8.12: Exercise Supplement

Exercise Supplement

Rational Expressions

For the following problems, find the domain of each rational expression.

Exercise $\PageIndex{1}$

$\dfrac{9}{x+4}$

Answer

$x \not = 4$

Exercise $\PageIndex{2}$

$\dfrac{10x}{x+6}$

Exercise $\PageIndex{3}$

$\dfrac{x+1}{2x - 5}$

Answer

$x \not = \dfrac{5}{2}$

Exercise $\PageIndex{4}$

$\dfrac{2a + 3}{7a + 5}$

Exercise $\PageIndex{5}$

$\dfrac{3m}{2m(m-1)}$

Answer

$m \not = 0, 1$

Exercise $\PageIndex{6}$

$\dfrac{5r + 6}{9r(2r + 1)}$

Exercise $\PageIndex{7}$

$\dfrac{s}{s(s+8)(4s + 7)}$

Answer

$s \not = -8, -\dfrac{7}{4}, 0$

Exercise $\PageIndex{8}$

$\dfrac{-11x}{x^2 - 9x + 18}$

Exercise $\PageIndex{9}$

$\dfrac{-y + 5}{12y^2 + 28y - 5}$

Answer

$y \not = \dfrac{1}{6}, -\dfrac{5}{2}$

Exercise $\PageIndex{10}$

$\dfrac{16}{12a^3 + 21a^2 - 6a}$

For the following problems, show that the fractions are equivalent.

Exercise $\PageIndex{11}$

$\dfrac{-4}{5}, -\dfrac{4}{5}$

Answer

$-(4 \cdot 5) = 20, -4(5) = -20$

Exercise $\PageIndex{12}$

$\dfrac{-3}{8}, -\dfrac{3}{8}$

Exercise $\PageIndex{13}$

$\dfrac{-7}{10}, -\dfrac{7}{10}$

Answer

$−(7 \cdot 10)=−70,−7(10)=−70$

For the following problems, fill in the missing term.

Exercise $\PageIndex{14}$

$-\dfrac{3}{y-5} = \dfrac{?}{y - 5}$

Exercise $\PageIndex{15}$

$-\dfrac{6a}{2a + 1} = \dfrac{?}{2a + 1}$

Answer

$−6a$

Exercise $\PageIndex{16}$

$-\dfrac{x+1}{x-3} = \dfrac{?}{x-3}$

Exercise $\PageIndex{17}$

$-\dfrac{9}{-a + 4} = \dfrac{?}{a - 4}$

Answer

$9$

Exercise $\PageIndex{18}$

$\dfrac{y + 3}{-y-5} = \dfrac{?}{y + 5}$

Exercise $\PageIndex{19}$

$\dfrac{-6m-7}{-5m-1} = \dfrac{6m + 7}{?}$

Answer

$5m+1$

Exercise $\PageIndex{20}$

$-\dfrac{2r - 5}{7r + 1} = \dfrac{2r - 5}{?}$

Reducing Rational Expressions

For the following problems, reduce the rational expressions to lowest terms.

