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Elementary Algebra 2e

Review Exercises

Elementary Algebra 2eReview 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 Solving 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 Trinomials of the Form x2+bx+c
    4. 7.3 Factor Trinomials of the Form ax2+bx+c
    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 in Two Variables
    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

Simplify Rational Expressions

Determine the Values for Which a Rational Expression is Undefined

In the following exercises, determine the values for which the rational expression is undefined.

513.

2 a + 1 3 a 2 2 a + 1 3 a 2

514.

b 3 b 2 16 b 3 b 2 16

515.

3 x y 2 5 y 3 x y 2 5 y

516.

u 3 u 2 u 30 u 3 u 2 u 30

Evaluate Rational Expressions

In the following exercises, evaluate the rational expressions for the given values.

517.

4 p 1 p 2 + 5 when p = −1 4 p 1 p 2 + 5 when p = −1

518.

q 2 5 q + 3 when q = 7 q 2 5 q + 3 when q = 7

519.

y 2 8 y 2 y 2 when y = 1 y 2 8 y 2 y 2 when y = 1

520.

z 2 + 2 4 z z 2 when z = 3 z 2 + 2 4 z z 2 when z = 3

Simplify Rational Expressions

In the following exercises, simplify.

521.

10 24 10 24

522.

8 m 4 16 m n 3 8 m 4 16 m n 3

523.

14 a 14 a 1 14 a 14 a 1

524.

b 2 + 7 b + 12 b 2 + 8 b + 16 b 2 + 7 b + 12 b 2 + 8 b + 16

Simplify Rational Expressions with Opposite Factors

In the following exercises, simplify.

525.

c 2 c 2 4 c 2 c 2 c 2 4 c 2

526.

d 16 16 d d 16 16 d

527.

7 v 35 25 v 2 7 v 35 25 v 2

528.

w 2 3 w 28 49 w 2 w 2 3 w 28 49 w 2

Multiply and Divide Rational Expressions

Multiply Rational Expressions

In the following exercises, multiply.

529.

3 8 · 2 15 3 8 · 2 15

530.

2 x y 2 8 y 3 · 16 y 24 x 2 x y 2 8 y 3 · 16 y 24 x

531.

3 a 2 + 21 a a 2 + 6 a 7 · a 1 a b 3 a 2 + 21 a a 2 + 6 a 7 · a 1 a b

532.

5 z 2 5 z 2 + 40 z + 35 · z 2 1 3 z 5 z 2 5 z 2 + 40 z + 35 · z 2 1 3 z

Divide Rational Expressions

In the following exercises, divide.

533.

t 2 4 t 12 t 2 + 8 t + 12 ÷ t 2 36 6 t t 2 4 t 12 t 2 + 8 t + 12 ÷ t 2 36 6 t

534.

r 2 16 4 ÷ r 3 64 2 r 2 + 8 r + 32 r 2 16 4 ÷ r 3 64 2 r 2 + 8 r + 32

535.

11 + w w 9 ÷ 121 w 2 9 w 11 + w w 9 ÷ 121 w 2 9 w

536.

3 y 2 12 y 63 4 y + 3 ÷ ( 6 y 2 42 y ) 3 y 2 12 y 63 4 y + 3 ÷ ( 6 y 2 42 y )

537.

c 2 64 3 c 2 + 26 c + 16 c 2 4 c 32 15 c + 10 c 2 64 3 c 2 + 26 c + 16 c 2 4 c 32 15 c + 10

538.

8 m 2 8 m m 4 · m 2 + 2 m 24 m 2 + 7 m + 10 ÷ 2 m 2 6 m m + 5 8 m 2 8 m m 4 · m 2 + 2 m 24 m 2 + 7 m + 10 ÷ 2 m 2 6 m m + 5

Add and Subtract Rational Expressions with a Common Denominator

Add Rational Expressions with a Common Denominator

In the following exercises, add.

539.

3 5 + 2 5 3 5 + 2 5

540.

4 a 2 2 a 1 1 2 a 1 4 a 2 2 a 1 1 2 a 1

541.

p 2 + 10 p p + 5 + 25 p + 5 p 2 + 10 p p + 5 + 25 p + 5

542.

3 x x 1 + 2 x 1 3 x x 1 + 2 x 1

Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

543.

d 2 d + 4 3 d + 28 d + 4 d 2 d + 4 3 d + 28 d + 4

544.

z 2 z + 10 100 z + 10 z 2 z + 10 100 z + 10

545.

4q2q+3q2+6q+5 3q2+q+6q2+6q+5 4q2q+3q2+6q+5 3q2+q+6q2+6q+5

546.

