Content check — Elementary Algebra, Chapter 8: Rational Expressions and Equations

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 8: Rational Expressions and Equations

529 problems · 28 figures

Be Prepared 8.1 §8.1 source ↗

Simplify:

\frac{90 y}{15 y^{2}} .

If you missed this problem, review Example 6.66.

Example 6.66 in the book ↗

Answer

\frac{6}{y}

Be Prepared 8.2 §8.1 source ↗

Factor:

6 x^{2} - 7 x + 2 .

If you missed this problem, review Example 7.34.

Example 7.34 in the book ↗

Answer

2 x - 1 3 x - 2

Be Prepared 8.3 §8.1 source ↗

Factor:

n^{3} + 8 .

If you missed this problem, review Example 7.54.

Example 7.54 in the book ↗

Answer

n + 2 n^{2} - 2 n + 4

Try It 8.1 §8.1 source ↗

Determine the values for which the rational expression is undefined:

ⓐ $\frac{3 y}{x}$ ⓑ $\frac{8 n - 5}{3 n + 1}$ ⓒ $\frac{a + 10}{a^{2} + 4 a + 3}$

Answer

ⓐ

x = 0

ⓑ

n = - \frac{1}{3}

ⓒ

a = −1, a = −3

Try It 8.2 §8.1 source ↗

Determine the values for which the rational expression is undefined:

ⓐ $\frac{4 p}{5 q}$ ⓑ $\frac{y - 1}{3 y + 2}$ ⓒ $\frac{m - 5}{m^{2} + m - 6}$

Answer

ⓐ

q = 0

ⓑ

y = - \frac{2}{3}

ⓒ

m = 2, m = −3

Try It 8.3 §8.1 source ↗

Evaluate $\frac{y + 1}{2 y - 3}$ for each value:

ⓐ $y = 1$ ⓑ $y = −3$ ⓒ $y = 0$

Answer

ⓐ

−2

ⓑ

\frac{2}{9}

ⓒ

- \frac{1}{3}

Try It 8.4 §8.1 source ↗

Evaluate $\frac{5 x - 1}{2 x + 1}$ for each value:

ⓐ $x = 1$ ⓑ $x = −1$ ⓒ $x = 0$

Answer

ⓐ

\frac{4}{3}

ⓑ

6

ⓒ

−1

Try It 8.5 §8.1 source ↗

Evaluate $\frac{x^{2} + 1}{x^{2} - 3 x + 2}$ for each value:

ⓐ $x = 0$ ⓑ $x = −1$ ⓒ $x = 3$

Answer

ⓐ

\frac{1}{2}

ⓑ

\frac{1}{3}

ⓒ 5

Try It 8.6 §8.1 source ↗

Evaluate $\frac{x^{2} + x - 6}{x^{2} - 9}$ for each value:

ⓐ $x = 0$ ⓑ $x = −2$ ⓒ $x = 1$

Answer

ⓐ

\frac{2}{3}

ⓑ

\frac{4}{5}

ⓒ

\frac{1}{2}

Try It 8.7 §8.1 source ↗

Evaluate $\frac{2 a^{3} b}{a^{2} + 2 a b + b^{2}}$ for each value:

ⓐ $a = −1, b = 2$ ⓑ $a = 0, b = −1$ ⓒ $a = 1, b = \frac{1}{2}$

Answer

ⓐ

−4

ⓑ

0

ⓒ

\frac{4}{9}

Try It 8.8 §8.1 source ↗

Evaluate $\frac{a^{2} - b^{2}}{8 a b^{3}}$ for each value:

ⓐ $a = 1, b = −1$ ⓑ $a = \frac{1}{2}, b = −1$ ⓒ $a = −2, b = 1$

Answer

ⓐ

0

ⓑ

\frac{3}{16}

ⓒ

- \frac{3}{16}

Try It 8.9 §8.1 source ↗

Simplify:

- \frac{45}{81} .

Answer

- \frac{5}{9} .

Try It 8.10 §8.1 source ↗

Simplify:

- \frac{42}{54} .

Answer

- \frac{7}{9}

Try It 8.11 §8.1 source ↗

Simplify:

\frac{4 x^{2} y}{12 x y^{2}} .

Answer

\frac{x}{3 y}

Try It 8.12 §8.1 source ↗

Simplify:

\frac{16 x^{2} y}{2 x y^{2}} .

Answer

\frac{8 x}{y}

Try It 8.13 §8.1 source ↗

Simplify:

\frac{3 x - 6}{2 x - 4} .

Answer

\frac{3}{2}

Try It 8.14 §8.1 source ↗

Simplify:

\frac{7 y + 35}{5 y + 25} .

Answer

\frac{7}{5}

Try It 8.15 §8.1 source ↗

Simplify:

\frac{x^{2} - x - 2}{x^{2} - 3 x + 2} .

Answer

\frac{x + 1}{x - 1}

Try It 8.16 §8.1 source ↗

Simplify:

\frac{x^{2} - 3 x - 10}{x^{2} + x - 2} .

Answer

\frac{x - 5}{x - 1}

Try It 8.17 §8.1 source ↗

Simplify:

\frac{x^{2} + x - 6}{x^{2} - 4} .

Answer

\frac{x + 3}{x + 2}

Try It 8.18 §8.1 source ↗

Simplify:

\frac{x^{2} + 8 x + 7}{x^{2} - 49} .

Answer

\frac{x + 1}{x - 7}

Try It 8.19 §8.1 source ↗

Simplify:

\frac{y^{3} - 3 y^{2} + y - 3}{y^{2} - y - 6} .

Answer

\frac{y^{2} + 1}{y + 2}

Try It 8.20 §8.1 source ↗

Simplify:

\frac{p^{3} - p^{2} + 2 p - 2}{p^{2} + 4 p - 5} .

Answer

\frac{p^{2} + 2}{p + 5}

Try It 8.21 §8.1 source ↗

Simplify:

\frac{2 n^{2} - 10 n}{4 n^{2} - 16 n - 20} .

Answer

\frac{n}{2 (n + 1)}

Try It 8.22 §8.1 source ↗

Simplify:

\frac{4 x^{2} - 16 x}{8 x^{2} - 16 x - 64} .

Answer

\frac{x}{2 (x + 2)}

Try It 8.23 §8.1 source ↗

Simplify:

\frac{2 x^{2} - 12 x + 18}{3 x^{2} - 27} .

