# 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?

One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?
rectangular field solutions
What is this?
Donna
the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is
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Choli
a rock is thrown directly upward with an initial velocity of 96feet per second from a cliff 190 feet above a beach. The hight of tha rock above the beach after t second is given by the equation h=_16t^2+96t+190
Usman
Stella bought a dinette set on sale for $725. The original price was$1,299. To the nearest tenth of a percent, what was the rate of discount?
44.19%
Scott
40.22%
Terence
44.2%
Orlando
I don't know
Donna
if you want the discounted price subtract $725 from$1299. then divide the answer by $1299. you get 0.4419... but as percent you get 44.19... but to the nearest tenth... round .19 to .2 and you get 44.2% Orlando you could also just divide$725/$1299 and then subtract it from 1. then you get the same answer. Orlando p mulripied-5 and add 30 to it Tausif Reply p mulripied-5 and add30 Tausif p mulripied-5 and addto30 Tausif Can you explain further Monica Reply p mulripied-5 and add to 30 Tausif How do you find divisible numbers without a calculator? Jacob Reply TAKE OFF THE LAST DIGIT AND MULTIPLY IT 9. SUBTRACT IT THE DIGITS YOU HAVE LEFT. IF THE ANSWER DIVIDES BY 13(OR IS ZERO), THEN YOUR ORIGINAL NUMBER WILL ALSO DIVIDE BY 13!IS DIVISIBLE BY 13 BAINAMA When she graduates college, Linda will owe$43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was $1,585. What is the amount of each loan? Ariana Reply Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 7 2/3 hours, what was the speed of the bus? Kirisma Reply 66miles/hour snigdha How did you work it out? Esther s=mi/hr 2/3~0.67 s=506mi/7.67hr = ~66 mi/hr Orlando hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused. Alicia Reply what do you need help in? Felix subtracting a negative....is adding!! Heather look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number... Felix for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer... Felix —12 Niazmohammad Thanks Felix.l also get confused with signs. Esther Thank you for this Shatey ty Graham think about it like you lost$19 (-19), then found $7(+7). Totally you lost just$12 (-12)
Annushka
I used to struggle a lot with negative numbers and math in general what I typically do is look at it in terms of money I have -$5 in my account I then take out 5 more dollars how much do I have in my account well-$10 ... I also for a long time would draw it out on a number line to visualize it
Meg
practicing with smaller numbers to understand then working with larger numbers helps too and the song/rhyme same sign add and keep opposite signs subtract keep the sign of the bigger # then you'll be exact
Meg
Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce
I don't see where the answers are.
Ed
Cindy and Richard leave their dorm in Charleston at the same time. Cindy rides her bicycle north at a speed of 18 miles per hour. Richard rides his bicycle south at a speed of 14 miles per hour. How long will it take them to be 96 miles apart?
3
Christopher
18t+14t=96 32t=96 32/96 3
Christopher
show that a^n-b^2n is divisible by a-b By Jonathan Long By Stephen Voron By Tod McGrath By Stephen Voron By Kevin Amaratunga By Janet Forrester By Jesenia Wofford By Jessica Collett By OpenStax By Brooke Delaney