13.9.4E: Exercises
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Practice Makes Perfect
In the following exercises, find the LCD.
Example \(\PageIndex{37}\)
\(\frac{5}{x^2−2x−8}\), \(\frac{2x}{x^2−x−12}\)
- Answer
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(x−4)(x+2)(x+3)
Example \(\PageIndex{38}\)
\(\frac{8}{y^2+12y+35}\), \(\frac{3y}{y^2+y−42}\)
Example \(\PageIndex{39}\)
\(\frac{9}{z^2+2z−8}\), \(\frac{4z}{z^2−49}\)
- Answer
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(z−2)(z+4)(z+2)
Example \(\PageIndex{39}\)
\(\frac{6}{a^2+14a+45}\), \(\frac{5a}{a^2−81}\)
Example \(\PageIndex{40}\)
\(\frac{4}{b^2+6b+9}\), \(\frac{2b}{b^2−2b−15}\)
- Answer
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(b+3)(b+3)(b−5)
Example \(\PageIndex{41}\)
\(\frac{5}{c^2−4c+4}\), \(\frac{3c}{c^2−10c+16}\)
Example \(\PageIndex{42}\)
\(\frac{2}{3d^2+14d−5}\), \(\frac{5d}{3d^2−19d+6}\)
- Answer
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(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)
- Answer
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\(\frac{5x+15}{(x−4)(x+2)(x+3)}\),
\(\frac{2x^2+4x}{(x−4)(x+2)(x+3)}\)
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)
- Answer
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\(\frac{9z+18}{(z−2)(z+4)(z+2)}\),
\(\frac{4z^2+16}{(z−2)(z+4)(z+2)}\)
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)
- Answer
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\(\frac{4b−20}{(b+3)(b+3)(b−5)}\),
\(\frac{2b^2+6b}{(b+3)(b+3)(b−5)}\)
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)
- Answer
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\(\frac{2d−12}{(3d−1)(d+5)(d−6)}\),
\(\frac{5d^2+25d}{(3d−1)(d+5)(d−6)}\)
Example \(\PageIndex{52}\)
\(\frac{3}{5m^2−3m−2}\), \(\frac{6m}{5m^2+17m+6}\)
LCD (5m+2)(m−1)(m+3)
In the following exercises, add.
Example \(\PageIndex{53}\)
\(\frac{5}{24}+\frac{11}{36}\)
- Answer
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\(\frac{37}{72}\)
Example \(\PageIndex{54}\)
\(\frac{7}{30}+\frac{13}{45}\)
Example \(\PageIndex{55}\)
\(\frac{9}{20}+\frac{11}{30}\)
- Answer
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\(\frac{49}{60}\)
Example \(\PageIndex{56}\)
\(\frac{8}{27}+\frac{7}{18}\)
Example \(\PageIndex{57}\)
\(\frac{7}{10x^{2}y}+\frac{4}{15xy^2}\)
- Answer
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\(\frac{21y+8x}{30x^{2}y^2}\)
Example \(\PageIndex{58}\)
\(\frac{1}{12a^{3}b^2}+\frac{5}{9a^{2}b^3}\)
Example \(\PageIndex{59}\)
\(\frac{1}{2m}+\frac{7}{8m^{2}n}\)
- Answer
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\(\frac{mn+14}{16m^{2}n}\)
Example \(\PageIndex{60}\)
\(\frac{5}{6p^{2}q}+\frac{1}{4p}\)
Example \(\PageIndex{61}\)
\(\frac{3}{r+4}+\frac{2}{r−5}\)
- Answer
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\(\frac{5r−7}{(r+4)(r−5)}\)
Example \(\PageIndex{62}\)
\(\frac{4}{s−7}+\frac{5}{s+3}\)
Example \(\PageIndex{63}\)
\(\frac{8}{t+5}+\frac{6}{t−5}\)
- Answer
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\(\frac{14t−10}{(t+5)(t−5)}\)
Example \(\PageIndex{64}\)
\(\frac{7}{v+5}+\frac{9}{v−5}\)
Example \(\PageIndex{65}\)
\(\frac{5}{3w−2}+\frac{2}{w+1}\)
- Answer
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\(\frac{11w+1}{(3w−2)(w+1)}\)
Example \(\PageIndex{66}\)
\(\frac{4}{2x+5}+\frac{2}{x−14}\)