Exercise $\PageIndex{21}$

$\dfrac{12}{6x + 24}$

Answer

$\dfrac{2}{x + 4}$

Exercise $\PageIndex{22}$

$\dfrac{16}{4y - 16}$

Exercise $\PageIndex{23}$

$\dfrac{5m + 25}{10m^2 + 15m}$

Answer

$\dfrac{m+5}{m(2m + 3)}$

Exercise $\PageIndex{24}$

$\dfrac{7 + 21r}{7r^2 + 28r}$

Exercise $\PageIndex{25}$

$\dfrac{3a^2 + 4a}{5a^3 + 6a^2}$

Answer

$\dfrac{3a + 4}{a(5a + 6)}$

Exercise $\PageIndex{26}$

$\dfrac{4x - 4}{x^2 + 2x - 3}$

Exercise $\PageIndex{27}$

$\dfrac{5y + 20}{y^2 - 16}$

Answer

$\dfrac{5}{y-4}$

Exercise $\PageIndex{28}$

$\dfrac{4y^3 - 12}{y^4 - 2y^2 - 3}$

Exercise $\PageIndex{29}$

$\dfrac{6a^9 - 12a^7}{2a^7 - 14a^5}$

Answer

$\dfrac{3a^2(a^2 - 2)}{a^2 - 7}$

Exercise $\PageIndex{30}$

$\dfrac{8x^4y^8 + 24x^3y^9}{4x^2y^5 - 12x^3y^6}$

Exercise $\PageIndex{31}$

$\dfrac{21y^8z^{10}w^2}{-7y^7w^2}$

Answer

$-3yz^{10}$

Exercise $\PageIndex{32}$

$\dfrac{-35a^5b^2c^4d^8}{-5abc^3d^6}$

Exercise $\PageIndex{33}$

$\dfrac{x^2 + 9x + 18}{x^3 + 3x^2}$

Answer

$\dfrac{x+6}{x^2}$

Exercise $\PageIndex{34}$

$\dfrac{a^2 - 12a + 35}{2a^4 - 14a^3}$

Exercise $\PageIndex{35}$

$\dfrac{y^2 - 7y + 12}{y^2 - 4y + 3}$

Answer

$\dfrac{y-4}{y-1}$

Exercise $\PageIndex{36}$

$\dfrac{m^2 - 6m - 16}{m^2 - 9m - 22}$

Exercise $\PageIndex{37}$

$\dfrac{12r^2 - 7r - 10}{4r^2 - 13r + 10}$

Answer

$\dfrac{3r + 2}{r - 2}$

Exercise $\PageIndex{38}$

$\dfrac{14a^2 - 5a - 1}{6a^2 + 9a - 6}$

Exercise $\PageIndex{39}$

$\dfrac{4a^4 - 8a^3}{4a^2}$

Answer

$a(a-2)$

Exercise $\PageIndex{40}$

$\dfrac{5m^2}{10m^3 + 5m^2}$

Exercise $\PageIndex{41}$

$\dfrac{-6a - 1}{-5a - 2}$

Answer

$\dfrac{6a + 1}{5a + 2}$

Exercise $\PageIndex{42}$

$\dfrac{-r}{-5r - 1}$

Multiplying and Dividing Rational Expressions - Adding and Subtracting Rational Expressions

For the following problems, perform the indicated operations.

