# 8.4E: Exercises

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• OpenStax
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## Practice Makes Perfect

In the following exercises, find the LCD.

##### Example $$\PageIndex{37}$$

$$\frac{5}{x^2−2x−8}$$, $$\frac{2x}{x^2−x−12}$$

(x−4)(x+2)(x+3)

##### Example $$\PageIndex{38}$$

$$\frac{8}{y^2+12y+35}$$, $$\frac{3y}{y^2+y−42}$$

##### Example $$\PageIndex{39}$$

$$\frac{9}{z^2+2z−8}$$, $$\frac{4z}{z^2−49}$$

(z−2)(z+4)(z+2)

##### Example $$\PageIndex{39}$$

$$\frac{6}{a^2+14a+45}$$, $$\frac{5a}{a^2−81}$$

##### Example $$\PageIndex{40}$$

$$\frac{4}{b^2+6b+9}$$, $$\frac{2b}{b^2−2b−15}$$

(b+3)(b+3)(b−5)

##### Example $$\PageIndex{41}$$

$$\frac{5}{c^2−4c+4}$$, $$\frac{3c}{c^2−10c+16}$$

##### Example $$\PageIndex{42}$$

$$\frac{2}{3d^2+14d−5}$$, $$\frac{5d}{3d^2−19d+6}$$

(3d−1)(d+5)(d−6)

##### Example $$\PageIndex{44}$$

$$\frac{3}{5m^2−3m−2}$$, $$\frac{6m}{5m^2+17m+6}$$

In the following exercises, write as equivalent rational expressions with the given LCD.

##### Example $$\PageIndex{45}$$

$$\frac{5}{x^2−2x−8}$$, $$\frac{2x}{x^2−x−12}$$
LCD (x−4)(x+2)(x+3)

$$\frac{5x+15}{(x−4)(x+2)(x+3)}$$,
$$\frac{2x^2+4x}{(x−4)(x+2)(x+3)}$$

##### Example $$\PageIndex{46}$$

$$\frac{8}{y^2+12y+35}$$, $$\frac{3y}{y^2+y−42}$$
LCD (y+7)(y+5)(y−6)

##### Example $$\PageIndex{47}$$

$$\frac{9}{z^2+2z−8}$$, $$\frac{4z}{z^2−49}$$
LCD (z−2)(z+4)(z+2)

$$\frac{9z+18}{(z−2)(z+4)(z+2)}$$,
$$\frac{4z^2+16}{(z−2)(z+4)(z+2)}$$

##### Example $$\PageIndex{48}$$

$$\frac{6}{a^2+14a+45}$$, $$\frac{5a}{a^2−81}$$
LCD (a+9)(a+5)(a−9)

##### Example $$\PageIndex{49}$$

$$\frac{4}{b^2+6b+9}$$, $$\frac{2b}{b^2−2b−15}$$
LCD (b+3)(b+3)(b−5)

$$\frac{4b−20}{(b+3)(b+3)(b−5)}$$,
$$\frac{2b^2+6b}{(b+3)(b+3)(b−5)}$$

##### Example $$\PageIndex{50}$$

$$\frac{5}{c^2−4c+4}$$, $$\frac{3c}{c^2−10c+10}$$
LCD (c−2)(c−2)(c−8)

##### Example $$\PageIndex{51}$$

$$\frac{2}{3d^2+14d−5}$$, $$\frac{5d}{3d^2−19d+6}$$
LCD (3d−1)(d+5)(d−6)

$$\frac{2d−12}{(3d−1)(d+5)(d−6)}$$,
$$\frac{5d^2+25d}{(3d−1)(d+5)(d−6)}$$

##### Example $$\PageIndex{52}$$

$$\frac{3}{5m^2−3m−2}$$, $$\frac{6m}{5m^2+17m+6}$$
LCD (5m+2)(m−1)(m+3)