Example \(\PageIndex{67}\)
\(\frac{2y}{y+3}+\frac{3}{y−12}\)
- Answer
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\(\frac{2y^2+y+9}{(y+3)(y−1)}\)
Example \(\PageIndex{68}\)
\(\frac{3z}{z−2}+\frac{1}{z+5}\)
Example \(\PageIndex{69}\)
\(\frac{5b}{a^2b−2a^2}+\frac{2b}{b^2−4}\)
- Answer
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\(\frac{b(5b+10+2a2)}{a^2(b−2)(b+2)}\)
Example \(\PageIndex{70}\)
\(\frac{4}{cd+3c}+\frac{1}{d^2−9}\)
Example \(\PageIndex{71}\)
\(\frac{2m}{3m−3}+\frac{5m}{m^2+3m−4}\)
- Answer
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\(\frac{2m^2+23m}{3(m−1)(m+4)}\)
Example \(\PageIndex{72}\)
\(\frac{3}{4n+4}+\frac{6}{n^2−n−2}\)
Example \(\PageIndex{73}\)
\(\frac{3}{n^2+3n−18}+\frac{4n}{n^2+8n+12}\)
- Answer
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\(\frac{4n^2−9n+6}{(n-3)(n+6)(n+2)}\)
Example \(\PageIndex{74}\)
\(\frac{6}{q^2−3q−10}+\frac{5q}{q^2−8q+15}\)
Example \(\PageIndex{75}\)
\(\frac{3r}{r^2+7r+6}+\frac{9}{r^2+4r+3}\)
- Answer
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\(\frac{3(r^2+6r+18)}{(r+1)(r+6)(r+3)}\)
Example \(\PageIndex{76}\)
\(\frac{2s}{s^2+2s−8}+\frac{4}{s^2+3s−10}\)
In the following exercises, subtract.
Example \(\PageIndex{77}\)
\(\frac{t}{t−6}−\frac{t−2}{t+6}\)
- Answer
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\(\frac{2(7t−6)}{(t−6)(t+6)}\)
Example \(\PageIndex{78}\)
\(\frac{v}{v−3}−\frac{v−6}{v+1}\)
Example \(\PageIndex{79}\)
\(\frac{w+2}{w+4}−\frac{w}{w−2}\)
- Answer
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\(\frac{−4(1+w)}{(w+4)(w−2)}\)
Example \(\PageIndex{80}\)
\(\frac{x−3}{x+6}−\frac{x}{x+3}\)
Example \(\PageIndex{81}\)
\(\frac{y−4}{y+1}−\frac{1}{y+7}\)
- Answer
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\(\frac{y^2+2y-29}{(y+1)(y+7)}\)
Example \(\PageIndex{82}\)
\(\frac{z+8}{z−3}−\frac{z}{z−2}\)
Example \(\PageIndex{83}\)
\(\frac{5a}{a+3}−\frac{a+2}{a+6}\)
- Answer
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\(\frac{4a^2+25a−6}{(a+3)(a+6)}\)
Example \(\PageIndex{84}\)
\(\frac{3b}{b−2}−\frac{b−6}{b−8}\)
Example \(\PageIndex{85}\)
\(\frac{6c}{c^2−25}−\frac{3}{c+5}\)
- Answer
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\(\frac{3}{c−5}\)
Example \(\PageIndex{86}\)
\(\frac{4d}{d^2−81}−\frac{2}{d+9}\)
Example \(\PageIndex{87}\)
\(\frac{6}{m+6}−\frac{12m}{m^2−36}\)
- Answer
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\(\frac{−6}{m−6}\)
Example \(\PageIndex{88}\)
\(\frac{4}{n+4}−\frac{8n}{n^2−16}\)
Example \(\PageIndex{89}\)
\(\frac{−9p−17}{p^2−4p−21}−\frac{p+1}{7−p}\)
- Answer
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\(\frac{p+2}{p+3}\)
Example \(\PageIndex{90}\)
\(\frac{7q+8}{q^2−2q−24}−\frac{q+2}{4−q}\)
Example \(\PageIndex{91}\)
\(\frac{−2r−16}{r^2+6r−16}−\frac{5}{2−r}\)
- Answer
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\(\frac{3}{r−2}\)
Example \(\PageIndex{92}\)
\(\frac{2t−30}{t^2+6t−27}−\frac{2}{3−t}\)
Example \(\PageIndex{93}\)
\(\frac{5v−2}{v+3}−4\)
- Answer
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\(\frac{−v−14}{v+3}\)
Example \(\PageIndex{94}\)
\(\frac{6w+5}{w−1}+2\)
Example \(\PageIndex{95}\)
\(\frac{2x+7}{10x−1}+3\)
- Answer
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\(\frac{4(8x+1)}{10x−1}\)
Example \(\PageIndex{96}\)
\(\frac{8y−4}{5y+2}−6\)
In the following exercises, add and subtract.
