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Elementary Algebra 2e

8.4 Add and Subtract Rational Expressions with Unlike Denominators

Elementary Algebra 2e8.4 Add and Subtract Rational Expressions with Unlike Denominators
  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:
  • Find the least common denominator of rational expressions
  • Find equivalent rational expressions
  • Add rational expressions with different denominators
  • Subtract rational expressions with different denominators
Be Prepared 8.13

Before you get started, take this readiness quiz.

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

Add: 710+815.710+815.
If you missed this problem, review Example 1.81.

Be Prepared 8.14

Subtract: 6(2x+1)4(x5).6(2x+1)4(x5).
If you missed this problem, review Example 1.139.

Be Prepared 8.15

Find the Greatest Common Factor of 9x2y39x2y3 and 12xy512xy5.
If you missed this problem, review Example 7.3.

Be Prepared 8.16

Factor completely −48n12−48n12.
If you missed this problem, review Example 7.11.

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 the example 712+518712+518 from Foundations. Since the denominators are not the same, the first step was to find the least common denominator (LCD). Remember, the LCD is the least common multiple of the denominators. It is the smallest number we can use as a common denominator.

To find the LCD of 12 and 18, we factored each number 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.

12=2·2·318=2·3·3LCD=2·2·3·3LCD=3612=2·2·318=2·3·3LCD=2·2·3·3LCD=36

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 expression completely.
  2. Step 2. List the factors of each expression. Match factors vertically when possible.
  3. Step 3. Bring down the columns.
  4. Step 4. Multiply the factors.

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

Example 8.38

Find the LCD for 8x22x3,3xx2+4x+38x22x3,3xx2+4x+3.

Try It 8.75

Find the LCD for 2x2x12,1x2162x2x12,1x216.

Try It 8.76

Find the LCD for xx2+8x+15,5x2+9x+18xx2+8x+15,5x2+9x+18.

Find Equivalent Rational Expressions

When we add numerical fractions, once we find the LCD, we rewrite each fraction as an equivalent fraction with the LCD.

The above image shows how to find the LCD (least common denominator) when adding numerical fractions in the example seven-twelfths plus five-eighteenths. The image shows 7 times 3 divided by 12 times 3 plus 5 times 2 plus 18 times 2. Below this is 21 divided by 36 plus 10 divided by 36. The image next to this shows that 12 equals 2 times 2 times 3. Below this shows 18 equals 2 times 3 times 3. A line is drawn. Below it is LCD equals 2 times 2 times 3 times 3. The line below this shows that the LCD equals 36.

We will do the same thing for rational expressions.

Example 8.39

Rewrite as equivalent rational expressions with denominator (x+1)(x3)(x+3)(x+1)(x3)(x+3): 8x22x3,3xx2+4x+3.8x22x3,3xx2+4x+3.

Try It 8.77

Rewrite as equivalent rational expressions with denominator (x+3)(x4)(x+4)(x+3)(x4)(x+4):
2x2x12,1x216.2x2x12,1x216.

Try It 8.78

Rewrite as equivalent rational expressions with denominator (x+3)(x+5)(x+6)(x+3)(x+5)(x+6):
xx2+8x+15,5x2+9x+18.xx2+8x+15,5x2+9x+18.

Add Rational Expressions with Different Denominators

Now we have all the steps we need to add rational expressions with different denominators. As we have done previously, we will do one example of adding numerical fractions first.

Example 8.40

Add: 712+518.712+518.

Try It 8.79

Add: 1130+712.1130+712.

Try It 8.80

Add: 38+920.38+920.

Now we will add rational expressions whose denominators are monomials.

Example 8.41

Add: 512x2y+421xy2.512x2y+421xy2.

Try It 8.81

Add: 215a2b+56ab2.215a2b+56ab2.

Try It 8.82

Add: 516c+38cd2.516c+38cd2.

Now we are ready to tackle polynomial denominators.

Example 8.42 How to Add Rational Expressions with Different Denominators

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

Try It 8.83

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

Try It 8.84

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

The steps to use to add rational expressions are summarized in the following procedure box.

How To

Add 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 the rational expressions.
  3. Step 3. Simplify, if possible.