Answer

\frac{2 (x - 3)}{3 (x + 3)}

Try It 8.24 §8.1 source ↗

Simplify:

\frac{5 y^{2} - 30 y + 25}{2 y^{2} - 50} .

Answer

\frac{5 (y - 1)}{2 (y + 5)}

Try It 8.25 §8.1 source ↗

Simplify:

\frac{p^{3} - 64}{p^{2} - 16} .

Answer

\frac{p^{2} + 4 p + 16}{p + 4}

Try It 8.26 §8.1 source ↗

Simplify:

\frac{x^{3} + 8}{x^{2} - 4} .

Answer

\frac{x^{2} - 2 x + 4}{x - 2}

Try It 8.27 §8.1 source ↗

Simplify:

\frac{y - 2}{2 - y} .

Answer

−1

Try It 8.28 §8.1 source ↗

Simplify:

\frac{n - 9}{9 - n} .

Answer

−1

Try It 8.29 §8.1 source ↗

Simplify:

\frac{10 - 2 y}{y^{2} - 25} .

Answer

- \frac{2}{y + 5} .

Try It 8.30 §8.1 source ↗

Simplify:

\frac{3 y - 27}{81 - y^{2}} .

Answer

- \frac{3}{9 + y}

Try It 8.31 §8.1 source ↗

Simplify:

\frac{x^{2} - 4 x - 5}{25 - x^{2}} .

Answer

- \frac{x + 1}{x + 5}

Try It 8.32 §8.1 source ↗

Simplify:

\frac{x^{2} + x - 2}{1 - x^{2}} .

Answer

- \frac{x + 2}{x + 1}

8.1-1 §8.1 source ↗

In the following exercises, determine the values for which the rational expression is undefined.

ⓐ $\frac{2 x}{z}$ ⓑ $\frac{4 p - 1}{6 p - 5}$ ⓒ $\frac{n - 3}{n^{2} + 2 n - 8}$

Answer

ⓐ

z = 0

ⓑ

p = \frac{5}{6}

ⓒ

n = −4, n = 2

8.1-3 §8.1 source ↗

In the following exercises, determine the values for which the rational expression is undefined.

ⓐ $\frac{4 x^{2} y}{3 y}$ ⓑ $\frac{3 x - 2}{2 x + 1}$ ⓒ $\frac{u - 1}{u^{2} - 3 u - 28}$

Answer

ⓐ

y = 0

ⓑ

x = - \frac{1}{2}

ⓒ

u = −4, u = 7

8.1-5 §8.1 source ↗

Evaluate Rational Expressions

In the following exercises, evaluate the rational expression for the given values.

$\frac{2 x}{x - 1}$

ⓐ $x = 0$ ⓑ $x = 2$ ⓒ $x = −1$

Answer

ⓐ

0

ⓑ

4

ⓒ

1

8.1-7 §8.1 source ↗

Evaluate Rational Expressions

In the following exercises, evaluate the rational expression for the given values.

$\frac{2 p + 3}{p^{2} + 1}$

ⓐ $p = 0$ ⓑ $p = 1$ ⓒ $p = −2$

Answer

ⓐ

3

ⓑ

\frac{5}{2}

ⓒ

- \frac{1}{5}

8.1-9 §8.1 source ↗

Evaluate Rational Expressions

In the following exercises, evaluate the rational expression for the given values.

$\frac{y^{2} + 5 y + 6}{y^{2} - 1}$

ⓐ $y = 0$ ⓑ $y = 2$ ⓒ $y = −2$

Answer

ⓐ

−6

ⓑ

\frac{20}{3}

ⓒ

0

8.1-11 §8.1 source ↗

Evaluate Rational Expressions

In the following exercises, evaluate the rational expression for the given values.

$\frac{a^{2} - 4}{a^{2} + 5 a + 4}$

ⓐ $a = 0$ ⓑ $a = 1$ ⓒ $a = −2$

Answer

ⓐ

−1

ⓑ

- \frac{3}{10}

ⓒ

0

8.1-13 §8.1 source ↗

Evaluate Rational Expressions

In the following exercises, evaluate the rational expression for the given values.

$\frac{x^{2} + 3 x y + 2 y^{2}}{2 x^{3} y}$

ⓐ $x = 1, y = −1$

ⓑ $x = 2, y = 1$

Answer

ⓐ

0

ⓑ

\frac{3}{4}

ⓒ

\frac{15}{4}

8.1-15 §8.1 source ↗

Evaluate Rational Expressions

In the following exercises, evaluate the rational expression for the given values.

$\frac{m^{2} - 4 n^{2}}{5 m n^{3}}$

ⓐ $m = 2, n = 1$

ⓑ $m = −1, n = −1$

Answer

ⓐ

0

ⓑ

- \frac{3}{5}

ⓒ

- \frac{7}{120}

8.1-17 §8.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify.

$- \frac{4}{52}$

Answer

- \frac{1}{13}

8.1-19 §8.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify.

$\frac{56}{63}$

Answer

\frac{8}{9}

8.1-21 §8.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify.

$\frac{6 a b^{2}}{12 a^{2} b}$

Answer

\frac{b}{2 a}

8.1-23 §8.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify.

$\frac{8 m^{3} n}{12 m n^{2}}$

Answer

\frac{2 m^{2}}{3 n}

8.1-25 §8.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify.

$\frac{3 a + 6}{4 a + 8}$

Answer

\frac{3}{4}

8.1-27 §8.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify.

$\frac{3 c - 9}{5 c - 15}$

Answer

\frac{3}{5}

8.1-29 §8.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify.

$\frac{7 m + 63}{5 m + 45}$

Answer

\frac{7}{5}

8.1-31 §8.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify.

$\frac{12 p - 240}{5 p - 100}$

Answer

\frac{12}{5}

8.1-33 §8.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify.

$\frac{a^{2} - a - 12}{a^{2} - 8 a + 16}$

Answer

\frac{a + 3}{a - 4}

8.1-35 §8.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify.

$\frac{y^{2} + 3 y - 4}{y^{2} - 6 y + 5}$

Answer

\frac{y + 4}{y - 5}

8.1-37 §8.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify.

$\frac{x^{2} - 25}{x^{2} + 2 x - 15}$

Answer

\frac{x - 5}{x - 3}

8.1-39 §8.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify.

$\frac{y^{2} - 2 y - 3}{y^{2} - 9}$

Answer

\frac{y + 1}{y + 3}

8.1-41 §8.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify.

