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

 Page 2 / 2

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

• Rational Expression Addition
• 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

In the following exercises, add.

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

$\frac{3}{5}$

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

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

$\frac{3}{4}$

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

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

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

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

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

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

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

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

$p+8$

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

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

$r+8$

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

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

$8t$

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

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

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

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

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

In the following exercises, add.

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

$4$

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

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

$x+2$

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

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?

#### Questions & Answers

The hypotenuse of a right triangle is 10cm long. One of the triangle’s legs is three times the length of the other leg. Find the lengths of the three sides of the triangle.
Edi Reply
Tickets for a show are $70 for adults and$50 for children. For one evening performance, a total of 300 tickets were sold and the receipts totaled $17,200. How many adult tickets and how many child tickets were sold? Mum Reply A 50% antifreeze solution is to be mixed with a 90% antifreeze solution to get 200 liters of a 80% solution. How many liters of the 50% solution and how many liters of the 90% solution will be used? Edi Reply June needs 45 gallons of punch for a party and has 2 different coolers to carry it in. The bigger cooler is 5 times as large as the smaller cooler. How many gallons can each cooler hold? Jesus Reply Washing his dad’s car alone, eight-year-old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself? Ronald Reply 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. Dojzae Reply LeBron needs 150 milliliters of a 30% solution of sulfuric acid for a lab experiment but only has access to a 25% and a 50% solution. How much of the 25% and how much of the 50% solution should he mix to make the 30% solution? Xona Reply 5% Michael hey everyone how to do algebra The Reply Felecia answer 1.5 hours before he reaches her Adriana Reply I would like to solve the problem -6/2x rachel Reply 12x Andrew how Christian Does the x represent a number or does it need to be graphed ? latonya -3/x Venugopal -3x is correct Atul Arnold invested$64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received $4,500 in interest in one year? Stephanie Reply Tickets for the community fair cost$12 for adults and $5 for children. On the first day of the fair, 312 tickets were sold for a total of$2204. How many adult tickets and how many child tickets were sold?
Alpha Reply
220
gayla
Three-fourths of the people at a concert are children. If there are 87 children, what is the total number of people at the concert?
Tsimmuaj Reply
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was$1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was$1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
? Is there anything wrong with this passage I found the total sum for 2 jobs, but found why elaborate on extra If I total one week from the store *4 would = the month than the total is = x than x can't calculate 10 month of a year
candido
what would be wong
candido
87 divided by 3 then multiply that by 4. 116 people total.
Melissa
the actual number that has 3 out of 4 of a whole pie
candido
was having a hard time finding
Teddy
use Matrices for the 2nd question
Daniel
One number is 11 less than the other number. If their sum is increased by 8, the result is 71. Find the numbers.
Tsimmuaj Reply
26 + 37 = 63 + 8 = 71
gayla
26+37=63+8=71
ziad
11+52=63+8=71
Thisha
how do we know the answer is correct?
Thisha
23 is 11 less than 37. 23+37=63. 63+8=71. that is what the question asked for.
gayla
23 +11 = 37. 23+37=63 63+8=71
Gayla
by following the question. one number is 11 less than the other number 26+11=37 so 26+37=63+8=71
Gayla
your answer did not fit the guidelines of the question 11 is 41 less than 52.
gayla
71-8-11 =52 is this correct?
Ruel
let the number is 'x' and the other number is "x-11". if their sum is increased means: x+(x-11)+8 result will be 71. so x+(x-11)+8=71 2x-11+8=71 2x-3=71 2x=71+3 2x=74 1/2(2x=74)1/2 x=37 final answer
tesfu
just new
Muwanga
Amara currently sells televisions for company A at a salary of $17,000 plus a$100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a$20 commission for each television she sells. How televisions would Amara need to sell for the options to be equal?
Tsimmuaj Reply
yes math
Kenneth
company A 13 company b 5. A 17,000+13×100=29,100 B 29,000+5×20=29,100
gayla
need help with math to do tsi test
Toocute
me too
Christian
have you tried the TSI practice test ***tsipracticetest.com
gayla

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