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Intermediate Algebra 2e

7.2 Add and Subtract Rational Expressions

Intermediate Algebra 2e7.2 Add and Subtract Rational Expressions
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. 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
    11. Chapter 11
    12. Chapter 12
  15. Index

Learning Objectives

By the end of this section, you will be able to:
  • Add and subtract rational expressions with a common denominator
  • Add and subtract rational expressions whose denominators are opposites
  • Find the least common denominator of rational expressions
  • Add and subtract rational expressions with unlike denominators
  • Add and subtract rational functions
Be Prepared 7.4

Before you get started, take this readiness quiz.

Add: 710+815.710+815.
If you missed this problem, review Example 1.29.

Be Prepared 7.5

Subtract: 3x489.3x489.
If you missed this problem, review Example 1.28.

Be Prepared 7.6

Subtract: 6(2x+1)4(x5).6(2x+1)4(x5).
If you missed this problem, review Example 1.56.

Add and Subtract 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 and Subtraction

If p, q, and r are polynomials where r0,r0, then

pr+qr=p+qrandprqr=pqrpr+qr=p+qrandprqr=pqr

To add or subtract rational expressions with a common denominator, add or subtract the numerators and place the result 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.

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

Example 7.13

Add: 11x+28x+4+x2x+4.11x+28x+4+x2x+4.

Try It 7.25

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

Try It 7.26

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

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. Be careful of the signs when you subtract a binomial or trinomial.

Example 7.14

Subtract: 5x27x+3x23x+184x2+x9x23x+18.5x27x+3x23x+184x2+x9x23x+18.

Try It 7.27

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

Try It 7.28

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. .

Be careful with the signs as you work with the opposites when the fractions are being subtracted.

Example 7.15

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

Try It 7.29

Subtract: y25yy246y64y2.y25yy246y64y2.

Try It 7.30

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

Find the Least Common Denominator of Rational Expressions

When we add or subtract rational expressions with unlike denominators, we will need to get common denominators. If we review the procedure we used with numerical fractions, we will know what to do with rational expressions.

Let’s look at this example: 712+518.712+518. Since the denominators are not the same, the first step was to find the least common denominator (LCD).

To find the LCD of the fractions, we factored 12 and 18 into primes, lining up any common primes in columns. Then we “brought down” one prime from each column. Finally, we multiplied the factors to find the LCD.

When we add numerical fractions, once we found the LCD, we rewrote each fraction as an equivalent fraction with the LCD by multiplying the numerator and denominator by the same number. We are now ready to add.

Seven-twelfths plus five-eighteenths. Write the prime factorizations of each denominator and line up the common factors. The denominator of the first fraction is 12. The prime factorization of 12 is 2 times 2 times 3. The denominator of the second fraction is 18. The prime factorization of 18 is 2 times 3 times 3. Bringing down a factor from each column, the lowest common denominator of 12 and 18 is 2 times 2 times 3 times 3, which is 36. Write both fractions using the lowest common denominator. To do this multiply the numerator and denominator of the first fraction by 3 and multiply the numerator and denominator of the second fraction by 2. The result is 7 times 3 all divided by 12 times 3 plus 5 times 2 all divided by 18 times 2. Simplify each fraction. 7 times 3 is 21 and 12 times 3 is 36. 5 times 2 is 10 and 18 times 2 is 36. The result is twenty-one thirty-sixths plus ten thirty-sixths.

We do the same thing for rational expressions. However, we leave the LCD in factored form.

How To

Find the least common denominator of rational expressions.

  1. Step 1. Factor each denominator completely.
  2. Step 2. List the factors of each denominator. Match factors vertically when possible.
  3. Step 3. Bring down the columns by including all factors, but do not include common factors twice.
  4. Step 4. Write the LCD as the product of the factors.

Remember, we always exclude values that would make the denominator zero. What values of xx should we exclude in this next example?

Example 7.16

Find the LCD for the expressions 8x22x3,3xx2+4x+38x22x3,3xx2+4x+3 and rewrite them as equivalent rational expressions with the lowest common denominator.

Try It 7.31

Find the LCD for the expressions 2x2x12,1x2162x2x12,1x216 rewrite them as equivalent rational expressions with the lowest common denominator.

Try It 7.32

Find the LCD for the expressions 3xx23x10,5x2+3x+23xx23x10,5x2+3x+2 rewrite them as equivalent rational expressions with the lowest common denominator.

Add and Subtract Rational Expressions with Unlike Denominators

Now we have all the steps we need to add or subtract rational expressions with unlike denominators.

Example 7.17 How to Add Rational Expressions with Unlike Denominators

Add: 3x3+2x2.3x3+2x2.

Try It 7.33

Add: 2x2+5x+3.2x2+5x+3.

