Add and Subtract Rational Expressions with a Common Denominator

8.3 Add and Subtract Rational Expressions with a Common Denominator

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: $\frac{y}{3} + \frac{9}{3} .$
If you missed this problem, review Example 1.77.

Be Prepared 8.10

Subtract: $\frac{10}{x} - \frac{2}{x} .$
If you missed this problem, review Example 1.79.

Be Prepared 8.11

Factor completely: $8 n^{5} - 20 n^{3} .$
If you missed this problem, review Example 7.59.

Be Prepared 8.12

Factor completely: $45 a^{3} - 5 a b^{2} .$
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 $p, q, and r$ are polynomials where $r \neq 0$ , then

\frac{p}{r} + \frac{q}{r} = \frac{p + q}{r}

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: $\frac{5}{18} + \frac{7}{18} .$

Solution

	$\frac{5}{18} + \frac{7}{18}$
The fractions have a common denominator, so add the numerators and place the sum over the common denominator.	$\frac{5 + 7}{18}$
Add in the numerator.	$\frac{12}{18}$
Factor the numerator and denominator to show the common factors.	$\frac{6 \cdot 2}{6 \cdot 3}$
Remove common factors.	$\frac{6 \cdot 2}{6 \cdot 3}$
Simplify.	$\frac{2}{3}$

Step-by-step process for adding two fractions with a common denominator and simplifying the result.

Try It 8.59

Add: $\frac{7}{16} + \frac{5}{16} .$

Try It 8.60

Add: $\frac{3}{10} + \frac{1}{10} .$

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

Example 8.31

Add: $\frac{3 y}{4 y - 3} + \frac{7}{4 y - 3} .$

Solution

	$\frac{3 y}{4 y - 3} + \frac{7}{4 y - 3}$
The fractions have a common denominator, so add the numerators and place the sum over the common denominator.	$\frac{3 y + 7}{4 y - 3}$

Demonstrates adding rational expressions with a common denominator by combining numerators over the shared denominator.

The numerator and denominator cannot be factored. The fraction is simplified.

Try It 8.61

Add: $\frac{5 x}{2 x + 3} + \frac{2}{2 x + 3} .$

Try It 8.62

Add: $\frac{x}{x - 2} + \frac{1}{x - 2} .$

Example 8.32

Add: $\frac{7 x + 12}{x + 3} + \frac{x^{2}}{x + 3} .$

Solution

	$\frac{7 x + 12}{x + 3} + \frac{x^{2}}{x + 3}$
The fractions have a common denominator, so add the numerators and place the sum over the common denominator.	$\frac{7 x + 12 + x^{2}}{x + 3}$
Write the degrees in descending order.	$\frac{x^{2} + 7 x + 12}{x + 3}$
Factor the numerator.	$\frac{(x + 3) (x + 4)}{x + 3}$
Simplify by removing common factors.	$\frac{(x + 3) (x + 4)}{x + 3}$
Simplify.	$x + 4$

Step-by-step solution for adding and simplifying rational expressions.

Try It 8.63

Add: $\frac{9 x + 14}{x + 7} + \frac{x^{2}}{x + 7} .$

Try It 8.64

Add: $\frac{x^{2} + 8 x}{x + 5} + \frac{15}{x + 5} .$

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 $p, q, and r$ are polynomials where $r \neq 0$ , then

\frac{p}{r} - \frac{q}{r} = \frac{p - q}{r}

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: $\frac{n^{2}}{n - 10} - \frac{100}{n - 10} .$

Solution

	$\frac{n^{2}}{n - 10} - \frac{100}{n - 10}$
The fractions have a common denominator, so subtract the numerators and place the difference over the common denominator.	$\frac{n^{2} - 100}{n - 10}$
Factor the numerator.	$\frac{(n - 10) (n + 10)}{n - 10}$
Simplify by removing common factors.	$\frac{(n - 10) (n + 10)}{n - 10}$
Simplify.	$n + 10$

Step-by-step simplification of a rational algebraic expression using common denominators and factoring.

Try It 8.65

Subtract: $\frac{x^{2}}{x + 3} - \frac{9}{x + 3} .$

Try It 8.66

Subtract: $\frac{4 x^{2}}{2 x - 5} - \frac{25}{2 x - 5} .$

Be careful of the signs when you subtract a binomial!

Example 8.34

Subtract: $\frac{y^{2}}{y - 6} - \frac{2 y + 24}{y - 6} .$

Solution

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

Step-by-step simplification of a rational algebraic expression involving subtraction and factorization.

Try It 8.67

Subtract: $\frac{n^{2}}{n - 4} - \frac{n + 12}{n - 4} .$

Try It 8.68

Subtract: $\frac{y^{2}}{y - 1} - \frac{9 y - 8}{y - 1} .$

Example 8.35

Subtract: $\frac{5 x^{2} - 7 x + 3}{x^{2} - 3 x - 18} - \frac{4 x^{2} + x - 9}{x^{2} - 3 x - 18} .$

Solution

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

This table illustrates the step-by-step process of subtracting two rational expressions, showing each mathematical transformation from initial problem to simplified solution.