##### Example $$\PageIndex{53}$$

$$\frac{5}{24}+\frac{11}{36}$$

$$\frac{37}{72}$$

##### Example $$\PageIndex{54}$$

$$\frac{7}{30}+\frac{13}{45}$$

##### Example $$\PageIndex{55}$$

$$\frac{9}{20}+\frac{11}{30}$$

$$\frac{49}{60}$$

##### Example $$\PageIndex{56}$$

$$\frac{8}{27}+\frac{7}{18}$$

##### Example $$\PageIndex{57}$$

$$\frac{7}{10x^{2}y}+\frac{4}{15xy^2}$$

$$\frac{21y+8x}{30x^{2}y^2}$$

##### Example $$\PageIndex{58}$$

$$\frac{1}{12a^{3}b^2}+\frac{5}{9a^{2}b^3}$$

##### Example $$\PageIndex{59}$$

$$\frac{1}{2m}+\frac{7}{8m^{2}n}$$

$$\frac{mn+14}{16m^{2}n}$$

##### Example $$\PageIndex{60}$$

$$\frac{5}{6p^{2}q}+\frac{1}{4p}$$

##### Example $$\PageIndex{61}$$

$$\frac{3}{r+4}+\frac{2}{r−5}$$

$$\frac{5r−7}{(r+4)(r−5)}$$

##### Example $$\PageIndex{62}$$

$$\frac{4}{s−7}+\frac{5}{s+3}$$

##### Example $$\PageIndex{63}$$

$$\frac{8}{t+5}+\frac{6}{t−5}$$

$$\frac{14t−10}{(t+5)(t−5)}$$

##### Example $$\PageIndex{64}$$

$$\frac{7}{v+5}+\frac{9}{v−5}$$

##### Example $$\PageIndex{65}$$

$$\frac{5}{3w−2}+\frac{2}{w+1}$$

$$\frac{11w+1}{(3w−2)(w+1)}$$

##### Example $$\PageIndex{66}$$

$$\frac{4}{2x+5}+\frac{2}{x−14}$$

##### Example $$\PageIndex{67}$$

$$\frac{2y}{y+3}+\frac{3}{y−12}$$

$$\frac{2y^2+y+9}{(y+3)(y−1)}$$

##### Example $$\PageIndex{68}$$

$$\frac{3z}{z−2}+\frac{1}{z+5}$$

##### Example $$\PageIndex{69}$$

$$\frac{5b}{a^2b−2a^2}+\frac{2b}{b^2−4}$$

$$\frac{b(5b+10+2a2)}{a^2(b−2)(b+2)}$$

##### Example $$\PageIndex{70}$$

$$\frac{4}{cd+3c}+\frac{1}{d^2−9}$$

##### Example $$\PageIndex{71}$$

$$\frac{2m}{3m−3}+\frac{5m}{m^2+3m−4}$$

$$\frac{2m^2+23m}{3(m−1)(m+4)}$$

##### Example $$\PageIndex{72}$$

$$\frac{3}{4n+4}+\frac{6}{n^2−n−2}$$

##### Example $$\PageIndex{73}$$

$$\frac{3}{n^2+3n−18}+\frac{4n}{n^2+8n+12}$$

$$\frac{4n^2−9n+6}{(n-3)(n+6)(n+2)}$$

##### Example $$\PageIndex{74}$$

$$\frac{6}{q^2−3q−10}+\frac{5q}{q^2−8q+15}$$

##### Example $$\PageIndex{75}$$

$$\frac{3r}{r^2+7r+6}+\frac{9}{r^2+4r+3}$$

$$\frac{3(r^2+6r+18)}{(r+1)(r+6)(r+3)}$$

##### Example $$\PageIndex{76}$$

$$\frac{2s}{s^2+2s−8}+\frac{4}{s^2+3s−10}$$

In the following exercises, subtract.