$\frac{y^{3} + y^{2} + y + 1}{y^{2} + 2 y + 1}$

Answer

\frac{y^{2} + 1}{y + 1}

8.1-43 §8.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify.

$\frac{x^{3} - 2 x^{2} - 25 x + 50}{x^{2} - 25}$

Answer

x - 2

8.1-45 §8.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify.

$\frac{3 a^{2} + 15 a}{6 a^{2} + 6 a - 36}$

Answer

\frac{a (a + 5)}{2 (a + 3) (a - 2)}

8.1-47 §8.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify.

$\frac{−5 c^{2} - 10 c}{−10 c^{2} + 30 c + 100}$

Answer

\frac{c}{2 (c - 5)}

8.1-49 §8.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify.

$\frac{3 m^{2} + 30 m + 75}{4 m^{2} - 100}$

Answer

\frac{3 (m + 5)}{4 (m - 5)}

8.1-51 §8.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify.

$\frac{5 r^{2} + 30 r - 35}{r^{2} - 49}$

Answer

\frac{5 (r - 1)}{r - 7}

8.1-53 §8.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify.

$\frac{t^{3} - 27}{t^{2} - 9}$

Answer

\frac{t^{2} + 3 t + 9}{t + 3}

8.1-55 §8.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify.

$\frac{w^{3} + 216}{w^{2} - 36}$

Answer

\frac{w^{2} - 6 w + 36}{w - 6}

8.1-57 §8.1 source ↗

Simplify Rational Expressions with Opposite Factors

In the following exercises, simplify each rational expression.

$\frac{a - 5}{5 - a}$

Answer

−1

8.1-59 §8.1 source ↗

Simplify Rational Expressions with Opposite Factors

In the following exercises, simplify each rational expression.

$\frac{11 - c}{c - 11}$

Answer

−1

8.1-61 §8.1 source ↗

Simplify Rational Expressions with Opposite Factors

In the following exercises, simplify each rational expression.

$\frac{12 - 2 x}{x^{2} - 36}$

Answer

- \frac{2}{x + 6}

8.1-63 §8.1 source ↗

Simplify Rational Expressions with Opposite Factors

In the following exercises, simplify each rational expression.

$\frac{4 v - 32}{64 - v^{2}}$

Answer

- \frac{4}{8 + v}

8.1-65 §8.1 source ↗

Simplify Rational Expressions with Opposite Factors

In the following exercises, simplify each rational expression.

$\frac{y^{2} - 11 y + 24}{9 - y^{2}}$

Answer

- \frac{(y - 8)}{3 + y}

8.1-67 §8.1 source ↗

Simplify Rational Expressions with Opposite Factors

In the following exercises, simplify each rational expression.

$\frac{a^{2} - 5 a - 36}{81 - a^{2}}$

Answer

- \frac{a + 4}{9 + a}

8.1-69 §8.1 source ↗

Simplify Rational Expressions with Opposite Factors

In the following exercises, simplify each rational expression.

Tax Rates For the tax year 2015, the amount of tax owed by a single person earning between $37,450 and $90,750, can be found by evaluating the formula $0.25 x -$ $4206.25,$ where x is income. The average tax rate for this income can be found by evaluating the formula $\frac{0.25 x - 4206.25}{x} .$ What would be the average tax rate for a single person earning $50,000?

Answer

16.6%

8.1-71 §8.1 source ↗

Simplify Rational Expressions with Opposite Factors

In the following exercises, simplify each rational expression.

Explain how you find the values of x for which the rational expression $\frac{x^{2} - x - 20}{x^{2} - 4}$ is undefined.

Answer

Answers will vary, but all should reference setting the denominator function to zero.

Be Prepared 8.4 §8.2 source ↗

Multiply:

\frac{14}{15} \cdot \frac{6}{35} .

If you missed this problem, review Example 1.68.

Example 1.68 in the book ↗

Answer

\frac{4}{25}

Be Prepared 8.5 §8.2 source ↗

Divide:

\frac{14}{15} \div \frac{6}{35} .

If you missed this problem, review Example 1.71.

Example 1.71 in the book ↗

Answer

\frac{49}{9}

Be Prepared 8.6 §8.2 source ↗

Factor completely:

2 x^{2} - 98 .

If you missed this problem, review Example 7.62.

Example 7.62 in the book ↗

Answer

2 x - 7 x + 7

Be Prepared 8.7 §8.2 source ↗

Factor completely:

10 n^{3} + 10 .

If you missed this problem, review Example 7.65.

Example 7.65 in the book ↗

Answer

10 n + 1 n^{2} - n + 1

Be Prepared 8.8 §8.2 source ↗

Factor completely:

10 p^{2} -

25 p q -

15 q^{2} .

If you missed this problem, review Example 7.68.

Example 7.68 in the book ↗

Answer

5 2 p + q p - 3 q

Try It 8.33 §8.2 source ↗

Mulitply:

\frac{6}{10} \cdot \frac{15}{12} .

Answer

\frac{3}{4}

Try It 8.34 §8.2 source ↗

Mulitply:

\frac{20}{15} \cdot \frac{6}{8} .

Answer

1

Try It 8.35 §8.2 source ↗

Mulitply:

\frac{3 p q}{q^{2}} \cdot \frac{5 p^{2} q}{6 p q} .

Answer

\frac{5 p^{2}}{2 q}

Try It 8.36 §8.2 source ↗

Mulitply:

\frac{6 x^{3} y}{7 x^{2}} \cdot \frac{2 x y^{3}}{x^{2} y} .

Answer

\frac{12 y^{3}}{7}

Try It 8.37 §8.2 source ↗

Mulitply:

\frac{5 x}{x^{2} + 5 x + 6} \cdot \frac{x^{2} - 4}{10 x} .

Answer

\frac{x - 2}{2 (x + 3)}

Try It 8.38 §8.2 source ↗

Mulitply:

\frac{9 x^{2}}{x^{2} + 11 x + 30} \cdot \frac{x^{2} - 36}{3 x^{2}} .

Answer

\frac{3 (x - 6)}{x + 5}

Try It 8.39 §8.2 source ↗

Multiply:

\frac{x^{2} - 25}{x^{2} - 3 x - 10} \cdot \frac{x + 2}{x} .

Answer

\frac{x + 5}{x}

Try It 8.40 §8.2 source ↗

Multiply:

\frac{x^{2} - 4 x}{x^{2} + 5 x + 6} \cdot \frac{x + 2}{x} .