Try It 8.69

Subtract: $\frac{4 x^{2} - 11 x + 8}{x^{2} - 3 x + 2} - \frac{3 x^{2} + x - 3}{x^{2} - 3 x + 2} .$

Try It 8.70

Subtract: $\frac{6 x^{2} - x + 20}{x^{2} - 81} - \frac{5 x^{2} + 11 x - 7}{x^{2} - 81} .$

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 $\frac{−1}{−1}$ .

Let’s see how this works.

	$A mathematical expression showing the sum of two fractions: 7 over d plus 5 over negative d.$
Multiply the second fraction by $\frac{−1}{−1}$ .	$An algebraic expression showing the sum of two fractions. The first term is 7/d, and the second term is a fraction with a numerator of (-1) to the power of 5 and a denominator of (-1)(-d).$
The denominators are the same.	$The image shows the mathematical expression 7/d + -5/d, representing the addition of two fractions with a common denominator 'd'.$
Simplify.	$A mathematical expression displaying the fraction 2 over d.$

Example 8.36

Add: $\frac{4 u - 1}{3 u - 1} + \frac{u}{1 - 3 u} .$

Solution

	$Algebraic expression: (4u-1)/(3u-1) + u/(1-3u). It's a sum of two fractions with denominators that are additive inverses of each other.$
The denominators are opposites, so multiply the second fraction by $\frac{−1}{−1}$ .	$A mathematical expression representing the sum of two algebraic fractions. The first term is (4u-1)/(3u-1), and the second is ((-1)u)/((-1)(1-3u)), illustrating a step in simplifying the expression.$
Simplify the second fraction.	$The image displays the sum of two algebraic fractions with a common denominator, written as (4u-1)/(3u-1) + (-u)/(3u-1).$
The denominators are the same. Add the numerators.	$A mathematical fraction displays '4u - 1 - u' as the numerator and '3u - 1' as the denominator, which simplifies to (3u - 1) / (3u - 1).$
Simplify.	$The mathematical expression showing the fraction (3u-1) divided by (3u-1). This expression simplifies to 1, assuming that (3u-1) is not equal to zero.$
Simplify.

Try It 8.71

Add: $\frac{8 x - 15}{2 x - 5} + \frac{2 x}{5 - 2 x} .$

Try It 8.72

Add: $\frac{6 y^{2} + 7 y - 10}{4 y - 7} + \frac{2 y^{2} + 2 y + 11}{7 - 4 y} .$

Example 8.37

Subtract: $\frac{m^{2} - 6 m}{m^{2} - 1} - \frac{3 m + 2}{1 - m^{2}} .$

Solution

	$A mathematical expression showing the subtraction of two algebraic fractions: (m^2 - 6m) / (m^2 - 1) - (3m + 2) / (1 - m^2).$
The denominators are opposites, so multiply the second fraction by $\frac{−1}{−1}$ .	$A mathematical expression displaying the subtraction of two algebraic fractions. The first fraction is (m^2 - 6m)/(m^2 - 1), and the second is -1(3m + 2)/-1(1 - m^2).$
Simplify the second fraction.	$A mathematical expression showing the subtraction of two algebraic fractions with a common denominator of m^2 - 1. The first fraction is (m^2 - 6m) / (m^2 - 1) and the second is (-3m - 2) / (m^2 - 1).$
The denominators are the same. Subtract the numerators.	$A mathematical expression showing the fraction (m^2 - 6m - (-3m - 2)) / (m^2 - 1).$
Distribute.	$\frac{m^{2} - 6 m + 3 m + 2}{m^{2} - 1}$
Combine like terms.	$A mathematical expression showing a fraction with the numerator m squared minus 3m plus 2, and the denominator m squared minus 1.$
Factor the numerator and denominator.	$A mathematical fraction displaying (m-1)(m-2) in the numerator and (m-1)(m+1) in the denominator, illustrating an algebraic expression suitable for simplification.$
Simplify by removing common factors.	$A fraction showing the expression (m-1)(m-2) divided by (m-1)(m+1), with the (m-1) terms crossed out in both the numerator and denominator, implying simplification to (m-2)/(m+1).$
Simplify.	$A mathematical fraction displays 'm minus 2' in the numerator and 'm plus 1' in the denominator, set against a plain white background.$

Try It 8.73

Subtract: $\frac{y^{2} - 5 y}{y^{2} - 4} - \frac{6 y - 6}{4 - y^{2}} .$

Try It 8.74

Subtract: $\frac{2 n^{2} + 8 n - 1}{n^{2} - 1} - \frac{n^{2} - 7 n - 1}{1 - n^{2}} .$

Section 8.3 Exercises

Practice Makes Perfect

Add Rational Expressions with a Common Denominator

In the following exercises, add.