Try It 7.34

Add:4m+3+3m+4.4m+3+3m+4.

The steps used to add rational expressions are summarized here.

How To

Add or subtract rational expressions.

  1. Step 1. Determine if the expressions have a common denominator.
    • Yes – go to step 2.
    • No – Rewrite each rational expression with the LCD.
      • Find the LCD.
      • Rewrite each rational expression as an equivalent rational expression with the LCD.
  2. Step 2. Add or subtract the rational expressions.
  3. Step 3. Simplify, if possible.

Avoid the temptation to simplify too soon. In the example above, we must leave the first rational expression as 3x6(x3)(x2)3x6(x3)(x2) to be able to add it to 2x6(x2)(x3).2x6(x2)(x3). Simplify only after you have combined the numerators.

Example 7.18

Add: 8x22x3+3xx2+4x+3.8x22x3+3xx2+4x+3.

Try It 7.35

Add: 1m2m2+5mm2+3m+2.1m2m2+5mm2+3m+2.

Try It 7.36

Add:2nn23n10+6n2+5n+6.2nn23n10+6n2+5n+6.

The process we use to subtract rational expressions with different denominators is the same as for addition. We just have to be very careful of the signs when subtracting the numerators.

Example 7.19

Subtract: 8yy2164y4.8yy2164y4.

Try It 7.37

Subtract: 2xx241x+2.2xx241x+2.

Try It 7.38

Subtract: 3z+36zz29.3z+36zz29.

There are lots of negative signs in the next example. Be extra careful.

Example 7.20

Subtract:−3n9n2+n6n+32n.−3n9n2+n6n+32n.

Try It 7.39

Subtract :3x1x25x626x.3x1x25x626x.

Try It 7.40

Subtract: −2y2y2+2y8y12y.−2y2y2+2y8y12y.

Things can get very messy when both fractions must be multiplied by a binomial to get the common denominator.

Example 7.21

Subtract: 4a2+6a+53a2+7a+10.4a2+6a+53a2+7a+10.

Try It 7.41

Subtract: 3b24b52b26b+5.3b24b52b26b+5.

Try It 7.42

Subtract: 4x243x2x2.4x243x2x2.

We follow the same steps as before to find the LCD when we have more than two rational expressions. In the next example, we will start by factoring all three denominators to find their LCD.

Example 7.22

Simplify: 2uu1+1u2u1u2u.2uu1+1u2u1u2u.

Try It 7.43

Simplify: vv+1+3v16v21.vv+1+3v16v21.

Try It 7.44

Simplify: 3ww+2+2w+717w+4w2+9w+14.3ww+2+2w+717w+4w2+9w+14.

Add and subtract rational functions

To add or subtract rational functions, we use the same techniques we used to add or subtract polynomial functions.

Example 7.23

Find R(x)=f(x)g(x)R(x)=f(x)g(x) where f(x)=x+5x2f(x)=x+5x2 and g(x)=5x+18x24.g(x)=5x+18x24.

Try It 7.45

Find R(x)=f(x)g(x)R(x)=f(x)g(x) where f(x)=x+1x+3f(x)=x+1x+3 and g(x)=x+17x2x12.g(x)=x+17x2x12.

Try It 7.46

Find R(x)=f(x)+g(x)R(x)=f(x)+g(x) where f(x)=x4x+3f(x)=x4x+3 and g(x)=4x+6x29.g(x)=4x+6x29.

Media Access Additional Online Resources

Access this online resource for additional instruction and practice with adding and subtracting rational expressions.

Section 7.2 Exercises

Practice Makes Perfect

Add and Subtract Rational Expressions with a Common Denominator

In the following exercises, add.

75.

215+715215+715

76.

724+1124724+1124

77.

3c4c5+54c53c4c5+54c5

78.

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

79.

2r22r1+15r82r12r22r1+15r82r1

80.

3s23s2+13s103s23s23s2+13s103s2

81.

2w2w216+8ww2162w2w216+8ww216

82.

7x2x29+21xx297x2x29+21xx29

In the following exercises, subtract.

83.

9a23a7493a79a23a7493a7

84.

25b25b6365b625b25b6365b6

85.

3m26m3021m306m303m26m3021m306m30

86.

2n24n3218n164n322n24n3218n164n32

87.

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

88.

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

89.

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

90.

7t2t4t2256t2+12t44t2257t2t4t2256t2+12t44t225

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add or subtract.

91.

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

92.

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

93.

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

94.

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

95.

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

96.

a2+3aa293a279a2a2+3aa293a279a2

97.

2b2+30b13b2492b25b849b22b2+30b13b2492b25b849b2

98.

c2+5c10c216c28c1016c2c2+5c10c216c28c1016c2

Find the Least Common Denominator of Rational Expressions

In the following exercises, find the LCD for the given rational expressions rewrite them as equivalent rational expressions with the lowest common denominator.

