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Elementary Algebra 2e

8.3 Add and Subtract Rational Expressions with a Common Denominator

Elementary Algebra 2e8.3 Add and Subtract Rational Expressions with a Common Denominator
  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. Key Terms
    13. Key Concepts
    14. 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. Key Terms
    10. Key Concepts
    11. 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. Key Terms
    9. Key Concepts
    10. 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. Key Terms
    10. Key Concepts
    11. 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. Key Terms
    9. Key Concepts
    10. 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. Key Terms
    10. Key Concepts
    11. 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. Key Terms
    9. Key Concepts
    10. 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. Key Terms
    12. Key Concepts
    13. 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. Key Terms
    11. Key Concepts
    12. 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. Key Terms
    8. Key Concepts
    9. 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

Learning Objectives

By the end of this section, you will be able to:
  • Add rational expressions with a common denominator
  • Subtract rational expressions with a common denominator
  • Add and subtract rational expressions whose denominators are opposites
Be Prepared 8.9

Before you get started, take this readiness quiz.

If you miss a problem, go back to the section listed and review the material.

Add: y3+93.y3+93.
If you missed this problem, review Example 1.77.

Be Prepared 8.10

Subtract: 10x2x.10x2x.
If you missed this problem, review Example 1.79.

Be Prepared 8.11

Factor completely: 8n520n3.8n520n3.
If you missed this problem, review Example 7.59.

Be Prepared 8.12

Factor completely: 45a35ab2.45a35ab2.
If you missed this problem, review Example 7.62.



Add Rational Expressions with a Common Denominator

What is the first step you take when you add numerical fractions? You check if they have a common denominator. If they do, you add the numerators and place the sum over the common denominator. If they do not have a common denominator, you find one before you add.

It is the same with rational expressions. To add rational expressions, they must have a common denominator. When the denominators are the same, you add the numerators and place the sum over the common denominator.

Rational Expression Addition

If p,q,andrp,q,andr are polynomials where r0r0, then

pr+qr=p+qrpr+qr=p+qr

To add rational expressions with a common denominator, add the numerators and place the sum over the common denominator.

We will add two numerical fractions first, to remind us of how this is done.

Example 8.30

Add: 518+718.518+718.

Try It 8.59

Add: 716+516.716+516.

Try It 8.60

Add: 310+110.310+110.

Remember, we do not allow values that would make the denominator zero. What value of yy should be excluded in the next example?

Example 8.31

Add: 3y4y3+74y3.3y4y3+74y3.

Try It 8.61

Add: 5x2x+3+22x+3.5x2x+3+22x+3.

Try It 8.62

Add: xx2+1x2.xx2+1x2.

Example 8.32

Add: 7x+12x+3+x2x+3.7x+12x+3+x2x+3.

Try It 8.63

Add: 9x+14x+7+x2x+7.9x+14x+7+x2x+7.

Try It 8.64

Add: x2+8xx+5+15x+5.x2+8xx+5+15x+5.

Subtract Rational Expressions with a Common Denominator

To subtract rational expressions, they must also have a common denominator. When the denominators are the same, you subtract the numerators and place the difference over the common denominator.

Rational Expression Subtraction

If p,q,andrp,q,andr are polynomials where r0r0, then

prqr=pqrprqr=pqr

To subtract rational expressions, subtract the numerators and place the difference over the common denominator.

We always simplify rational expressions. Be sure to factor, if possible, after you subtract the numerators so you can identify any common factors.

Example 8.33

Subtract: n2n10100n10.n2n10100n10.

Try It 8.65

Subtract: x2x+39x+3.x2x+39x+3.

Try It 8.66

Subtract: 4x22x5252x5.4x22x5252x5.

Be careful of the signs when you subtract a binomial!

Example 8.34

Subtract: y2y62y+24y6.y2y62y+24y6.

Try It 8.67

Subtract: n2n4n+12n4.n2n4n+12n4.

Try It 8.68

Subtract: y2y19y8y1.y2y19y8y1.

Example 8.35

Subtract: 5x27x+3x23x184x2+x9x23x18.5x27x+3x23x184x2+x9x23x18.

Try It 8.69

Subtract: 4x211x+8x23x+23x2+x3x23x+2.4x211x+8x23x+23x2+x3x23x+2.

Try It 8.70

Subtract: 6x2x+20x2815x2+11x7x281.6x2x+20x2815x2+11x7x281.

Add and Subtract Rational Expressions whose Denominators are Opposites

When the denominators of two rational expressions are opposites, it is easy to get a common denominator. We just have to multiply one of the fractions by −1−1−1−1.

Let’s see how this works.