Example 8.43

Add: 2a2ab+b2+3a4a2b2.2a2ab+b2+3a4a2b2.

Try It 8.85

Add: 5xxyy2+2xx2y2.5xxyy2+2xx2y2.

Try It 8.86

Add: 72m+6+4m2+4m+3.72m+6+4m2+4m+3.

Avoid the temptation to simplify too soon! In the example above, we must leave the first rational expression as 2a(2ab)b(2a+b)(2ab)2a(2ab)b(2a+b)(2ab) to be able to add it to 3a·b(2a+b)(2ab)·b3a·b(2a+b)(2ab)·b. Simplify only after you have combined the numerators.

Example 8.44

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

Try It 8.87

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

Try It 8.88

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

Subtract Rational Expressions with Different Denominators

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 8.45 How to Subtract Rational Expressions with Different Denominators

Subtract: xx3x2x+3.xx3x2x+3.

Try It 8.89

Subtract: yy+4y2y5.yy+4y2y5.

Try It 8.90

Subtract: z+3z+2zz+3.z+3z+2zz+3.

The steps to take to subtract rational expressions are listed below.

How To

Subtract rational expressions.

  1. Step 1. Determine if they 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. Subtract the rational expressions.
  3. Step 3. Simplify, if possible.

Example 8.46

Subtract: 8yy2164y4.8yy2164y4.

Try It 8.91

Subtract: 2xx241x+2.2xx241x+2.

Try It 8.92

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

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

Example 8.47

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

Try It 8.93

Subtract: 3x1x25x626x.3x1x25x626x.

Try It 8.94

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

When one expression is not in fraction form, we can write it as a fraction with denominator 1.

Example 8.48

Subtract: 5c+4c23.5c+4c23.

Try It 8.95

Subtract: 2x+1x73.2x+1x73.

Try It 8.96

Subtract: 4y+32y15.4y+32y15.

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.

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 8.49

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

Try It 8.97

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

Try It 8.98

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

Section 8.4 Exercises

Practice Makes Perfect

In the following exercises, find the LCD.

169.

5x22x8,2xx2x125x22x8,2xx2x12

170.

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

171.

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

172.

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

173.

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

174.

5c24c+4,3cc210c+165c24c+4,3cc210c+16

175.

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

176.

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

In the following exercises, write as equivalent rational expressions with the given LCD.

177.

5x22x8,2xx2x125x22x8,2xx2x12
LCD (x4)(x+2)(x+3)(x4)(x+2)(x+3)

178.

8y2+12y+35,3yy2+y428y2+12y+35,3yy2+y42
LCD (y+7)(y+5)(y6)(y+7)(y+5)(y6)

179.

9z2+2z8,4zz249z2+2z8,4zz24
LCD (z2)(z+4)(z+2)(z2)(z+4)(z+2)

180.

6a2+14a+45,5aa2816a2+14a+45,5aa281
LCD (a+9)(a+5)(a9)(a+9)(a+5)(a9)

181.

4b2+6b+9,2bb22b154b2+6b+9,2bb22b15
LCD (b+3)(b+3)(b5)(b+3)(b+3)(b5)

182.

5c24c+4,3cc210c+165c24c+4,3cc210c+16
LCD (c2)(c2)(c8)(c2)(c2)(c8)

183.

23d2+14d5,5d3d219d+623d2+14d5,5d3d219d+6
LCD (3d1)(d+5)(d6)(3d1)(d+5)(d6)

184.

35m23m2,6m5m2+17m+635m23m2,6m5m2+17m+6
LCD (5m+2)(m1)(m+3)(5m+2)(m1)(m+3)

In the following exercises, add.

185.

524+1136524+1136

186.

730+1345730+1345

187.

920+1130920+1130

188.

827+718827+718

189.

710x2y+415xy2710x2y+415xy2

190.

112a3b2+59a2b3112a3b2+59a2b3

191.

12m+78m2n12m+78m2n

192.

56p2q+14p56p2q+14p

193.

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

194.

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

195.

8t+5+6t58t+5+6t5

196.

7v+5+9v57v+5+9v5

197.

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

198.

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

199.

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

200.

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

201.

