# 8.3 Add and subtract rational expressions with a common denominator  (Page 2/2)

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

$x-3$

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

$2x+5$

Be careful of the signs when you subtract a binomial!

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

## Solution

$\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{{y}^{2}}{y-6}-\frac{2y+24}{y-6}\hfill \\ \\ \\ \begin{array}{c}\text{The fractions have a common}\hfill \\ \text{denominator, so subtract the numerators}\hfill \\ \text{and place the difference over the}\hfill \\ \text{common denominator.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{{y}^{2}-\left(2y+24\right)}{y-6}\hfill \\ \\ \\ \text{Distribute the sign in the numerator.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{{y}^{2}-2y-24}{y-6}\hfill \\ \\ \\ \text{Factor the numerator.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left(y-6\right)\left(y+4\right)}{y-6}\hfill \\ \\ \\ \text{Remove common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\overline{)\left(y-6\right)}\left(y+4\right)}{\overline{)y-6}}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}y+4\hfill \end{array}$

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

$n+3$

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

$y-8$

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

## Solution

$\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{5{x}^{2}-7x+3}{{x}^{2}-3x+18}-\frac{4{x}^{2}+x-9}{{x}^{2}-3x+18}\hfill \\ \\ \\ \begin{array}{c}\text{Subtract the numerators and place the}\hfill \\ \text{difference over the common}\hfill \\ \text{denominator.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{5{x}^{2}-7x+3-\left(4{x}^{2}+x-9\right)}{{x}^{2}-3x+18}\hfill \\ \\ \\ \text{Distribute the sign in the numerator.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{5{x}^{2}-7x+3-4{x}^{2}-x+9}{{x}^{2}-3x-18}\hfill \\ \\ \\ \text{Combine like terms.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{{x}^{2}-8x+12}{{x}^{2}-3x-18}\hfill \\ \\ \\ \begin{array}{c}\text{Factor the numerator and the}\hfill \\ \text{denominator.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left(x-2\right)\left(x-6\right)}{\left(x+3\right)\left(x-6\right)}\hfill \\ \\ \\ \text{Simplify by removing common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left(x-2\right)\overline{)\left(x-6\right)}}{\left(x+3\right)\overline{)\left(x-6\right)}}\hfill \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left(x-2\right)}{\left(x+3\right)}\hfill \end{array}$

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

$\frac{x-11}{x-2}$

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

$\frac{x-3}{x+9}$

## 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.

 Multiply the second fraction by $\frac{-1}{-1}$ . The denominators are the same. Simplify.

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

## Solution

 The denominators are opposites, so multiply the second fraction by $\frac{-1}{-1}$ . Simplify the second fraction. The denominators are the same. Add the numerators. Simplify. Simplify.

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

$3$

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

$y+3$

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

## Solution

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

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

$\frac{y+3}{y+2}$

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

$\frac{3n-2}{n-1}$

## Key concepts

• If $p,q,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}r$ are polynomials where $r\ne 0$ , 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.
• Rational Expression Subtraction
• If $p,q,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}r$ are polynomials where $r\ne 0$ , 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.

## Practice makes perfect

Add Rational Expressions with a Common Denominator

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

$\frac{3}{5}$

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

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

$\frac{3}{4}$

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

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

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

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

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

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

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

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

$p+8$

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

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

$r+8$

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

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

$8t$

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

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

$\frac{2w}{w-4}$

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

Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

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

$y-8$

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

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

$3a+7$

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

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

$c+2$

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

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

$\frac{m-2}{3}$

$\frac{2{n}^{2}}{4n-32}-\frac{30n-16}{4n-32}$

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

$\frac{p+3}{p+5}$

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

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

$\frac{r+9}{r+7}$

$\frac{7{t}^{2}-t-4}{{t}^{2}-25}-\frac{6{t}^{2}+2t-1}{{t}^{2}-25}$

Add and Subtract Rational Expressions whose Denominators are Opposites

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

$4$

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

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

$x+2$

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

In the following exercises, subtract.

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

$\frac{z+4}{z-5}$

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

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

$\frac{4b-3}{b-7}$

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

## 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 $\frac{8}{r}$ 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.

$\frac{32r+32}{r\left(r+4\right)}$

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

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

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

## Self check

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

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

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