5 t + 4 t + 3 t 2 25 4 t 2 8 t 32 t 2 25 5 t + 4 t + 3 t 2 25 4 t 2 8 t 32 t 2 25

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add and subtract.

547.

18 w 6 w 1 + 3 w 2 1 6 w 18 w 6 w 1 + 3 w 2 1 6 w

548.

a 2 + 3 a a 2 16 3 a + 8 16 a 2 a 2 + 3 a a 2 16 3 a + 8 16 a 2

549.

2 b 2 + 3 b 15 b 2 49 b 2 + 16 b 1 49 b 2 2 b 2 + 3 b 15 b 2 49 b 2 + 16 b 1 49 b 2

550.

8 y 2 10 y + 7 2 y 5 + 2 y 2 + 7 y + 2 5 2 y 8 y 2 10 y + 7 2 y 5 + 2 y 2 + 7 y + 2 5 2 y

Add and Subtract Rational Expressions With Unlike Denominators

Find the Least Common Denominator of Rational Expressions

In the following exercises, find the LCD.

551.

4 m 2 3 m 10 , 2 m m 2 m 20 4 m 2 3 m 10 , 2 m m 2 m 20

552.

6 n 2 4 , 2 n n 2 4 n + 4 6 n 2 4 , 2 n n 2 4 n + 4

553.

5 3 p 2 + 19 p + 6 , 2 p 3 p 2 + 25 p + 8 5 3 p 2 + 19 p + 6 , 2 p 3 p 2 + 25 p + 8

Find Equivalent Rational Expressions

In the following exercises, rewrite as equivalent rational expressions with the given denominator.

554.

Rewrite as equivalent rational expressions with denominator (m+2)(m5)(m+4)(m+2)(m5)(m+4):

4 m 2 3 m 10 , 2 m m 2 m 20 . 4 m 2 3 m 10 , 2 m m 2 m 20 .
555.

Rewrite as equivalent rational expressions with denominator (n2)(n2)(n+2)(n2)(n2)(n+2):

6 n 2 4 n + 4 , 2 n n 2 4 . 6 n 2 4 n + 4 , 2 n n 2 4 .
556.

Rewrite as equivalent rational expressions with denominator (3p+1)(p+6)(p+8)(3p+1)(p+6)(p+8):

5 3 p 2 + 19 p + 6 , 7 p 3 p 2 + 25 p + 8 5 3 p 2 + 19 p + 6 , 7 p 3 p 2 + 25 p + 8

Add Rational Expressions with Different Denominators

In the following exercises, add.

557.

2 3 + 3 5 2 3 + 3 5

558.

7 5 a + 3 2 b 7 5 a + 3 2 b

559.

2 c 2 + 9 c + 3 2 c 2 + 9 c + 3

560.

3 d d 2 9 + 5 d 2 + 6 d + 9 3 d d 2 9 + 5 d 2 + 6 d + 9

561.

2 x x 2 + 10 x + 24 + 3 x x 2 + 8 x + 16 2 x x 2 + 10 x + 24 + 3 x x 2 + 8 x + 16

562.

5 q p 2 q p 2 + 4 q q 2 1 5 q p 2 q p 2 + 4 q q 2 1

Subtract Rational Expressions with Different Denominators

In the following exercises, subtract and add.

563.

3 v v + 2 v + 2 v + 8 3 v v + 2 v + 2 v + 8

564.

−3 w 15 w 2 + w 20 w + 2 4 w −3 w 15 w 2 + w 20 w + 2 4 w

565.

7 m + 3 m + 2 5 7 m + 3 m + 2 5

566.

n n + 3 + 2 n 3 n 9 n 2 9 n n + 3 + 2 n 3 n 9 n 2 9

567.

8 d d 2 64 4 d + 8 8 d d 2 64 4 d + 8

568.

5 12 x 2 y + 7 20 x y 3 5 12 x 2 y + 7 20 x y 3

Simplify Complex Rational Expressions

Simplify a Complex Rational Expression by Writing it as Division

In the following exercises, simplify.

569.

5 a a + 2 10 a 2 a 2 4 5 a a + 2 10 a 2 a 2 4

570.

2 5 + 5 6 1 3 + 1 4 2 5 + 5 6 1 3 + 1 4

571.

x 3 x x + 5 1 x + 5 + 1 x 5 x 3 x x + 5 1 x + 5 + 1 x 5

572.

2 m + m n n m 1 n 2 m + m n n m 1 n

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

573.

6 + 2 q 4 5 q + 4 6 + 2 q 4 5 q + 4

574.

3 a 2 1 b 1 a + 1 b 2 3 a 2 1 b 1 a + 1 b 2

575.

