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8.4E: Exercises

  • Page ID
    30269
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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

    (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

    (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

    (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

    (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

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

    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

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

    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

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

    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

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

    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

    \(\frac{37}{72}\)

    Example \(\PageIndex{54}\)

    \(\frac{7}{30}+\frac{13}{45}\)

    Example \(\PageIndex{55}\)

    \(\frac{9}{20}+\frac{11}{30}\)

    Answer

    \(\frac{49}{60}\)

    Example \(\PageIndex{56}\)

    \(\frac{8}{27}+\frac{7}{18}\)

    Example \(\PageIndex{57}\)

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

    Answer

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

    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

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

    Example \(\PageIndex{60}\)

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

    Example \(\PageIndex{61}\)

    \(\frac{3}{r+4}+\frac{2}{r−5}\)

    Answer

    \(\frac{5r−7}{(r+4)(r−5)}\)

    Example \(\PageIndex{62}\)

    \(\frac{4}{s−7}+\frac{5}{s+3}\)

    Example \(\PageIndex{63}\)

    \(\frac{8}{t+5}+\frac{6}{t−5}\)

    Answer

    \(\frac{14t−10}{(t+5)(t−5)}\)

    Example \(\PageIndex{64}\)

    \(\frac{7}{v+5}+\frac{9}{v−5}\)

    Example \(\PageIndex{65}\)

    \(\frac{5}{3w−2}+\frac{2}{w+1}\)

    Answer

    \(\frac{11w+1}{(3w−2)(w+1)}\)

    Example \(\PageIndex{66}\)

    \(\frac{4}{2x+5}+\frac{2}{x−14}\)

    Example \(\PageIndex{67}\)

    \(\frac{2y}{y+3}+\frac{3}{y−12}\)

    Answer

    \(\frac{2y^2+y+9}{(y+3)(y−1)}\)

    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

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

    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

    \(\frac{2m^2+23m}{3(m−1)(m+4)}\)

    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

    \(\frac{4n^2−9n+6}{(n-3)(n+6)(n+2)}\)

    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

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

    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

    \(\frac{2(7t−6)}{(t−6)(t+6)}\)

    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

    \(\frac{−4(1+w)}{(w+4)(w−2)}\)

    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

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

    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

    \(\frac{4a^2+25a−6}{(a+3)(a+6)}\)

    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

    \(\frac{3}{c−5}\)​​​​​​​

    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

    \(\frac{−6}{m−6}\)

    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

    \(\frac{p+2}{p+3}\)

    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

    \(\frac{3}{r−2}\)

    Example \(\PageIndex{92}\)

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

    Example \(\PageIndex{93}\)

    \(\frac{5v−2}{v+3}−4\)

    Answer

    \(\frac{−v−14}{v+3}\)

    Example \(\PageIndex{94}\)

    \(\frac{6w+5}{w−1}+2\)

    Example \(\PageIndex{95}\)

    \(\frac{2x+7}{10x−1}+3\)

    Answer

    \(\frac{4(8x+1)}{10x−1}\)

    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

    \(\frac{5a^2+7a−36}{a(a−2)}\)

    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

    \(\frac{c−5}{c+2}\)

    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

    \(\frac{3a(6a−b)}{b(3a+b)(3a−b)}\)

    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

    \(\frac{3}{d+8}\)

    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

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

    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

    \(\frac{n+1}{n+3}\)​​​​​​​

    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

    \(\frac{42q+25}{90pq}\)

    Example \(\PageIndex{110}\)

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

    Example \(\PageIndex{111}\)

    \(\frac{4}{x−2}+\frac{3}{x+5}\)

    Answer

    \(\frac{7(x+2)}{(x−2)(x+5)}\)

    Example \(\PageIndex{112}\)

    \(\frac{6}{m+4}+\frac{9}{m−8}\)

    Example \(\PageIndex{113}\)

    \(\frac{2q+7}{y+4}−2\)

    Answer

    \(\frac{17q+2}{3q−1}\)

    Example \(\PageIndex{114}\)

    \(\frac{3y−1}{y+4}−2\)

    Example \(\PageIndex{115}\)

    \(\frac{z+2}{z−5}−\frac{z}{z+1}\)

    Answer

    \(\frac{8z+2}{(z−5)(z+1)}\)​​​​​​​

    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

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

    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 \(\frac{1}{t}+\frac{1}{t+3}\).
    2. Evaluate your answer to part (a) when t=5.
    Answer
    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 \(\frac{1}{x}+\frac{1}{y}\).
    Answer

    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.

    This is a table that has five rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “find the least common denominator of rational expressions,” “find equivalent rational expressions,” “add rational expressions with different denominators,” and “subtract rational expressions with different denominators.” The rest of the cells are blank.

    ⓑ 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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