Exercise $\PageIndex{43}$

$\dfrac{x^2}{18} \cdot \dfrac{3}{x^3}$

Answer

$\dfrac{1}{6x}$

Exercise $\PageIndex{44}$

$\dfrac{4a^2b^3}{15x^4y^5} \cdot \dfrac{10x^6y^3}{ab^2}$

Exercise $\PageIndex{45}$

$\dfrac{x+6}{x-1} \cdot \dfrac{x+7}{x+6}$

Answer

$\dfrac{x+7}{x-1}$

Exercise $\PageIndex{46}$

$\dfrac{8a - 12}{3a + 3} \div \dfrac{(a+1)^2}{4a - 6}$

Exercise $\PageIndex{47}$

$\dfrac{10m^4 - 5m^2}{4r^7 + 20r^3} \div \dfrac{m}{16r^8 + 80r^4}$

Answer

$20mr(2m^2 - 1)$

Exercise $\PageIndex{48}$

$\dfrac{5}{r + 7} - \dfrac{3}{r + 7}$

Exercise $\PageIndex{49}$

$\dfrac{2a}{3a - 1} - \dfrac{9a}{3a - 1}$

Answer

$\dfrac{-7a}{3a - 1}$

Exercise $\PageIndex{50}$

$\dfrac{9x + 7}{4x - 6} + \dfrac{3x + 2}{4x - 6}$

Exercise $\PageIndex{51}$

$\dfrac{15y - 4}{8y + 1} - \dfrac{2y + 1}{8y + 1}$

Answer

$\dfrac{13y - 5}{8y + 1}$

Exercise $\PageIndex{52}$

$\dfrac{4}{a + 3} + \dfrac{6}{a - 5}$

Exercise $\PageIndex{53}$

$\dfrac{7a}{a + 6} + \dfrac{5a}{a-8}$

Answer

$\dfrac{2a(6a - 13)}{(a+6)(a-8)}$

Exercise $\PageIndex{54}$

$\dfrac{x+4}{x-2} - \dfrac{y+6}{y+1}$

Exercise $\PageIndex{55}$

$\dfrac{2y + 1}{y + 4} - \dfrac{y + 6}{y + 1}$

Answer

$\dfrac{y^2 - 7y - 23}{(y+4)(y+1)}$

Exercise $\PageIndex{56}$

$\dfrac{x-3}{(x+2)(x+4)} + \dfrac{2x - 1}{x + 4}$

Exercise $\PageIndex{57}$

$\dfrac{6a + 5}{(2a + 1)(4a - 3)} + \dfrac{4a + 1}{2a + 1}$

Answer

$\dfrac{2(8a^2 - a + 1)}{(2a + 1)(4a - 3)}$

Exercise $\PageIndex{58}$

$\dfrac{4}{x^2 + 3x + 2} + \dfrac{9}{x^2 + 6x + 8}$

Exercise $\PageIndex{59}$

$\dfrac{6r}{r^2 + 7r - 18} - \dfrac{-3r}{r^2 - 3r + 2}$

Answer

$\dfrac{3r(3r + 7)}{(r-1)(r-2)(r+9)}$

Exercise $\PageIndex{60}$

$\dfrac{y+3}{y^2 - 11y + 10} - \dfrac{y + 1}{y^2 + 3y - 4}$

Exercise $\PageIndex{61}$

$\dfrac{2a + 5}{16a^2 - 1} - \dfrac{6a + 7}{16a^2 - 12a + 2}$

Answer

$\dfrac{-16a^2 - 18a - 17}{2(4a-1)(4a+1)(2a-1)}$

Exercise $\PageIndex{62}$

$\dfrac{7y + 4}{6y^2 - 32y + 32} + \dfrac{6y - 10}{2y^2 - 18y + 40}$

Exercise $\PageIndex{63}$

$\dfrac{x^2 - x - 12}{x^2 - 3x + 2} \cdot \dfrac{x^2 + 3x - 4}{x^2 - 3x - 18}$

Answer

$\dfrac{(x+4)(x-4)}{(x-2)(x-6)}$

Exercise $\PageIndex{64}$

$(r+3)^4 \cdot \dfrac{r+4}{(r+3)^3}$

Answer

$(r+3)(r+4)$

Exercise $\PageIndex{65}$

$(b+5)^3 \cdot \dfrac{(b+1)^2}{(b+5)^2}$

Exercise $\PageIndex{66}$

$(x-7)^4 \div \dfrac{(x-7)^3}{x+1}$

Answer

$(x-7)(x+1)$

Exercise $\PageIndex{67}$

$(4x + 9)^6 \div \dfrac{(4x + 9)^2}{(3x + 1)^4}$

Exercise $\PageIndex{68}$

$5x + \dfrac{2x^2 + 1}{x-4}$

Answer

$\dfrac{7x^2 - 20x + 1}{(x-4)}$

Exercise $\PageIndex{69}$

$2y + \dfrac{4y^2 + 5}{y - 1}$

Exercise $\PageIndex{70}$

$\dfrac{y^2 + 4y + 4}{y^2 + 10y + 21} \div (y+2)$

Answer

$\dfrac{(y+2)}{(y+3)(y+7)}$

Exercise $\PageIndex{71}$

$2x - 3 + \dfrac{4x^2 + x - 1}{x - 1}$

Exercise $\PageIndex{72}$

$\dfrac{3 x+1}{x^{2}+3 x+2}+\dfrac{5 x+6}{x^{2}+6 x+5}-\dfrac{3 x-7}{x^{2}-2 x-35}$

Answer

$\dfrac{5 x^{3}-26 x^{2}-192 x-105}{\left(x^{2}-2 x-35\right)(x+1)(x+2)}$

Exercise $\PageIndex{73}$

$\dfrac{5 a+3 b}{8 a^{2}+2 a b-b^{2}}-\dfrac{3 a-b}{4 a^{2}-9 a b+2 b^{2}}-\dfrac{a+5 b}{4 a^{2}+3 a b-b^{2}}$