Example \(\PageIndex{97}\)
\(\frac{5a}{a−2}+\frac{9}{a}−\frac{2a+18}{a^2−2a}\)
- Answer
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\(\frac{5a^2+7a−36}{a(a−2)}\)
Example \(\PageIndex{98}\)
\(\frac{2b}{b−5}+\frac{3}{2b}−\frac{2}{b−15}\)
Example \(\PageIndex{99}\)
\(\frac{c}{c+2}+\frac{5}{c−2}−\frac{10c}{c^2−4}\)
- Answer
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\(\frac{c−5}{c+2}\)
Example \(\PageIndex{100}\)
\(\frac{6d}{d−5}+\frac{1}{d+4}−\frac{7d−5}{d^2−d−20}\)
In the following exercises, simplify.
Example \(\PageIndex{101}\)
\(\frac{6a}{3ab+b^2}+\frac{3a}{9a^2−b^2}\)
- Answer
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\(\frac{3a(6a−b)}{b(3a+b)(3a−b)}\)
Example \(\PageIndex{102}\)
\(\frac{2c}{2c+10}+\frac{7c}{c^2+9c+20}\)
Example \(\PageIndex{103}\)
\(\frac{6d}{d^2−64}−\frac{3}{d−8}\)
- Answer
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\(\frac{3}{d+8}\)
Example \(\PageIndex{104}\)
\(\frac{5}{n+7}−\frac{10n}{n^2−49}\)
Example \(\PageIndex{105}\)
\(\frac{4m}{m^2+6m−7}+\frac{2}{m^2+10m+21}\)
- Answer
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\(\frac{2(2m^2+7m−1)}{(m+7)(m−1)(m+3)}\)
Example \(\PageIndex{106}\)
\(\frac{3p}{p^2+4p−12}+\frac{1}{p^2+p−30}\)
Example \(\PageIndex{107}\)
\(\frac{−5n−5}{n^2+n−6}+\frac{n+1}{2−n}\)
- Answer
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\(\frac{n+1}{n+3}\)
Example \(\PageIndex{108}\)
\(\frac{−4b−24}{b^2+b−30}+\frac{b+7}{5−b}\)
Example \(\PageIndex{109}\)
\(\frac{7}{15p}+\frac{5}{18pq}\)
- Answer
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\(\frac{42q+25}{90pq}\)
Example \(\PageIndex{110}\)
\(\frac{3}{20a^2}+\frac{11}{12ab^2}\)
Example \(\PageIndex{111}\)
\(\frac{4}{x−2}+\frac{3}{x+5}\)
- Answer
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\(\frac{7(x+2)}{(x−2)(x+5)}\)
Example \(\PageIndex{112}\)
\(\frac{6}{m+4}+\frac{9}{m−8}\)
Example \(\PageIndex{113}\)
\(\frac{2q+7}{y+4}−2\)
- Answer
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\(\frac{17q+2}{3q−1}\)
Example \(\PageIndex{114}\)
\(\frac{3y−1}{y+4}−2\)
Example \(\PageIndex{115}\)
\(\frac{z+2}{z−5}−\frac{z}{z+1}\)
- Answer
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\(\frac{8z+2}{(z−5)(z+1)}\)
Example \(\PageIndex{116}\)
\(\frac{t}{t−5}−\frac{t−1}{t+5}\)
Example \(\PageIndex{117}\)
\(\frac{3d}{d+2}+\frac{4}{d}−\frac{d+8}{d^2+2d}\)
- Answer
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\(\frac{3(d+1)}{d+2}\)
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.
- Find the fraction of the decorating job that Victoria and her sister, working together, would complete in one hour by adding the rational expressions \(\frac{1}{t}+\frac{1}{t+3}\).
- Evaluate your answer to part (a) when t=5.
- Answer
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- \(\frac{2t+3}{t(t+3)}\)
- \(\frac{13}{40}\)
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.
- Find an expression for the number of hours it would take Trina to kayak 5 miles up the river and then return by adding \(\frac{5}{3−c}+\frac{5}{3+c}\).
- 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 \(\frac{2}{x+y}\).
- Choose numerical values for x and y and evaluate \(\frac{1}{x}+\frac{1}{y}\).
- Evaluate \(\frac{2}{x+y}\) for the same values of x and y you used in part (a).
- Explain why Felipe is wrong.
- Find the correct expression for \(\frac{1}{x}+\frac{1}{y}\).
- Answer
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Answers may vary.
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?