + 5 ( 8 k − 6 ) ]
= 8 ( 3 − 4 k ) − 32 = 8 ( 3 − 4 k ) − 32

Classify Equations

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

584.

17 y − 3 ( 4 − 2 y ) = 11 ( y − 1 ) 17 y − 3 ( 4 − 2 y ) = 11 ( y − 1 )
+ 12 y − 1 + 12 y − 1

585.

9 u + 32 = 15 ( u − 4 ) 9 u + 32 = 15 ( u − 4 )
− 3 ( 2 u + 21 ) − 3 ( 2 u + 21 )

586.

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

587.

21 ( c − 1 ) − 19 ( c + 1 ) 21 ( c − 1 ) − 19 ( c + 1 )
= 2 ( c − 20 ) = 2 ( c − 20 )

Solve Equations with Fractions and Decimals

Solve Equations with Fraction Coefficients

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

588.

2 5 n − 1 10 = 7 10 2 5 n − 1 10 = 7 10

589.

1 3 x + 1 5 x = 8 1 3 x + 1 5 x = 8

590.

3 4 a − 1 3 = 1 2 a − 5 6 3 4 a − 1 3 = 1 2 a − 5 6

591.

1 2 ( k − 3 ) = 1 3 ( k + 16 ) 1 2 ( k − 3 ) = 1 3 ( k + 16 )

592.

3 x − 2 5 = 3 x + 4 8 3 x − 2 5 = 3 x + 4 8

593.

5 y − 1 3 + 4 = −8 y + 4 6 5 y − 1 3 + 4 = −8 y + 4 6

Solve Equations with Decimal Coefficients

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

594.

0.8 x − 0.3 = 0.7 x + 0.2 0.8 x − 0.3 = 0.7 x + 0.2

595.

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

596.

0.6 p − 1.9 = 0.78 p + 1.7 0.6 p − 1.9 = 0.78 p + 1.7

597.

0.6 p − 1.9 = 0.78 p + 1.7 0.6 p − 1.9 = 0.78 p + 1.7

Solve a Formula for a Specific Variable

Use the Distance, Rate, and Time Formula

In the following exercises, solve.

598.

Natalie drove for 712712 hours at 60 miles per hour. How much distance did she travel?

599.

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?

600.

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?

601.

Link rode his bike at a steady rate of 15 miles per hour for 212212 hours. How much distance did he travel?

Solve a Formula for a Specific Variable

In the following exercises, solve.

602.

Use the formula. d=rtd=rt to solve for t
ⓐ when d=510d=510 and r=60r=60
ⓑ in general

603.

Use the formula. d=rtd=rt to solve for rr
ⓐ when when d=451d=451 and t=5.5t=5.5
ⓑ in general

604.

Use the formula A=12bhA=12bh to solve for bb
ⓐ when A=390A=390 and h=26h=26
ⓑ in general

605.

Use the formula A=12bhA=12bh to solve for hh
ⓐ when A=153A=153 and b=18b=18
ⓑ in general

606.

Use the formula I=PrtI=Prt to solve for the principal, P for
ⓐ I=$2,501,r=4.1%,I=$2,501,r=4.1%,
t=5yearst=5years
ⓑ in general

607.

Solve the formula 4x+3y=64x+3y=6 for y
ⓐ when x=−2x=−2
ⓑ in general

608.

Solve 180=a+b+c180=a+b+c for cc.

609.

Solve the formula V=LWHV=LWH for HH.

Solve Linear Inequalities

Graph Inequalities on the Number Line

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

610.


ⓐ x≤4x≤4
ⓑ x>−2x>−2
ⓒ x<1x<1

611.


ⓐ x>0x>0
ⓑ x<−3x<−3
ⓒ x≥−1x≥−1

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

612.


ⓐ x<−1x<−1
ⓑ x≥−2.5x≥−2.5
ⓒ x≤54x≤54

613.


ⓐ x>2x>2
ⓑ x≤−1.5x≤−1.5
ⓒ x≥53x≥53

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.

614.

n − 12 ≤ 23 n − 12 ≤ 23

615.

m + 14 ≤ 56 m + 14 ≤ 56

616.

a + 2 3 ≥ 7 12 a + 2 3 ≥ 7 12

617.

b − 7 8 ≥ − 1 2 b − 7 8 ≥ − 1 2

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.

618.

9 x > 54 9 x > 54

619.

−12 d ≤ 108 −12 d ≤ 108

620.

5 2 j < − 60 5 2 j < − 60

621.

q − 2 ≥ − 24 q − 2 ≥ − 24

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.

622.

6 p > 15 p − 30 6 p > 15 p − 30

623.

9 h − 7 ( h − 1 ) ≤ 4 h − 23 9 h − 7 ( h − 1 ) ≤ 4 h − 23

624.

5 n − 15 ( 4 − n ) < 10 ( n − 6 ) + 10 n 5 n − 15 ( 4 − n ) < 10 ( n − 6 ) + 10 n

625.

3 8 a − 1 12 a > 5 12 a + 3 4 3 8 a − 1 12 a > 5 12 a + 3 4

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.

626.

Five more than z is at most 19.

627.

Three less than c is at least 360.

628.

Nine times n exceeds 42.

629.

Negative two times a is no more than 8.

Everyday Math
630.

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

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