Answer

\frac{x - 4}{x + 3}

Try It 8.41 §8.2 source ↗

Multiply:

\frac{12 x - 6 x^{2}}{x^{2} + 8 x} \cdot \frac{x^{2} + 11 x + 24}{x^{2} - 4} .

Answer

- \frac{6 (x + 3)}{x + 2}

Try It 8.42 §8.2 source ↗

Multiply:

\frac{9 v - 3 v^{2}}{9 v + 36} \cdot \frac{v^{2} + 7 v + 12}{v^{2} - 9} .

Answer

- \frac{v}{3}

Try It 8.43 §8.2 source ↗

Multiply:

\frac{3 a - 21}{a^{2} - 9 a + 14} \cdot \frac{a^{2} - 4}{3 a + 6} .

Answer

1

Try It 8.44 §8.2 source ↗

Multiply:

\frac{b^{2} - b}{b^{2} + 9 b - 10} \cdot \frac{b^{2} - 100}{b^{2} - 10 b} .

Answer

Try It 8.45 §8.2 source ↗

Divide:

\frac{c + 3}{5 - c} \div \frac{c^{2} - 9}{c - 5} .

Answer

- \frac{1}{c - 3}

Try It 8.46 §8.2 source ↗

Divide:

\frac{2 - d}{d - 4} \div \frac{4 - d^{2}}{4 - d} .

Answer

- \frac{1}{2 + d}

Try It 8.47 §8.2 source ↗

Divide:

\frac{2 m^{2}}{m^{2} - 8 m} \div \frac{8 m^{2} + 24 m}{m^{2} + m - 6} .

Answer

\frac{(m - 2)}{4 (m - 8)}

Try It 8.48 §8.2 source ↗

Divide:

\frac{15 n^{2}}{3 n^{2} + 33 n} \div \frac{5 n - 5}{n^{2} + 9 n - 22} .

Answer

\frac{n (n - 2)}{n - 1}

Try It 8.49 §8.2 source ↗

Divide:

\frac{3 a^{2} - 8 a - 3}{a^{2} - 25} \div \frac{3 a^{2} - 14 a - 5}{a^{2} + 10 a + 25} .

Answer

\frac{(a - 3) (a + 5)}{(a - 5) (a - 5)}

Try It 8.50 §8.2 source ↗

Divide:

\frac{4 b^{2} + 7 b - 2}{1 - b^{2}} \div \frac{4 b^{2} + 15 b - 4}{b^{2} - 2 b + 1} .

Answer

-

\frac{(b + 2) (b - 1)}{(1 + b) (b + 4)}

Try It 8.51 §8.2 source ↗

Divide:

\frac{x^{3} - 8}{3 x^{2} - 6 x + 12} \div \frac{x^{2} - 4}{6} .

Answer

\frac{2 (x^{2} + 2 x + 4)}{(x + 2) (x^{2} - 2 x + 4)}

Try It 8.52 §8.2 source ↗

Divide:

\frac{2 z^{2}}{z^{2} - 1} \div \frac{z^{3} - z^{2} + z}{z^{3} - 1} .

Answer

\frac{2 z (z^{2} + z + 1)}{(z + 1) (z^{2} - z + 1)}

Try It 8.53 §8.2 source ↗

Divide:

\frac{2 x^{2} - 14 x - 16}{4} \div (x^{2} + 2 x + 1) .

Answer

\frac{x - 8}{2 (x + 1)}

Try It 8.54 §8.2 source ↗

Divide:

\frac{y^{2} - 6 y + 8}{y^{2} - 4 y} \div (3 y^{2} - 12 y) .

Answer

\frac{y - 2}{3 y^{2} (y - 4)}

Try It 8.55 §8.2 source ↗

Divide:

\frac{\frac{3 x^{2} + 7 x + 2}{4 x + 24}}{\frac{3 x^{2} - 14 x - 5}{x^{2} + x - 30}} .

Answer

\frac{x + 2}{4}

Try It 8.56 §8.2 source ↗

Divide:

\frac{\frac{y^{2} - 36}{2 y^{2} + 11 y - 6}}{\frac{2 y^{2} - 2 y - 60}{8 y - 4}} .

Answer

\frac{2}{y + 5}

Try It 8.57 §8.2 source ↗

Divide:

\frac{4 m + 4}{3 m - 15} \cdot \frac{m^{2} - 3 m - 10}{m^{2} - 4 m - 32} \div \frac{12 m - 36}{6 m - 48} .

Answer

\frac{2 (m + 1) (m + 2)}{3 (m + 4) (m - 3)}

Try It 8.58 §8.2 source ↗

Divide:

\frac{2 n^{2} + 10 n}{n - 1} \div \frac{n^{2} + 10 n + 24}{n^{2} + 8 n - 9} \cdot \frac{n + 4}{8 n^{2} + 12 n} .

Answer

\frac{(n + 5) (n + 9)}{2 (n + 6) (2 n + 3)}

8.2-73 §8.2 source ↗

Multiply Rational Expressions

In the following exercises, multiply.

$\frac{12}{16} \cdot \frac{4}{10}$

Answer

\frac{3}{10}

8.2-75 §8.2 source ↗

Multiply Rational Expressions

In the following exercises, multiply.

$\frac{18}{10} \cdot \frac{4}{30}$

Answer

\frac{6}{25}

8.2-77 §8.2 source ↗

Multiply Rational Expressions

In the following exercises, multiply.

$\frac{5 x^{2} y^{4}}{12 x y^{3}} \cdot \frac{6 x^{2}}{20 y^{2}}$

Answer

\frac{x^{3}}{8 y}

8.2-79 §8.2 source ↗

Multiply Rational Expressions

In the following exercises, multiply.

$\frac{12 a^{3} b}{b^{2}} \cdot \frac{2 a b^{2}}{9 b^{3}}$

Answer

\frac{8 a^{4}}{3 b^{2}}

8.2-81 §8.2 source ↗

Multiply Rational Expressions

In the following exercises, multiply.

$\frac{5 p^{2}}{p^{2} - 5 p - 36} \cdot \frac{p^{2} - 16}{10 p}$

Answer

\frac{p (p - 4)}{2 (p - 9)}

8.2-83 §8.2 source ↗

Multiply Rational Expressions

In the following exercises, multiply.

$\frac{4 r}{r^{2} - 3 r - 10} \cdot \frac{r^{2} - 25}{8 r^{2}}$

Answer

\frac{r + 5}{2 r (r + 2)}

8.2-85 §8.2 source ↗

Multiply Rational Expressions

In the following exercises, multiply.