Exercise $\PageIndex{74}$

$\dfrac{3 x^{2}+6 x+10}{10 x^{2}+11 x-6}+\dfrac{2 x^{2}-4 x+15}{2 x^{2}-11 x-21}$

Answer

$\dfrac{13 x^{3}-39 x^{2}+51 x-100}{(2 x+3)(x-7)(5 x-2)}$

Exercise $\PageIndex{75}$

$\dfrac{y^2 - 1}{y^2 + 9y + 20} \div \dfrac{y^2 + 5y - 6}{y^2 - 16}$

Rational Equations

For the following problems, solve the rational equations.

Exercise $\PageIndex{76}$

$\dfrac{4x}{5} + \dfrac{3x - 1}{15} = \dfrac{29}{25}$

Exercise $\PageIndex{77}$

$\dfrac{6a}{7} + \dfrac{2a - 3}{21} = \dfrac{77}{21}$

Answer

$a = 4$

Exercise $\PageIndex{78}$

$\dfrac{5x - 1}{6} + \dfrac{3x + 4}{9} = \dfrac{-8}{9}$

Exercise $\PageIndex{79}$

$\dfrac{4y - 5}{4} + \dfrac{8y + 1}{6} = \dfrac{-69}{12}$

Answer

$y=−2$

Exercise $\PageIndex{80}$

$\dfrac{4}{x-1} + \dfrac{7}{x+2} = \dfrac{43}{x^2 + x - 2}$

Exercise $\PageIndex{81}$

$\dfrac{5}{a + 3} + \dfrac{6}{a - 4} = \dfrac{9}{a^2 - a - 12}$

Answer

$a = 1$

Exercise $\PageIndex{82}$

$\dfrac{-5}{y - 3} + \dfrac{2}{y - 3} = \dfrac{3}{y - 3}$

Exercise $\PageIndex{83}$

$\dfrac{2m + 5}{m - 8} + \dfrac{9}{m - 8} = \dfrac{30}{m - 8}$

Answer

No solution; $m=8$ is excluded.

Exercise $\PageIndex{84}$

$\dfrac{r + 6}{r - 1} - \dfrac{3r + 2}{r - 1} = \dfrac{-6}{r - 1}$

Exercise $\PageIndex{85}$

$\dfrac{8b + 1}{b-7} - \dfrac{b + 5}{b - 7} = \dfrac{45}{b - 7}$

Answer

No solution; $b=7$ is excluded.

Exercise $\PageIndex{86}$

Solve $z = \dfrac{x - \hat{x}}{x}$ for $s$

Exercise $\PageIndex{87}$

Solve $A = P(1 + rt)$ for $t$.

Answer

$t = \dfrac{A-P}{Pr}$

Exercise $\PageIndex{88}$

Solve $\dfrac{1}{R} = \dfrac{1}{E} + \dfrac{1}{F}$ for $E$.

Exercise $\PageIndex{89}$

Solve $Q = \dfrac{2mn}{s + t}$ for $t$

Answer

$t = \dfrac{2mn - Qs}{Q}$

Exercise $\PageIndex{90}$

Solve $I = \dfrac{E}{R + r}$ for $r$

Applications

For the following problems, find the solution.

Exercise $\PageIndex{91}$

When the same number is subtracted from both terms of the fraction $\dfrac{7}{12}$, the result is $\dfrac{1}{2}$. What is the number?

Answer

$2$

Exercise $\PageIndex{92}$

When the same number is added to both terms of the fraction $\dfrac{13}{15}$, the result is $\dfrac{8}{9}$. What is the number?

Exercise $\PageIndex{93}$

When three-fourths of a number is added to the reciprocal of the number, the result is $\dfrac{173}{16}$. What is the number?

Answer

No rational solution.

Exercise $\PageIndex{94}$

When one-third of a number is added to the reciprocal of the number, the result is $\dfrac{127}{90}$. What is the number?

Exercise $\PageIndex{95}$

Person A working alone can complete a job in 9 hours. Person B working alone can complete the same job in 7 hours. How long will it take both people to complete the job working together?