99.

5x22x8,2xx2x125x22x8,2xx2x12

100.

8y2+12y+35,3yy2+y428y2+12y+35,3yy2+y42

101.

9z2+2z8,4zz249z2+2z8,4zz24

102.

6a2+14a+45,5aa2816a2+14a+45,5aa281

103.

4b2+6b+9,2bb22b154b2+6b+9,2bb22b15

104.

5c24c+4,3cc27c+105c24c+4,3cc27c+10

105.

23d2+14d5,5d3d219d+623d2+14d5,5d3d219d+6

106.

35m23m2,6m5m2+17m+635m23m2,6m5m2+17m+6

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

107.

710x2y+415xy2710x2y+415xy2

108.

112a3b2+59a2b3112a3b2+59a2b3

109.

3r+4+2r53r+4+2r5

110.

4s7+5s+34s7+5s+3

111.

53w2+2w+153w2+2w+1

112.

42x+5+2x142x+5+2x1

113.

2yy+3+3y12yy+3+3y1

114.

3zz2+1z+53zz2+1z+5

115.

5ba2b2a2+2bb245ba2b2a2+2bb24

116.

4cd+3c+1d294cd+3c+1d29

117.

−3m3m3+5mm2+3m4−3m3m3+5mm2+3m4

118.

84n+4+6n2n284n+4+6n2n2

119.

3rr2+7r+6+9r2+4r+33rr2+7r+6+9r2+4r+3

120.

2ss2+2s8+4s2+3s102ss2+2s8+4s2+3s10

121.

tt6t2t+6tt6t2t+6

122.

x3x+6xx+3x3x+6xx+3

123.

5aa+3a+2a+65aa+3a+2a+6

124.

3bb2b6b83bb2b6b8

125.

6m+612mm2366m+612mm236

126.

4n+48nn2164n+48nn216

127.

−9p17p24p21p+17p−9p17p24p21p+17p

128.

13q8q2+2q24q+24q13q8q2+2q24q+24q

129.

−2r16r2+6r1652r−2r16r2+6r1652r

130.

2t30t2+6t2723t2t30t2+6t2723t

131.

2x+710x1+32x+710x1+3

132.

8y45y+268y45y+26

133.

3x23x42x25x+43x23x42x25x+4

134.

4x26x+53x27x+104x26x+53x27x+10

135.

5x2+8x94x2+10x+95x2+8x94x2+10x+9

136.

32x2+5x+212x2+3x+132x2+5x+212x2+3x+1

137.

5aa2+9a2a+18a22a5aa2+9a2a+18a22a

138.

2bb5+32b2b152b210b2bb5+32b2b152b210b

139.

cc+2+5c210cc24cc+2+5c210cc24

140.

6dd5+1d+47d5d2d206dd5+1d+47d5d2d20

141.

3dd+2+4dd+8d2+2d3dd+2+4dd+8d2+2d

142.

2qq+5+3q313q+15q2+2q152qq+5+3q313q+15q2+2q15

Add and Subtract Rational Functions

In the following exercises, find R(x)=f(x)+g(x)R(x)=f(x)+g(x) R(x)=f(x)g(x).R(x)=f(x)g(x).

143.

f(x)=−5x5x2+x6f(x)=−5x5x2+x6 and
g(x)=x+12xg(x)=x+12x

144.

f(x)=−4x24x2+x30f(x)=−4x24x2+x30 and
g(x)=x+75xg(x)=x+75x

145.

f(x)=6xx264f(x)=6xx264 and
g(x)=3x8g(x)=3x8

146.

f(x)=5x+7f(x)=5x+7 and
g(x)=10xx249g(x)=10xx249

Writing Exercises

147.

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

148.

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

149.

Felipe thinks 1x+1y1x+1y is 2x+y.2x+y. Choose numerical values for x and y and evaluate 1x+1y.1x+1y. Evaluate 2x+y2x+y for the same values of x and y you used in part . Explain why Felipe is wrong. Find the correct expression for 1x+1y.1x+1y.

150.

Simplify the expression 4n2+6n+91n294n2+6n+91n29 and explain all your steps.

Self Check

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

This table has four columns and six rows. The first row is a header and it labels each column, “I can…”, “Confidently,” “With some help,” and “No-I don’t get it!” In row 2, the I can was add and subtract rational expressions with a common denominator. In row 3, the I can was add and subtract rational expressions with denominators that are opposites. In row 4, the I can find the least common denominator of rational expressions. In row 5, the I can was add and subtract rational expressions with unlike denominators. In row 6, the I can was add or subtract rational functions. There is the nothing in the other columns.

After reviewing this checklist, what will you do to become confident for all objectives?

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