$\frac{x^{2} - 7 x}{x^{2} + 6 x + 9} \cdot \frac{x + 3}{4 x}$

Answer

\frac{x - 7}{4 (x + 3)}

8.2-87 §8.2 source ↗

Multiply Rational Expressions

In the following exercises, multiply.

$\frac{z^{2} + 3 z}{z^{2} - 3 z - 4} \cdot \frac{z - 4}{z^{2}}$

Answer

\frac{z + 3}{z (z + 1)}

8.2-89 §8.2 source ↗

Multiply Rational Expressions

In the following exercises, multiply.

$\frac{28 - 4 b}{3 b - 3} \cdot \frac{b^{2} + 8 b - 9}{b^{2} - 49}$

Answer

-

\frac{4 (b + 9)}{3 (b + 7)}

8.2-91 §8.2 source ↗

Multiply Rational Expressions

In the following exercises, multiply.

$\frac{35 d - 7 d^{2}}{d^{2} + 7 d} \cdot \frac{d^{2} + 12 d + 35}{d^{2} - 25}$

Answer

−7

8.2-93 §8.2 source ↗

Multiply Rational Expressions

In the following exercises, multiply.

$\frac{4 n + 20}{n^{2} + n - 20} \cdot \frac{n^{2} - 16}{4 n + 16}$

Answer

1

8.2-95 §8.2 source ↗

Multiply Rational Expressions

In the following exercises, multiply.

$\frac{q^{2} - 2 q}{q^{2} + 6 q - 16} \cdot \frac{q^{2} - 64}{q^{2} - 8 q}$

Answer

1

8.2-97 §8.2 source ↗

Divide Rational Expressions

In the following exercises, divide.

$\frac{t - 6}{3 - t} \div \frac{t - 5}{t^{2} - 9}$

Answer

\frac{6 - t t + 3}{t - 5}

8.2-99 §8.2 source ↗

Divide Rational Expressions

In the following exercises, divide.

$\frac{10 + w}{w - 8} \div \frac{100 - w^{2}}{8 - w}$

Answer

- \frac{1}{10 - w}

8.2-101 §8.2 source ↗

Divide Rational Expressions

In the following exercises, divide.

$\frac{27 y^{2}}{3 y - 21} \div \frac{3 y^{2} + 18}{y^{2} + 13 y + 42}$

Answer

\frac{3 y^{2} (y + 6) (y + 7)}{(y - 7) (y^{2} + 6)}

8.2-103 §8.2 source ↗

Divide Rational Expressions

In the following exercises, divide.

$\frac{16 a^{2}}{4 a + 36} \div \frac{4 a^{2} - 24 a}{a^{2} + 4 a - 45}$

Answer

\frac{a (a - 5)}{a - 6}

8.2-105 §8.2 source ↗

Divide Rational Expressions

In the following exercises, divide.

$\frac{5 c^{2} + 9 c - 2}{c^{2} - 4} \div \frac{5 c^{2} - 16 c + 3}{c^{2} + 4 c + 4}$

Answer

\frac{(c + 2) (c + 2)}{(c - 2) (c - 3)}

8.2-107 §8.2 source ↗

Divide Rational Expressions

In the following exercises, divide.

$\frac{6 m^{2} - 11 m - 2}{9 - m^{2}} \div \frac{6 m^{2} + 25 m + 4}{m^{2} - 6 m + 9}$

Answer

-

\frac{(m - 2) (m - 3)}{(3 + m) (m + 4)}

8.2-109 §8.2 source ↗

Divide Rational Expressions

In the following exercises, divide.

$\frac{3 s^{2}}{s^{2} - 16} \div \frac{s^{3} + 4 s^{2} + 16 s}{s^{3} - 64}$

Answer

\frac{3 s}{s + 4}

8.2-111 §8.2 source ↗

Divide Rational Expressions

In the following exercises, divide.

$\frac{p^{3} + q^{3}}{3 p^{2} + 3 p q + 3 q^{2}} \div \frac{p^{2} - q^{2}}{12}$

Answer

\frac{4 (p^{2} - p q + q^{2})}{(p - q) (p^{2} + p q + q^{2})}

8.2-113 §8.2 source ↗

Divide Rational Expressions

In the following exercises, divide.

$\frac{t^{2} - 9}{2 t} \div (t^{2} - 6 t + 9)$

Answer

\frac{t + 3}{2 t (t - 3)}

8.2-115 §8.2 source ↗

Divide Rational Expressions

In the following exercises, divide.

$\frac{2 y^{2} - 10 y z - 48 z^{2}}{2 y - 1} \div (4 y^{2} - 32 y z)$

Answer

\frac{y + 3 z}{2 y (2 y - 1)}

8.2-117 §8.2 source ↗

Divide Rational Expressions

In the following exercises, divide.

$\frac{\frac{2 a^{2} - a - 21}{5 a + 20}}{\frac{a^{2} + 7 a + 12}{a^{2} + 8 a + 16}}$

Answer

\frac{2 a - 7}{5}

8.2-119 §8.2 source ↗

Divide Rational Expressions

In the following exercises, divide.

$\frac{\frac{12 c^{2} - 12}{2 c^{2} - 3 c + 1}}{\frac{4 c + 4}{6 c^{2} - 13 c + 5}}$

Answer

3 (3 c - 5)

8.2-121 §8.2 source ↗

Divide Rational Expressions

In the following exercises, divide.

$\frac{10 m^{2} + 80 m}{3 m - 9} \cdot \frac{m^{2} + 4 m - 21}{m^{2} - 9 m + 20}$
$\div \frac{5 m^{2} + 10 m}{2 m - 10}$

Answer

\frac{4 (m + 8) (m + 7)}{3 (m - 4) (m + 2)}

8.2-123 §8.2 source ↗

Divide Rational Expressions

In the following exercises, divide.

$\frac{12 p^{2} + 3 p}{p + 3} \div \frac{p^{2} + 2 p - 63}{p^{2} - p - 12}$
$\cdot \frac{p - 7}{9 p^{3} - 9 p^{2}}$

Answer

\frac{(4 p + 1) (p - 4)}{3 p (p + 9) (p - 1)}

8.2-125 §8.2 source ↗

Divide Rational Expressions

In the following exercises, divide.