.
Multiply the second fraction by −1−1−1−1. .
The denominators are the same. .
Simplify. .

Example 8.36

Add: 4u13u1+u13u.4u13u1+u13u.

Try It 8.71

Add: 8x152x5+2x52x.8x152x5+2x52x.

Try It 8.72

Add: 6y2+7y104y7+2y2+2y+1174y.6y2+7y104y7+2y2+2y+1174y.

Example 8.37

Subtract: m26mm213m+21m2.m26mm213m+21m2.

Try It 8.73

Subtract: y25yy246y64y2.y25yy246y64y2.

Try It 8.74

Subtract: 2n2+8n1n21n27n11n2.2n2+8n1n21n27n11n2.

Section 8.3 Exercises

Practice Makes Perfect

Add Rational Expressions with a Common Denominator

In the following exercises, add.

129.

215+715215+715

130.

421+321421+321

131.

724+1124724+1124

132.

736+1336736+1336

133.

3aab+1ab3aab+1ab

134.

3c4c5+54c53c4c5+54c5

135.

dd+8+5d+8dd+8+5d+8

136.

7m2m+n+42m+n7m2m+n+42m+n

137.

p2+10pp+2+16p+2p2+10pp+2+16p+2

138.

q2+12qq+3+27q+3q2+12qq+3+27q+3

139.

2r22r1+15r82r12r22r1+15r82r1

140.

3s23s2+13s103s23s23s2+13s103s2

141.

8t2t+4+32tt+48t2t+4+32tt+4

142.

6v2v+5+30vv+56v2v+5+30vv+5

143.

2w2w216+8ww2162w2w216+8ww216

144.

7x2x29+21xx297x2x29+21xx29

Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

145.

y2y+864y+8y2y+864y+8

146.

z2z+24z+2z2z+24z+2

147.

9a23a7493a79a23a7493a7

148.

25b25b6365b625b25b6365b6

149.

c2c86c+16c8c2c86c+16c8

150.

d2d96d+27d9d2d96d+27d9

151.

3m26m3021m306m303m26m3021m306m30

152.

2n24n3218n164n322n24n3218n164n32

153.

6p2+3p+4p2+4p55p2+p+7p2+4p56p2+3p+4p2+4p55p2+p+7p2+4p5

154.

5q2+3q9q2+6q+84q2+9q+7q2+6q+85q2+3q9q2+6q+84q2+9q+7q2+6q+8

155.

5r2+7r33r2494r2+5r+30r2495r2+7r33r2494r2+5r+30r249

156.

7t2t4t2256t2+12t44t2257t2t4t2256t2+12t44t225

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add.

157.

10v2v1+2v+412v10v2v1+2v+412v

158.

20w5w2+5w+625w20w5w2+5w+625w

159.

10x2+16x78x3+2x2+3x138x10x2+16x78x3+2x2+3x138x

160.

6y2+2y113y7+3y23y+1773y6y2+2y113y7+3y23y+1773y

In the following exercises, subtract.

161.

z2+6zz2253z+2025z2z2+6zz2253z+2025z2

162.

a2+3aa293a279a2a2+3aa293a279a2

163.

2b2+30b13b2492b25b849b22b2+30b13b2492b25b849b2

164.

c2+5c10c216c28c1016c2c2+5c10c216c28c1016c2

Everyday Math

165.

Sarah ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. If rr represents Sarah’s speed when she ran, then her running time is modeled by the expression 8r8r and her biking time is modeled by the expression 24r+4.24r+4. Add the rational expressions 8r+24r+48r+24r+4 to get an expression for the total amount of time Sarah ran and biked.

166.

If Pete can paint a wall in pp hours, then in one hour he can paint 1p1p of the wall. It would take Penelope 3 hours longer than Pete to paint the wall, so in one hour she can paint 1p+31p+3 of the wall. Add the rational expressions 1p+1p+31p+1p+3 to get an expression for the part of the wall Pete and Penelope would paint in one hour if they worked together.

Writing Exercises

167.

Donald thinks that 3x+4x3x+4x is 72x.72x. Is Donald correct? Explain.

168.

Explain how you find the Least Common Denominator of x2+5x+4x2+5x+4 and x216.x216.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

The above image is a table with four columns and four rows. The first row is the header row. The first header is labeled “I can…”, the second “Confidently”, the third, “With some help”, and the fourth “No – I don’t get it!”. In the first column under “I can”, the next row reads “add rational expressions with a common denominator.”, the next row reads “subtract rational expressions with a common denominator.”, the next row reads, “add and subtract rational expressions whose denominators are opposites.”, the last row reads “What does this checklist tell you about your mastery of this section? What steps will you take to improve?” The remaining columns are blank.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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