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Subtract: x 2 x + 3 9 x + 3 .

x 3

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Subtract: 4 x 2 2 x 5 25 2 x 5 .

2 x + 5

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Be careful of the signs when you subtract a binomial!

Subtract: y 2 y 6 2 y + 24 y 6 .

Solution

y 2 y 6 2 y + 24 y 6 The fractions have a common denominator, so subtract the numerators and place the difference over the common denominator. y 2 ( 2 y + 24 ) y 6 Distribute the sign in the numerator. y 2 2 y 24 y 6 Factor the numerator. ( y 6 ) ( y + 4 ) y 6 Remove common factors. ( y 6 ) ( y + 4 ) y 6 Simplify. y + 4

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Subtract: n 2 n 4 n + 12 n 4 .

n + 3

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Subtract: y 2 y 1 9 y 8 y 1 .

y 8

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Subtract: 5 x 2 7 x + 3 x 2 3 x + 18 4 x 2 + x 9 x 2 3 x + 18 .

Solution

5 x 2 7 x + 3 x 2 3 x + 18 4 x 2 + x 9 x 2 3 x + 18 Subtract the numerators and place the difference over the common denominator. 5 x 2 7 x + 3 ( 4 x 2 + x 9 ) x 2 3 x + 18 Distribute the sign in the numerator. 5 x 2 7 x + 3 4 x 2 x + 9 x 2 3 x 18 Combine like terms. x 2 8 x + 12 x 2 3 x 18 Factor the numerator and the denominator. ( x 2 ) ( x 6 ) ( x + 3 ) ( x 6 ) Simplify by removing common factors. ( x 2 ) ( x 6 ) ( x + 3 ) ( x 6 ) Simplify. ( x 2 ) ( x + 3 )

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Subtract: 4 x 2 11 x + 8 x 2 3 x + 2 3 x 2 + x 3 x 2 3 x + 2 .

x 11 x 2

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Subtract: 6 x 2 x + 20 x 2 81 5 x 2 + 11 x 7 x 2 81 .

x 3 x + 9

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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 .

Let’s see how this works.

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

Add: 4 u 1 3 u 1 + u 1 3 u .

Solution

.
The denominators are opposites, so multiply the second fraction by −1 −1 . .
Simplify the second fraction. .
The denominators are the same. Add the numerators. .
Simplify. .
Simplify. .

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Add: 8 x 15 2 x 5 + 2 x 5 2 x .

3

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Add: 6 y 2 + 7 y 10 4 y 7 + 2 y 2 + 2 y + 11 7 4 y .

y + 3

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Subtract: m 2 6 m m 2 1 3 m + 2 1 m 2 .

Solution

.
The denominators are opposites, so multiply the second fraction by −1 −1 . .
Simplify the second fraction. .
The denominators are the same. Subtract the numerators. .
Distribute. m 2 6 m + 3 m + 2 m 2 1
Combine like terms. .
Factor the numerator and denominator. .
Simplify by removing common factors. .
Simplify. .

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Subtract: y 2 5 y y 2 4 6 y 6 4 y 2 .

y + 3 y + 2

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Subtract: 2 n 2 + 8 n 1 n 2 1 n 2 7 n 1 1 n 2 .

3 n 2 n 1

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Key concepts

  • Rational Expression Addition
    • If p , q , and r are polynomials where r 0 , then
      p r + q r = p + q r
    • To add rational expressions with a common denominator, add the numerators and place the sum over the common denominator.
  • Rational Expression Subtraction
    • If p , q , and r are polynomials where r 0 , then
      p r q r = p q r
    • To subtract rational expressions, subtract the numerators and place the difference over the common denominator.

Practice makes perfect

Add Rational Expressions with a Common Denominator

In the following exercises, add.

3 a a b + 1 a b

3 a + 1 a + b

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3 c 4 c 5 + 5 4 c 5

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d d + 8 + 5 d + 8

d + 5 d + 8

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p 2 + 10 p p + 2 + 16 p + 2

p + 8

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q 2 + 12 q q + 3 + 27 q + 3

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2 r 2 2 r 1 + 15 r 8 2 r 1

r + 8

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3 s 2 3 s 2 + 13 s 10 3 s 2

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8 t 2 t + 4 + 32 t t + 4

8 t

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6 v 2 v + 5 + 30 v v + 5

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2 w 2 w 2 16 + 8 w w 2 16

2 w w 4

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7 x 2 x 2 9 + 21 x x 2 9

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Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

y 2 y + 8 64 y + 8

y 8

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9 a 2 3 a 7 49 3 a 7

3 a + 7

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25 b 2 5 b 6 36 5 b 6

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c 2 c 8 6 c + 16 c 8

c + 2

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d 2 d 9 6 d + 27 d 9

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3 m 2 6 m 30 21 m 30 6 m 30

m 2 3

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2 n 2 4 n 32 30 n 16 4 n 32

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6 p 2 + 3 p + 4 p 2 + 4 p 5 5 p 2 + p + 7 p 2 + 4 p 5

p + 3 p + 5

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5 q 2 + 3 q 9 q 2 + 6 q + 8 4 q 2 + 9 q + 7 q 2 + 6 q + 8

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5 r 2 + 7 r 33 r 2 49 4 r 2 5 r 30 r 2 49

r + 9 r + 7

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7 t 2 t 4 t 2 25 6 t 2 + 2 t 1 t 2 25

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Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add.

10 v 2 v 1 + 2 v + 4 1 2 v

4

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20 w 5 w 2 + 5 w + 6 2 5 w

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10 x 2 + 16 x 7 8 x 3 + 2 x 2 + 3 x 1 3 8 x

x + 2

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6 y 2 + 2 y 11 3 y 7 + 3 y 2 3 y + 17 7 3 y

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In the following exercises, subtract.

z 2 + 6 z z 2 25 3 z + 20 25 z 2

z + 4 z 5

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a 2 + 3 a a 2 9 3 a 27 9 a 2

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2 b 2 + 30 b 13 b 2 49 2 b 2 5 b 8 49 b 2

4 b 3 b 7

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c 2 + 5 c 10 c 2 16 c 2 8 c 10 16 c 2

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Everyday math

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

32 r + 32 r ( r + 4 )

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

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Writing exercises

Donald thinks that 3 x + 4 x is 7 2 x . Is Donald correct? Explain.

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Explain how you find the Least Common Denominator of x 2 + 5 x + 4 and x 2 16 .

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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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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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