Probability The director of large company is interviewing applicants for two identical jobs. If $w =$ the number of women applicants and $m =$ the number of men applicants, then the probability that two women are selected for the jobs is $\frac{w}{w + m} \cdot \frac{w - 1}{w + m - 1} .$

ⓐ Simplify the probability by multiplying the two rational expressions.

ⓑ Find the probability that two women are selected when $w = 5$ and $m = 10$ .

Answer

ⓐ

\frac{w (w - 1)}{(w + m) (w + m - 1)}

ⓑ

\frac{2}{21}

8.2-127 §8.2 source ↗

Divide Rational Expressions

In the following exercises, divide.

ⓐ Multiply $\frac{7}{4} \cdot \frac{9}{10}$ and explain all your steps.

ⓑ Multiply $\frac{n}{n - 3} \cdot \frac{9}{n + 3}$ and explain all your steps.

Answer

Answers will vary.

Be Prepared 8.9 §8.3 source ↗

Add:

\frac{y}{3} + \frac{9}{3} .

If you missed this problem, review Example 1.77.

Example 1.77 in the book ↗

Answer

\frac{y + 9}{3}

Be Prepared 8.10 §8.3 source ↗

Subtract:

\frac{10}{x} - \frac{2}{x} .

If you missed this problem, review Example 1.79.

Example 1.79 in the book ↗

Answer

\frac{8}{x}

Be Prepared 8.12 §8.3 source ↗

Factor completely:

45 a^{3} - 5 a b^{2} .

If you missed this problem, review Example 7.62.

Example 7.62 in the book ↗

Answer

5 a 3 a - b 3 a + b

Try It 8.59 §8.3 source ↗

Add:

\frac{7}{16} + \frac{5}{16} .

Answer

\frac{3}{4}

Try It 8.60 §8.3 source ↗

Add:

\frac{3}{10} + \frac{1}{10} .

Answer

\frac{2}{5}

Try It 8.61 §8.3 source ↗

Add:

\frac{5 x}{2 x + 3} +

\frac{2}{2 x + 3} .

Answer

\frac{5 x + 2}{2 x + 3} .

Try It 8.62 §8.3 source ↗

Add:

\frac{x}{x - 2} + \frac{1}{x - 2} .

Answer

\frac{x + 1}{x - 2}

Try It 8.63 §8.3 source ↗

Add:

\frac{9 x + 14}{x + 7} +

\frac{x^{2}}{x + 7} .

Answer

x + 2

Try It 8.64 §8.3 source ↗

Add:

\frac{x^{2} + 8 x}{x + 5} +

\frac{15}{x + 5} .

Answer

x + 3

Try It 8.65 §8.3 source ↗

Subtract:

\frac{x^{2}}{x + 3} - \frac{9}{x + 3} .

Answer

x - 3

Try It 8.66 §8.3 source ↗

Subtract:

\frac{4 x^{2}}{2 x - 5} -

\frac{25}{2 x - 5} .

Answer

2 x + 5

Try It 8.67 §8.3 source ↗

Subtract:

\frac{n^{2}}{n - 4} -

\frac{n + 12}{n - 4} .

Answer

n + 3

Try It 8.68 §8.3 source ↗

Subtract:

\frac{y^{2}}{y - 1} -

\frac{9 y - 8}{y - 1} .

Answer

y - 8

Try It 8.69 §8.3 source ↗

Subtract:

\frac{4 x^{2} - 11 x + 8}{x^{2} - 3 x + 2} -

\frac{3 x^{2} + x - 3}{x^{2} - 3 x + 2} .

Answer

\frac{x - 11}{x - 2}

Try It 8.70 §8.3 source ↗

Subtract:

\frac{6 x^{2} - x + 20}{x^{2} - 81} -

\frac{5 x^{2} + 11 x - 7}{x^{2} - 81} .

Answer

\frac{x - 3}{x + 9}

Try It 8.71 §8.3 source ↗

Add:

\frac{8 x - 15}{2 x - 5} +

\frac{2 x}{5 - 2 x} .

Answer

3

Try It 8.72 §8.3 source ↗

Add:

\frac{6 y^{2} + 7 y - 10}{4 y - 7} +

\frac{2 y^{2} + 2 y + 11}{7 - 4 y} .

Answer

y + 3

Try It 8.73 §8.3 source ↗

Subtract:

\frac{y^{2} - 5 y}{y^{2} - 4} -

\frac{6 y - 6}{4 - y^{2}} .

Answer

\frac{y + 3}{y + 2}

Try It 8.74 §8.3 source ↗

Subtract:

\frac{2 n^{2} + 8 n - 1}{n^{2} - 1} -

\frac{n^{2} - 7 n - 1}{1 - n^{2}} .

Answer

\frac{3 n - 2}{n - 1}

8.3-129 §8.3 source ↗

Add Rational Expressions with a Common Denominator

In the following exercises, add.

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

Answer

\frac{3}{5}

8.3-131 §8.3 source ↗

Add Rational Expressions with a Common Denominator

In the following exercises, add.

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

Answer

\frac{3}{4}

8.3-133 §8.3 source ↗

Add Rational Expressions with a Common Denominator

In the following exercises, add.

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

Answer

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

8.3-135 §8.3 source ↗

Add Rational Expressions with a Common Denominator

In the following exercises, add.

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

Answer

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

8.3-137 §8.3 source ↗

Add Rational Expressions with a Common Denominator

In the following exercises, add.

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

Answer

p + 8

8.3-139 §8.3 source ↗

Add Rational Expressions with a Common Denominator

In the following exercises, add.

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

Answer

r + 8

8.3-141 §8.3 source ↗

Add Rational Expressions with a Common Denominator

In the following exercises, add.

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

Answer

8 t

8.3-143 §8.3 source ↗

Add Rational Expressions with a Common Denominator

In the following exercises, add.

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

Answer

\frac{2 w}{w - 4}

8.3-145 §8.3 source ↗

Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

$\frac{y^{2}}{y + 8} - \frac{64}{y + 8}$

Answer

y - 8

8.3-147 §8.3 source ↗

Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

$\frac{9 a^{2}}{3 a - 7} -$ $\frac{49}{3 a - 7}$

Answer

3 a + 7

8.3-149 §8.3 source ↗

Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

$\frac{c^{2}}{c - 8} -$ $\frac{6 c + 16}{c - 8}$

Answer

c + 2

8.3-151 §8.3 source ↗

Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

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

Answer

\frac{m - 2}{2}

8.3-153 §8.3 source ↗

Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

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

Answer

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

8.3-155 §8.3 source ↗

Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

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

Answer

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

8.3-157 §8.3 source ↗

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add.