129.

$\frac{2}{15} + \frac{7}{15}$

130.

$\frac{4}{21} + \frac{3}{21}$

131.

$\frac{7}{24} + \frac{11}{24}$

132.

$\frac{7}{36} + \frac{13}{36}$

133.

$\frac{3 a}{a - b} + \frac{1}{a - b}$

134.

$\frac{3 c}{4 c - 5} + \frac{5}{4 c - 5}$

135.

$\frac{d}{d + 8} + \frac{5}{d + 8}$

136.

$\frac{7 m}{2 m + n} + \frac{4}{2 m + n}$

137.

$\frac{p^{2} + 10 p}{p + 2} + \frac{16}{p + 2}$

138.

$\frac{q^{2} + 12 q}{q + 3} + \frac{27}{q + 3}$

139.

$\frac{2 r^{2}}{2 r - 1} + \frac{15 r - 8}{2 r - 1}$

140.

$\frac{3 s^{2}}{3 s - 2} + \frac{13 s - 10}{3 s - 2}$

141.

$\frac{8 t^{2}}{t + 4} + \frac{32 t}{t + 4}$

142.

$\frac{6 v^{2}}{v + 5} + \frac{30 v}{v + 5}$

143.

$\frac{2 w^{2}}{w^{2} - 16} + \frac{8 w}{w^{2} - 16}$

144.

$\frac{7 x^{2}}{x^{2} - 9} + \frac{21 x}{x^{2} - 9}$

Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

145.

$\frac{y^{2}}{y + 8} - \frac{64}{y + 8}$

146.

$\frac{z^{2}}{z + 2} - \frac{4}{z + 2}$

147.

$\frac{9 a^{2}}{3 a - 7} - \frac{49}{3 a - 7}$

148.

$\frac{25 b^{2}}{5 b - 6} - \frac{36}{5 b - 6}$

149.

$\frac{c^{2}}{c - 8} - \frac{6 c + 16}{c - 8}$

150.

$\frac{d^{2}}{d - 9} - \frac{6 d + 27}{d - 9}$

151.

$\frac{3 m^{2}}{6 m - 30} - \frac{21 m - 30}{6 m - 30}$

152.

$\frac{2 n^{2}}{4 n - 32} - \frac{18 n - 16}{4 n - 32}$

153.

$\frac{6 p^{2} + 3 p + 4}{p^{2} + 4 p - 5} - \frac{5 p^{2} + p + 7}{p^{2} + 4 p - 5}$

154.

$\frac{5 q^{2} + 3 q - 9}{q^{2} + 6 q + 8} - \frac{4 q^{2} + 9 q + 7}{q^{2} + 6 q + 8}$

155.

$\frac{5 r^{2} + 7 r - 33}{r^{2} - 49} - \frac{4 r^{2} + 5 r + 30}{r^{2} - 49}$

156.

$\frac{7 t^{2} - t - 4}{t^{2} - 25} - \frac{6 t^{2} + 12 t - 44}{t^{2} - 25}$

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add.

157.

$\frac{10 v}{2 v - 1} + \frac{2 v + 4}{1 - 2 v}$

158.

$\frac{20 w}{5 w - 2} + \frac{5 w + 6}{2 - 5 w}$

159.

$\frac{10 x^{2} + 16 x - 7}{8 x - 3} + \frac{2 x^{2} + 3 x - 1}{3 - 8 x}$

160.

$\frac{6 y^{2} + 2 y - 11}{3 y - 7} + \frac{3 y^{2} - 3 y + 17}{7 - 3 y}$

In the following exercises, subtract.

161.

$\frac{z^{2} + 6 z}{z^{2} - 25} - \frac{3 z + 20}{25 - z^{2}}$

162.

$\frac{a^{2} + 3 a}{a^{2} - 9} - \frac{3 a - 27}{9 - a^{2}}$

163.

$\frac{2 b^{2} + 30 b - 13}{b^{2} - 49} - \frac{2 b^{2} - 5 b - 8}{49 - b^{2}}$

164.

$\frac{c^{2} + 5 c - 10}{c^{2} - 16} - \frac{c^{2} - 8 c - 10}{16 - c^{2}}$

Everyday Math

165.

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 $\frac{8}{r}$ and her biking time is modeled by the expression $\frac{24}{r + 4} .$ Add the rational expressions $\frac{8}{r} + \frac{24}{r + 4}$ to get an expression for the total amount of time Sarah ran and biked.

166.

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

Writing Exercises

167.

Donald thinks that $\frac{3}{x} + \frac{4}{x}$ is $\frac{7}{2 x} .$ Is Donald correct? Explain.

168.

Explain how you find the Least Common Denominator of $x^{2} + 5 x + 4$ and $x^{2} - 16 .$

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?