5ba2b2a2+2bb245ba2b2a2+2bb24

202.

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

203.

2m3m3+5mm2+3m42m3m3+5mm2+3m4

204.

34n+4+6n2n234n+4+6n2n2

205.

3n2+3n18+4nn2+8n+123n2+3n18+4nn2+8n+12

206.

6q23q10+5qq28q+156q23q10+5qq28q+15

207.

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

208.

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

In the following exercises, subtract.

209.

tt6t2t+6tt6t2t+6

210.

vv3v6v+1vv3v6v+1

211.

w+2w+4ww2w+2w+4ww2

212.

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

213.

y4y+11y+7y4y+11y+7

214.

z+8z3zz2z+8z3zz2

215.

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

216.

3bb2b6b83bb2b6b8

217.

6cc2253c+56cc2253c+5

218.

4dd2812d+94dd2812d+9

219.

6m+612mm2366m+612mm236

220.

4n+48nn2164n+48nn216

221.

−9p17p24p21p+17p−9p17p24p21p+17p

222.

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

223.

−2r16r2+6r1652r−2r16r2+6r1652r

224.

2t30t2+6t2723t2t30t2+6t2723t

225.

5v2v+345v2v+34

226.

6w+5w1+26w+5w1+2

227.

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

228.

8y45y+268y45y+26

In the following exercises, add and subtract.

229.

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

230.

2bb5+32b2b152b210b2bb5+32b2b152b210b

231.

cc+2+5c210cc24cc+2+5c210cc24

232.

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

In the following exercises, simplify.

233.

6a3ab+b2+3a9a2b26a3ab+b2+3a9a2b2

234.

2c2c+10+7cc2+9c+202c2c+10+7cc2+9c+20

235.

6dd2643d86dd2643d8

236.

5n+710nn2495n+710nn249

237.

4mm2+6m7+2m2+10m+214mm2+6m7+2m2+10m+21

238.

3pp2+4p12+1p2+p303pp2+4p12+1p2+p30

239.

−5n5n2+n6+n+12n−5n5n2+n6+n+12n

240.

−4b24b2+b30+b+75b−4b24b2+b30+b+75b

241.

715p+518pq715p+518pq

242.

320a2+1112ab2320a2+1112ab2

243.

4x2+3x+54x2+3x+5

244.

6m+4+9m86m+4+9m8

245.

2q+7q+422q+7q+42

246.

3y1y+423y1y+42

247.

z+2z5zz+1z+2z5zz+1

248.

tt5t1t+5tt5t1t+5

249.

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

250.

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

Everyday Math

251.

Decorating cupcakes Victoria can decorate an order of cupcakes for a wedding in tt hours, so in 1 hour she can decorate 1t1t of the cupcakes. It would take her sister 3 hours longer to decorate the same order of cupcakes, so in 1 hour she can decorate 1t+31t+3 of the cupcakes.

  1. Find the fraction of the decorating job that Victoria and her sister, working together, would complete in one hour by adding the rational expressions 1t+1t+3.1t+1t+3.
  2. Evaluate your answer to part (a) when t=5.t=5.
252.

Kayaking When Trina kayaks upriver, it takes her 53c53c hours to go 5 miles, where cc is the speed of the river current. It takes her 53+c53+c hours to kayak 5 miles down the river.

  1. Find an expression for the number of hours it would take Trina to kayak 5 miles up the river and then return by adding 53c+53+c.53c+53+c.
  2. Evaluate your answer to part (a) when c=1c=1 to find the number of hours it would take Trina if the speed of the river current is 1 mile per hour.

Writing Exercises

253.

Felipe thinks 1x+1y1x+1y is 2x+y.2x+y.

  1. Choose numerical values for x and y and evaluate 1x+1y.1x+1y.
  2. Evaluate 2x+y2x+y for the same values of x and y you used in part (a).
  3. Explain why Felipe is wrong.
  4. Find the correct expression for 1x+1y.1x+1y.
254.

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 is a table that has five rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “find the least common denominator of rational expressions,” “find equivalent rational expressions,” “add rational expressions with different denominators,” and “subtract rational expressions with different denominators.” The rest of the cells are blank.

On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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