$\frac{10 v}{2 v - 1} +$ $\frac{2 v + 4}{1 - 2 v}$

Answer

4

8.3-159 §8.3 source ↗

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add.

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

Answer

x + 2

8.3-161 §8.3 source ↗

In the following exercises, subtract.

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

Answer

\frac{z + 4}{z - 5}

8.3-163 §8.3 source ↗

In the following exercises, subtract.

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

Answer

\frac{4 b - 3}{b - 7}

8.3-165 §8.3 source ↗

In the following exercises, subtract.

Sarah ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. If $r$ 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{32 (r + 1)}{r (r + 4)}

Be Prepared 8.13 §8.4 source ↗

Add:

\frac{7}{10} + \frac{8}{15} .

If you missed this problem, review Example 1.81.

Example 1.81 in the book ↗

Answer

\frac{37}{30}

Be Prepared 8.14 §8.4 source ↗

Subtract:

6 (2 x + 1) -

4 (x - 5) .

If you missed this problem, review Example 1.139.

Example 1.139 in the book ↗

Answer

8 x + 26

Be Prepared 8.15 §8.4 source ↗

Find the Greatest Common Factor of

9 x^{2} y^{3}

and

12 x y^{5}

.
If you missed this problem, review Example 7.3.

Example 7.3 in the book ↗

Answer

3 x y^{3}

Be Prepared 8.16 §8.4 source ↗

Factor completely

−48 n - 12

.
If you missed this problem, review Example 7.11.

Example 7.11 in the book ↗

Answer

- 12 4 n + 1

Try It 8.75 §8.4 source ↗

Find the LCD for

\frac{2}{x^{2} - x - 12},

\frac{1}{x^{2} - 16}

Answer

(x - 4) (x + 4) (x + 3)

Try It 8.76 §8.4 source ↗

Find the LCD for

\frac{x}{x^{2} + 8 x + 15},

\frac{5}{x^{2} + 9 x + 18}

Answer

(x + 3) (x + 6) (x + 5)

Try It 8.77 §8.4 source ↗

Rewrite as equivalent rational expressions with denominator

(x + 3) (x - 4) (x + 4)

\frac{2}{x^{2} - x - 12},

\frac{1}{x^{2} - 16} .

Answer

\frac{2 x + 8}{(x - 4) (x + 3) (x + 4)},

\frac{x + 3}{(x - 4) (x + 3) (x + 4)}

Try It 8.78 §8.4 source ↗

Rewrite as equivalent rational expressions with denominator

(x + 3) (x + 5) (x + 6)

\frac{x}{x^{2} + 8 x + 15},

\frac{5}{x^{2} + 9 x + 18} .

Answer

\frac{x^{2} + 6 x}{(x + 3) (x + 5) (x + 6)},

\frac{5 x + 25}{(x + 3) (x + 5) (x + 6)}

Try It 8.79 §8.4 source ↗

Add:

\frac{11}{30} + \frac{7}{12} .

Answer

\frac{19}{20}

Try It 8.80 §8.4 source ↗

Add:

\frac{3}{8} + \frac{9}{20} .

Answer

\frac{33}{40}

Try It 8.81 §8.4 source ↗

Add:

\frac{2}{15 a^{2} b} +

\frac{5}{6 a b^{2}} .

Answer

\frac{4 b + 25 a}{30 a^{2} b^{2}}

Try It 8.82 §8.4 source ↗

Add:

\frac{5}{16 c} + \frac{3}{8 c d^{2}} .

Answer

\frac{5 d^{2} + 6}{16 c d^{2}}

Try It 8.83 §8.4 source ↗

Add:

\frac{2}{x - 2} + \frac{5}{x + 3} .

Answer

\frac{7 x - 4}{(x + 3) (x - 2)}

Try It 8.84 §8.4 source ↗

Add:

\frac{4}{m + 3} + \frac{3}{m + 4} .

Answer

\frac{7 m + 25}{(m + 3) (m + 4)}

Try It 8.85 §8.4 source ↗

Add:

\frac{5 x}{x y - y^{2}} +

\frac{2 x}{x^{2} - y^{2}} .

Answer

\frac{x (5 x + 7 y)}{y (x - y) (x + y)}

Try It 8.86 §8.4 source ↗

Add:

\frac{7}{2 m + 6} +

\frac{4}{m^{2} + 4 m + 3} .

Answer

\frac{7 m + 15}{2 (m + 3) (m + 1)}

Try It 8.87 §8.4 source ↗

Add:

\frac{1}{m^{2} - m - 2} +

\frac{5 m}{m^{2} + 3 m + 2} .

Answer

\frac{5 m^{2} - 9 m + 2}{(m - 2) (m + 1) (m + 2)}

Try It 8.88 §8.4 source ↗

Add:

\frac{2 n}{n^{2} - 3 n - 10} +

\frac{6}{n^{2} + 5 n + 6} .

Answer

\frac{2 (n^{2} + 6 n - 15)}{(n + 2) (n - 5) (n + 3)}

Try It 8.89 §8.4 source ↗

Subtract:

\frac{y}{y + 4} -

\frac{y - 2}{y - 5} .

Answer

\frac{−7 y + 8}{(y + 4) (y - 5)}

Try It 8.90 §8.4 source ↗

Subtract:

\frac{z + 3}{z + 2} -

\frac{z}{z + 3} .

Answer

\frac{4 z + 9}{(z + 2) (z + 3)}

Try It 8.91 §8.4 source ↗

Subtract:

\frac{2 x}{x^{2} - 4} -

\frac{1}{x + 2} .

Answer

\frac{1}{x - 2}

Try It 8.92 §8.4 source ↗

Subtract:

\frac{3}{z + 3} -

\frac{6 z}{z^{2} - 9} .

Answer

\frac{−3}{z - 3}

Try It 8.93 §8.4 source ↗

Subtract:

\frac{3 x - 1}{x^{2} - 5 x - 6} -

\frac{2}{6 - x} .

Answer

\frac{5 x + 1}{x - 6 x + 1}

Try It 8.94 §8.4 source ↗

Subtract:

\frac{−2 y - 2}{y^{2} + 2 y - 8} -

\frac{y - 1}{2 - y} .

Answer

\frac{y + 3}{y + 4}

Try It 8.95 §8.4 source ↗

Subtract:

\frac{2 x + 1}{x - 7} - 3 .