2 z 2 49 + 1 z + 7 9 z + 7 + 12 z 7 2 z 2 49 + 1 z + 7 9 z + 7 + 12 z 7

576.

3 y 2 4 y 32 2 y 8 + 1 y + 4 3 y 2 4 y 32 2 y 8 + 1 y + 4

Solve Rational Equations

Solve Rational Equations

In the following exercises, solve.

577.

1 2 + 2 3 = 1 x 1 2 + 2 3 = 1 x

578.

1 2 m = 8 m 2 1 2 m = 8 m 2

579.

1 b 2 + 1 b + 2 = 3 b 2 4 1 b 2 + 1 b + 2 = 3 b 2 4

580.

3 q + 8 2 q 2 = 1 3 q + 8 2 q 2 = 1

581.

v 15 v 2 9 v + 18 = 4 v 3 + 2 v 6 v 15 v 2 9 v + 18 = 4 v 3 + 2 v 6

582.

z 12 + z + 3 3 z = 1 z z 12 + z + 3 3 z = 1 z

Solve a Rational Equation for a Specific Variable

In the following exercises, solve for the indicated variable.

583.

V l = h w for l V l = h w for l

584.

1 x 2 y = 5 for y 1 x 2 y = 5 for y

585.

x = y + 5 z 7 for z x = y + 5 z 7 for z

586.

P = k V for V P = k V for V

Solve Proportion and Similar Figure Applications Similarity

Solve Proportions

In the following exercises, solve.

587.

x 4 = 3 5 x 4 = 3 5

588.

3 y = 9 5 3 y = 9 5

589.

s s + 20 = 3 7 s s + 20 = 3 7

590.

t 3 5 = t + 2 9 t 3 5 = t + 2 9

In the following exercises, solve using proportions.

591.

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

592.

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.

593.

∆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.
594.

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. Paris to Rome
  2. 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.
595.

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.

596.

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?

Solve Uniform Motion and Work Applications Problems

Solve Uniform Motion Applications

In the following exercises, solve.

597.

When making the 5-hour drive home from visiting her parents, Lisa ran into bad weather. She was able to drive 176 miles while the weather was good, but then driving 10 mph slower, went 81 miles in the bad weather. How fast did she drive when the weather was bad?

598.

Mark is riding on a plane that can fly 490 miles with a tailwind of 20 mph in the same time that it can fly 350 miles against a tailwind of 20 mph. What is the speed of the plane?

599.

John can ride his bicycle 8 mph faster than Luke can ride his bike. It takes Luke 3 hours longer than John to ride 48 miles. How fast can John ride his bike?

600.

Mark was training for a triathlon. He ran 8 kilometers and biked 32 kilometers in a total of 3 hours. His running speed was 8 kilometers per hour less than his biking speed. What was his running speed?

Solve Work Applications

In the following exercises, solve.

601.

Jerry can frame a room in 1 hour, while Jake takes 4 hours. How long could they frame a room working together?

602.

Lisa takes 3 hours to mow the lawn while her cousin, Barb, takes 2 hours. How long will it take them working together?

603.

Jeffrey can paint a house in 6 days, but if he gets a helper he can do it in 4 days. How long would it take the helper to paint the house alone?

604.

Sue and Deb work together writing a book that takes them 90 days. If Sue worked alone it would take her 120 days. How long would it take Deb to write the book alone?

Use Direct and Inverse Variation

Solve Direct Variation Problems

In the following exercises, solve.

605.

If yy varies directly as xx, when y=9y=9 and x=3x=3, find xx when y=21.y=21.

606.

If yy varies inversely as xx, when y=20y=20 and x=2x=2 find yy when x=4.x=4.

607.

If mm varies inversely with the square of nn, when m=4m=4 and n=6n=6 find mm when n=2.n=2.

608.

Vanessa is traveling to see her fiancé. The distance, d, varies directly with the speed, v, she drives. If she travels 258 miles driving 60 mph, how far would she travel going 70 mph?

609.

If the cost of a pizza varies directly with its diameter, and if an 8” diameter pizza costs $12, how much would a 6” diameter pizza cost?

610.

The distance to stop a car varies directly with the square of its speed. It takes 200 feet to stop a car going 50 mph. How many feet would it take to stop a car going 60 mph?

Solve Inverse Variation Problems

In the following exercises, solve.

611.

The number of tickets for a music fundraiser varies inversely with the price of the tickets. If Madelyn has just enough money to purchase 12 tickets for $6, how many tickets can Madelyn afford to buy if the price increased to $8?

612.

On a string instrument, the length of a string varies inversely with the frequency of its vibrations. If an 11-inch string on a violin has a frequency of 360 cycles per second, what frequency does a 12 inch string have?

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