##### Example $$\PageIndex{77}$$

$$\frac{t}{t−6}−\frac{t−2}{t+6}$$

$$\frac{2(7t−6)}{(t−6)(t+6)}$$

##### Example $$\PageIndex{78}$$

$$\frac{v}{v−3}−\frac{v−6}{v+1}$$

##### Example $$\PageIndex{79}$$

$$\frac{w+2}{w+4}−\frac{w}{w−2}$$

$$\frac{−4(1+w)}{(w+4)(w−2)}$$

##### Example $$\PageIndex{80}$$

$$\frac{x−3}{x+6}−\frac{x}{x+3}$$

##### Example $$\PageIndex{81}$$

$$\frac{y−4}{y+1}−\frac{1}{y+7}$$

$$\frac{y^2+2y-29}{(y+1)(y+7)}$$

##### Example $$\PageIndex{82}$$

$$\frac{z+8}{z−3}−\frac{z}{z−2}$$

##### Example $$\PageIndex{83}$$

$$\frac{5a}{a+3}−\frac{a+2}{a+6}$$

$$\frac{4a^2+25a−6}{(a+3)(a+6)}$$

##### Example $$\PageIndex{84}$$

$$\frac{3b}{b−2}−\frac{b−6}{b−8}$$

##### Example $$\PageIndex{85}$$

$$\frac{6c}{c^2−25}−\frac{3}{c+5}$$

$$\frac{3}{c−5}$$​​​​​​​

##### Example $$\PageIndex{86}$$

$$\frac{4d}{d^2−81}−\frac{2}{d+9}$$

##### Example $$\PageIndex{87}$$

$$\frac{6}{m+6}−\frac{12m}{m^2−36}$$

$$\frac{−6}{m−6}$$

##### Example $$\PageIndex{88}$$

$$\frac{4}{n+4}−\frac{8n}{n^2−16}$$

##### Example $$\PageIndex{89}$$

$$\frac{−9p−17}{p^2−4p−21}−\frac{p+1}{7−p}$$

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

##### Example $$\PageIndex{90}$$

$$\frac{7q+8}{q^2−2q−24}−\frac{q+2}{4−q}$$

##### Example $$\PageIndex{91}$$

$$\frac{−2r−16}{r^2+6r−16}−\frac{5}{2−r}$$

$$\frac{3}{r−2}$$

##### Example $$\PageIndex{92}$$

$$\frac{2t−30}{t^2+6t−27}−\frac{2}{3−t}$$

##### Example $$\PageIndex{93}$$

$$\frac{5v−2}{v+3}−4$$

$$\frac{−v−14}{v+3}$$

##### Example $$\PageIndex{94}$$

$$\frac{6w+5}{w−1}+2$$

##### Example $$\PageIndex{95}$$

$$\frac{2x+7}{10x−1}+3$$

$$\frac{4(8x+1)}{10x−1}$$

##### Example $$\PageIndex{96}$$

$$\frac{8y−4}{5y+2}−6$$

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

##### Example $$\PageIndex{97}$$

$$\frac{5a}{a−2}+\frac{9}{a}−\frac{2a+18}{a^2−2a}$$​​​​​​​

$$\frac{5a^2+7a−36}{a(a−2)}$$

##### Example $$\PageIndex{98}$$

$$\frac{2b}{b−5}+\frac{3}{2b}−\frac{2}{b−15}$$

##### Example $$\PageIndex{99}$$

$$\frac{c}{c+2}+\frac{5}{c−2}−\frac{10c}{c^2−4}$$

$$\frac{c−5}{c+2}$$

##### Example $$\PageIndex{100}$$

$$\frac{6d}{d−5}+\frac{1}{d+4}−\frac{7d−5}{d^2−d−20}$$

​​​​​​​In the following exercises, simplify.

