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Be careful of the signs when you subtract a binomial!
Subtract: $\frac{{y}^{2}}{y-6}-\frac{2y+24}{y-6}.$
$\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{{y}^{2}}{y-6}-\frac{2y+24}{y-6}\hfill \\ \\ \\ \begin{array}{c}\text{The fractions have a common}\hfill \\ \text{denominator, so subtract the numerators}\hfill \\ \text{and place the difference over the}\hfill \\ \text{common denominator.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{{y}^{2}-\left(2y+24\right)}{y-6}\hfill \\ \\ \\ \text{Distribute the sign in the numerator.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{{y}^{2}-2y-24}{y-6}\hfill \\ \\ \\ \text{Factor the numerator.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left(y-6\right)\left(y+4\right)}{y-6}\hfill \\ \\ \\ \text{Remove common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\overline{)\left(y-6\right)}\left(y+4\right)}{\overline{)y-6}}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}y+4\hfill \end{array}$
Subtract: $\frac{5{x}^{2}-7x+3}{{x}^{2}-3x+18}-\frac{4{x}^{2}+x-9}{{x}^{2}-3x+18}.$
$\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{5{x}^{2}-7x+3}{{x}^{2}-3x+18}-\frac{4{x}^{2}+x-9}{{x}^{2}-3x+18}\hfill \\ \\ \\ \begin{array}{c}\text{Subtract the numerators and place the}\hfill \\ \text{difference over the common}\hfill \\ \text{denominator.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{5{x}^{2}-7x+3-\left(4{x}^{2}+x-9\right)}{{x}^{2}-3x+18}\hfill \\ \\ \\ \text{Distribute the sign in the numerator.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{5{x}^{2}-7x+3-4{x}^{2}-x+9}{{x}^{2}-3x-18}\hfill \\ \\ \\ \text{Combine like terms.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{{x}^{2}-8x+12}{{x}^{2}-3x-18}\hfill \\ \\ \\ \begin{array}{c}\text{Factor the numerator and the}\hfill \\ \text{denominator.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left(x-2\right)\left(x-6\right)}{\left(x+3\right)\left(x-6\right)}\hfill \\ \\ \\ \text{Simplify by removing common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left(x-2\right)\overline{)\left(x-6\right)}}{\left(x+3\right)\overline{)\left(x-6\right)}}\hfill \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left(x-2\right)}{\left(x+3\right)}\hfill \end{array}$
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: $\frac{6{x}^{2}-x+20}{{x}^{2}-81}-\frac{5{x}^{2}+11x-7}{{x}^{2}-81}.$
$\frac{x-3}{x+9}$
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 $\frac{\mathrm{-1}}{\mathrm{-1}}$ .
Let’s see how this works.
Multiply the second fraction by $\frac{\mathrm{-1}}{\mathrm{-1}}$ . | |
The denominators are the same. | |
Simplify. |
Add: $\frac{4u-1}{3u-1}+\frac{u}{1-3u}.$
The denominators are opposites, so multiply the second fraction by $\frac{\mathrm{-1}}{\mathrm{-1}}$ . | |
Simplify the second fraction. | |
The denominators are the same. Add the numerators. | |
Simplify. | |
Simplify. |
Add: $\frac{6{y}^{2}+7y-10}{4y-7}+\frac{2{y}^{2}+2y+11}{7-4y}.$
$y+3$
Subtract: $\frac{{m}^{2}-6m}{{m}^{2}-1}-\frac{3m+2}{1-{m}^{2}}.$
The denominators are opposites, so multiply the second fraction by $\frac{\mathrm{-1}}{\mathrm{-1}}$ . | |
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: $\frac{2{n}^{2}+8n-1}{{n}^{2}-1}-\frac{{n}^{2}-7n-1}{1-{n}^{2}}.$
$\frac{3n-2}{n-1}$
Add Rational Expressions with a Common Denominator
In the following exercises, add.
$\frac{4}{21}+\frac{3}{21}$
$\frac{7}{36}+\frac{13}{36}$
$\frac{3c}{4c-5}+\frac{5}{4c-5}$
$\frac{7m}{2m+n}+\frac{4}{2m+n}$
$\frac{{q}^{2}+12q}{q+3}+\frac{27}{q+3}$
$\frac{3{s}^{2}}{3s-2}+\frac{13s-10}{3s-2}$
$\frac{6{v}^{2}}{v+5}+\frac{30v}{v+5}$
$\frac{2{w}^{2}}{{w}^{2}-16}+\frac{8w}{{w}^{2}-16}$
$\frac{2w}{w-4}$
$\frac{7{x}^{2}}{{x}^{2}-9}+\frac{21x}{{x}^{2}-9}$
Subtract Rational Expressions with a Common Denominator
In the following exercises, subtract.
$\frac{{z}^{2}}{z+2}-\frac{4}{z+2}$
$\frac{25{b}^{2}}{5b-6}-\frac{36}{5b-6}$
$\frac{{d}^{2}}{d-9}-\frac{6d+27}{d-9}$
$\frac{3{m}^{2}}{6m-30}-\frac{21m-30}{6m-30}$
$\frac{m-2}{3}$
$\frac{2{n}^{2}}{4n-32}-\frac{30n-16}{4n-32}$
$\frac{6{p}^{2}+3p+4}{{p}^{2}+4p-5}-\frac{5{p}^{2}+p+7}{{p}^{2}+4p-5}$
$\frac{p+3}{p+5}$
$\frac{5{q}^{2}+3q-9}{{q}^{2}+6q+8}-\frac{4{q}^{2}+9q+7}{{q}^{2}+6q+8}$
$\frac{5{r}^{2}+7r-33}{{r}^{2}-49}-\frac{4{r}^{2}-5r-30}{{r}^{2}-49}$
$\frac{r+9}{r+7}$
$\frac{7{t}^{2}-t-4}{{t}^{2}-25}-\frac{6{t}^{2}+2t-1}{{t}^{2}-25}$
Add and Subtract Rational Expressions whose Denominators are Opposites
In the following exercises, add.
$\frac{20w}{5w-2}+\frac{5w+6}{2-5w}$
$\frac{10{x}^{2}+16x-7}{8x-3}+\frac{2{x}^{2}+3x-1}{3-8x}$
$x+2$
$\frac{6{y}^{2}+2y-11}{3y-7}+\frac{3{y}^{2}-3y+17}{7-3y}$
In the following exercises, subtract.
$\frac{{z}^{2}+6z}{{z}^{2}-25}-\frac{3z+20}{25-{z}^{2}}$
$\frac{z+4}{z-5}$
$\frac{{a}^{2}+3a}{{a}^{2}-9}-\frac{3a-27}{9-{a}^{2}}$
$\frac{2{b}^{2}+30b-13}{{b}^{2}-49}-\frac{2{b}^{2}-5b-8}{49-{b}^{2}}$
$\frac{4b-3}{b-7}$
$\frac{{c}^{2}+5c-10}{{c}^{2}-16}-\frac{{c}^{2}-8c-10}{16-{c}^{2}}$
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.
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 ${x}^{2}-16.$
ⓐ 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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