13.9.3E: Exersices

Practice Makes Perfect

Add Rational Expressions with a Common Denominator

In the following exercises, add.

Example $\PageIndex{25}$

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

Answer

$\frac{3}{5}$

Example $\PageIndex{26}$

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

Example $\PageIndex{27}$

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

Answer

$\frac{3}{4}$

Example $\PageIndex{28}$

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

Example $\PageIndex{29}$

$\frac{3a}{a−b}+\frac{1}{a−b}$

Answer

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

Example $\PageIndex{30}$

$\frac{3c}{4c−5}+\frac{5}{4c−5}$

Example $\PageIndex{31}$

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

Answer

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

Example $\PageIndex{32}$

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

Example $\PageIndex{33}$

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

Answer

p+8

Example $\PageIndex{34}$

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

Example $\PageIndex{35}$

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

Answer

r+8

Example $\PageIndex{36}$

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

Example $\PageIndex{37}$

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

Answer

8t

Example $\PageIndex{38}$

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

Example $\PageIndex{39}$

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

Answer

$\frac{2w}{w−4}$

Example $\PageIndex{40}$

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

​​​​​​​Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

Example $\PageIndex{41}$

$\frac{y^2}{y+8}−\frac{64}{y+8}$

Answer

y−8​​​​​​​

Example $\PageIndex{42}$

$\frac{z^2}{z+2}−\frac{4}{z+2}$​​​​​​​

Example $\PageIndex{43}$

$\frac{9a^2}{3a−7}−\frac{49}{3a−7}$

Answer

3a+7

Example $\PageIndex{44}$

$\frac{25b^2}{5b−6}−\frac{36}{5b−6}$

Example $\PageIndex{45}$

$\frac{c^2}{c−8}−\frac{6c+16}{c−8}$

Answer

c+2

Example $\PageIndex{46}$

$\frac{d^2}{d−9}−\frac{6d+27}{d−9}$

Example $\PageIndex{47}$

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

Answer

$\frac{m−2}{3}$

Example $\PageIndex{48}$

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

Example $\PageIndex{49}$

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

Answer

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

Example $\PageIndex{50}$

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

Example $\PageIndex{51}$

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

Answer

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

Example $\PageIndex{52}$

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

​​​​​​​Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add.

Example $\PageIndex{53}$

$\frac{10v^2}{v−1}+\frac{2v+4}{1−2v}$

Answer

4​​​​​​​

Example $\PageIndex{54}$

$\frac{20w}{5w−2}+\frac{5w+6}{2−5w}$

Example $\PageIndex{55}$

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

Answer

x+2

Example $\PageIndex{56}$

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

​​​​​​​In the following exercises, subtract.

Example $\PageIndex{57}$

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

Answer

$\frac{z+4}{z−5}$

Example $\PageIndex{58}$

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

Example $\PageIndex{59}$

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

Answer

$\frac{4b−3}{b−7}$

Example $\PageIndex{60}$

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

Everyday Math

Example $\PageIndex{61}$

Sarah ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. If rr 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.

Answer

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

Example $\PageIndex{62}$

If Pete can paint a wall in pp 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

Example $\PageIndex{63}$

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

Example $\PageIndex{64}$

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?