##### Example $$\PageIndex{101}$$

$$\frac{6a}{3ab+b^2}+\frac{3a}{9a^2−b^2}$$

$$\frac{3a(6a−b)}{b(3a+b)(3a−b)}$$

##### Example $$\PageIndex{102}$$

$$\frac{2c}{2c+10}+\frac{7c}{c^2+9c+20}$$

##### Example $$\PageIndex{103}$$

$$\frac{6d}{d^2−64}−\frac{3}{d−8}$$

$$\frac{3}{d+8}$$

##### Example $$\PageIndex{104}$$

$$\frac{5}{n+7}−\frac{10n}{n^2−49}$$

##### Example $$\PageIndex{105}$$

$$\frac{4m}{m^2+6m−7}+\frac{2}{m^2+10m+21}$$

$$\frac{2(2m^2+7m−1)}{(m+7)(m−1)(m+3)}$$

##### Example $$\PageIndex{106}$$

$$\frac{3p}{p^2+4p−12}+\frac{1}{p^2+p−30}$$

##### Example $$\PageIndex{107}$$

$$\frac{−5n−5}{n^2+n−6}+\frac{n+1}{2−n}$$

$$\frac{n+1}{n+3}$$​​​​​​​

##### Example $$\PageIndex{108}$$

$$\frac{−4b−24}{b^2+b−30}+\frac{b+7}{5−b}$$​​​​​​​

##### Example $$\PageIndex{109}$$

$$\frac{7}{15p}+\frac{5}{18pq}$$

$$\frac{42q+25}{90pq}$$

##### Example $$\PageIndex{110}$$

$$\frac{3}{20a^2}+\frac{11}{12ab^2}$$

##### Example $$\PageIndex{111}$$

$$\frac{4}{x−2}+\frac{3}{x+5}$$

$$\frac{7(x+2)}{(x−2)(x+5)}$$

##### Example $$\PageIndex{112}$$

$$\frac{6}{m+4}+\frac{9}{m−8}$$

##### Example $$\PageIndex{113}$$

$$\frac{2q+7}{y+4}−2$$

$$\frac{17q+2}{3q−1}$$

##### Example $$\PageIndex{114}$$

$$\frac{3y−1}{y+4}−2$$

##### Example $$\PageIndex{115}$$

$$\frac{z+2}{z−5}−\frac{z}{z+1}$$

$$\frac{8z+2}{(z−5)(z+1)}$$​​​​​​​

##### Example $$\PageIndex{116}$$

$$\frac{t}{t−5}−\frac{t−1}{t+5}$$​​​​​​​

##### Example $$\PageIndex{117}$$

$$\frac{3d}{d+2}+\frac{4}{d}−\frac{d+8}{d^2+2d}$$

$$\frac{3(d+1)}{d+2}$$

##### Example $$\PageIndex{118}$$

$$\frac{2q}{q+5}+\frac{3}{q−3}−\frac{13q+15}{q^2+2q−15}$$

## Everyday Math

##### Example $$\PageIndex{119}$$

Decorating cupcakes Victoria can decorate an order of cupcakes for a wedding in tt hours, so in 1 hour she can decorate $$\frac{1}{t}$$ of the cupcakes. It would take her sister 3 hours longer to decorate the same order of cupcakes, so in 1 hour she can decorate $$\frac{1}{t+3}$$ of the cupcakes.

1. Find the fraction of the decorating job that Victoria and her sister, working together, would complete in one hour by adding the rational expressions $$\frac{1}{t}+\frac{1}{t+3}$$.
1. $$\frac{2t+3}{t(t+3)}$$
2. $$\frac{13}{40}$$
##### Example $$\PageIndex{120}$$

Kayaking When Trina kayaks upriver, it takes her $$\frac{5}{3−c}$$ hours to go 5 miles, where cc is the speed of the river current. It takes her $$\frac{5}{3+c}$$ hours to kayak 5 miles down the river.

1. Find an expression for the number of hours it would take Trina to kayak 5 miles up the river and then return by adding $$\frac{5}{3−c}+\frac{5}{3+c}$$.
2. Evaluate your answer to part (a) when c=1 to find the number of hours it would take Trina if the speed of the river current is 1 mile per hour.​​​​​​

## Writing Exercises

##### Example $$\PageIndex{121}$$

Felipe thinks $$\frac{1}{x}+\frac{1}{y}$$ is $$\frac{2}{x+y}$$.

1. Choose numerical values for x and y and evaluate $$\frac{1}{x}+\frac{1}{y}$$.
2. Evaluate $$\frac{2}{x+y}$$ for the same values of x and y you used in part (a).
3. Explain why Felipe is wrong.
4. Find the correct expression for $$\frac{1}{x}+\frac{1}{y}$$.

##### Example $$\PageIndex{122}$$

Simplify the expression $$\frac{4}{n^2+6n+9}−\frac{1}{n^2−9}$$ and explain all your steps.

## Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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