Answer

$3\dfrac{15}{16}$ hrs.

Exercise $\PageIndex{96}$

Debbie can complete an algebra assignment in $\dfrac{3}{4}$ of an hour. Sandi, who plays her radio while working, can complete the same assignment in $1\dfrac{1}{4}$ hours. If Debbie and Sandi work together, how long will it take them to complete the assignment?

Exercise $\PageIndex{97}$

An inlet pipe can fill a tank in 6 hours and an outlet pipe can drain the tank in 8 hours. If both pipes are open, how long will it take to fill the tank?

Answer

24 hrs

Exercise $\PageIndex{98}$

Two pipes can fill a tank in 4 and 5 hours, respectively. How long will it take both pipes to fill the tank?

Exercise $\PageIndex{99}$

The pressure due to surface tension in a spherical bubble is given by $P = \dfrac{4T}{r}$, where $T$ is the surface tension of the liquid, and $r$ is the radius of the bubble.
(a) Determine the pressure due to surface tension within a soap bubble of radius $\dfrac{1}{2}$ inch and surface tension 22.
(b) Determine the radius of a bubble if the pressure due to surface tension is 57.6 and the surface tension is 18.

Answer

a) 176 units of pressure

b) $\dfrac{5}{4}$ units of length.

Exercise $\PageIndex{100}$

The equation $\dfrac{1}{p} + \dfrac{1}{q} = \dfrac{1}{f}$ relates an object's distance $p$ from a lens and the image distance $q$ from the lens to the focal length $f$ of the lens.
(a) Determine the focal length of a lens in which an object 8 feet away produces an image 6 feet away.
(b) Determine how far an object is from a lens if the focal length of the lens is 10 inches and the image distance is 10 inches.
(c) Determine how far an object will be from a lens that has a focal length of $\dfrac{7}{8}$ cm and the object distance is 3 cm away from the lens.

Dividing Polynomials

For the following problems, divide the polynomials.

Exercise $\PageIndex{101}$

$a^2 + 9a + 18$ by $a + 3$

Answer

$a+6$

Exercise $\PageIndex{102}$

$c^2 + 3c - 88$ by $c - 8$

Exercise $\PageIndex{103}$

$x^3 + 9x^2 + 18x + 28$ by $x + 7$.

Answer

$x^2 + 2x + 4$

Exercise $\PageIndex{104}$

$9y^3 - 2y^2 - 49y - 6$ by $y + 6$

Exercise $\PageIndex{105}$

$m^4 + 2m^3 - 8m^2 - m + 2$ by $m - 2$.

Answer

$m^3 + 4m^2 - 1$

Exercise $\PageIndex{106}$

$3r^2 - 17r - 27$ by $r - 7$

Exercise $\PageIndex{107}$

$a^3 - 3a^2 - 56a + 10$ by $a-9$.

Answer

$a^2 + 6a - 2 - \dfrac{8}{a-9}$

Exercise $\PageIndex{108}$

$x^3 - x + 1$ by $x + 3$

Exercise $\PageIndex{109}$

$y^3 + y^2 - y$ by $y + 4$

Answer

$y^2 - 3y + 11 - \dfrac{44}{y + 4}$

Exercise $\PageIndex{110}$

$5x^6 + 5x^5 - 2x^4 + 5x^3 - 7x^2 - 8x + 6$ by $x^2 + x - 1$

Exercise $\PageIndex{111}$

$y^{10} - y^7 + 3y^4 - 3y$ by $y^4 - y$

Answer

$y^6 + 3$

Exercise $\PageIndex{112}$

$-4b^7 - 3b^6 - 22b^5 - 19b^4 + 12b^3 - 6b^2 + b + 4$ by $b^2 + 6$.

Exercise $\PageIndex{113}$

$x^3 + 1$ by $x + 1$

Answer

$x^2 - x + 1$

Exercise $\PageIndex{114}$

$a^4 + 6a^3 + 4a^2 + 12a + 8$ by $a^2 + 3a + 2$

Exercise $\PageIndex{115}$

$y^{10} + 6y^5 + 9$ by $y^5 + 3$

Answer

$y^5 + 3$