Answer

\frac{− x + 22}{x - 7}

Try It 8.96 §8.4 source ↗

Subtract:

\frac{4 y + 3}{2 y - 1} - 5 .

Answer

\frac{−2 (3 y - 4)}{2 y - 1}

Try It 8.97 §8.4 source ↗

Simplify:

\frac{v}{v + 1} +

\frac{3}{v - 1} -

\frac{6}{v^{2} - 1} .

Answer

\frac{v + 3}{v + 1}

Try It 8.98 §8.4 source ↗

Simplify:

\frac{3 w}{w + 2} +

\frac{2}{w + 7} -

\frac{17 w + 4}{w^{2} + 9 w + 14} .

Answer

\frac{3 w}{w + 7}

8.4-169 §8.4 source ↗

In the following exercises, find the LCD.

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

Answer

(x - 4) (x + 2) (x + 3)

8.4-171 §8.4 source ↗

In the following exercises, find the LCD.

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

Answer

(z - 2) (z + 4) (z + 2)

8.4-173 §8.4 source ↗

In the following exercises, find the LCD.

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

Answer

(b + 3) (b + 3) (b - 5)

8.4-175 §8.4 source ↗

In the following exercises, find the LCD.

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

Answer

(3 d - 1) (d + 5) (d - 6)

8.4-177 §8.4 source ↗

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

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

Answer

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

\frac{2 x^{2} + 4 x}{(x - 4) (x + 2) (x + 3)}

8.4-179 §8.4 source ↗

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

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

Answer

\frac{9 z + 18}{(z - 2) (z + 4) (z + 2)},

\frac{4 z^{2} + 16 z}{(z - 2) (z + 4) (z + 2)}

8.4-181 §8.4 source ↗

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

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

Answer

\frac{4 b - 20}{(b + 3) (b + 3) (b - 5)},

\frac{2 b^{2} + 6 b}{(b + 3) (b + 3) (b - 5)}

8.4-183 §8.4 source ↗

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

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

Answer

\frac{2 d - 12}{(3 d - 1) (d + 5) (d - 6)},

\frac{5 d^{2} + 25 d}{(3 d - 1) (d + 5) (d - 6)}

8.4-185 §8.4 source ↗

In the following exercises, add.

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

Answer

\frac{37}{72}

8.4-187 §8.4 source ↗

In the following exercises, add.

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

Answer

\frac{49}{60}

8.4-189 §8.4 source ↗

In the following exercises, add.

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

Answer

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

8.4-191 §8.4 source ↗

In the following exercises, add.

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

Answer

\frac{4 m n + 7}{8 m^{2} n}

8.4-193 §8.4 source ↗

In the following exercises, add.

$\frac{3}{r + 4} + \frac{2}{r - 5}$

Answer

\frac{5 r - 7}{(r + 4) (r - 5)}

8.4-195 §8.4 source ↗

In the following exercises, add.

$\frac{8}{t + 5} + \frac{6}{t - 5}$

Answer

\frac{14 t - 10}{(t + 5) (t - 5)}

8.4-197 §8.4 source ↗

In the following exercises, add.

$\frac{5}{3 w - 2} + \frac{2}{w + 1}$

Answer

\frac{11 w + 1}{(3 w - 2) (w + 1)}

8.4-199 §8.4 source ↗

In the following exercises, add.

$\frac{2 y}{y + 3} + \frac{3}{y - 1}$

Answer

\frac{2 y^{2} + y + 9}{(y + 3) (y - 1)}

8.4-201 §8.4 source ↗

In the following exercises, add.

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

Answer

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

8.4-203 §8.4 source ↗

In the following exercises, add.

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

Answer

\frac{2 m^{2} + 23 m}{3 (m - 1) (m + 4)}

8.4-205 §8.4 source ↗

In the following exercises, add.

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

Answer

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

8.4-207 §8.4 source ↗

In the following exercises, add.

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

Answer

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

8.4-209 §8.4 source ↗

In the following exercises, subtract.

$\frac{t}{t - 6} - \frac{t - 2}{t + 6}$

Answer

\frac{2 (7 t - 6)}{(t - 6) (t + 6)}

8.4-211 §8.4 source ↗

In the following exercises, subtract.

$\frac{w + 2}{w + 4} - \frac{w}{w - 2}$

Answer

\frac{−4 (1 + w)}{(w + 4) (w - 2)}

8.4-213 §8.4 source ↗

In the following exercises, subtract.

$\frac{y - 4}{y + 1} - \frac{1}{y + 7}$

Answer

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

8.4-215 §8.4 source ↗

In the following exercises, subtract.

$\frac{5 a}{a + 3} -$ $\frac{a + 2}{a + 6}$

Answer

\frac{4 a^{2} + 25 a - 6}{(a + 3) (a + 6)}

8.4-217 §8.4 source ↗

In the following exercises, subtract.

$\frac{6 c}{c^{2} - 25} -$ $\frac{3}{c + 5}$

Answer

\frac{3}{c - 5}

8.4-219 §8.4 source ↗

In the following exercises, subtract.

$\frac{6}{m + 6} -$ $\frac{12 m}{m^{2} - 36}$

Answer

\frac{−6}{m - 6}

8.4-221 §8.4 source ↗

In the following exercises, subtract.

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

Answer

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

8.4-223 §8.4 source ↗

In the following exercises, subtract.

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

Answer

\frac{3}{r - 2}

8.4-225 §8.4 source ↗

In the following exercises, subtract.

$\frac{5 v - 2}{v + 3} - 4$

Answer

\frac{v - 14}{v + 3}

8.4-227 §8.4 source ↗

In the following exercises, subtract.

$\frac{2 x + 7}{10 x - 1} + 3$

Answer

\frac{4 (8 x + 1)}{10 x - 1}

8.4-229 §8.4 source ↗

In the following exercises, add and subtract.

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

Answer

\frac{5 a^{2} + 7 a - 36}{a (a - 2)}

8.4-231 §8.4 source ↗

In the following exercises, add and subtract.

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

Answer

\frac{c - 5}{c + 2}

8.4-233 §8.4 source ↗

In the following exercises, simplify.

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

Answer

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

8.4-235 §8.4 source ↗

In the following exercises, simplify.

$\frac{6 d}{d^{2} - 64} -$ $\frac{3}{d - 8}$

Answer

\frac{3}{d + 8}

8.4-237 §8.4 source ↗

In the following exercises, simplify.

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

Answer

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

8.4-239 §8.4 source ↗

In the following exercises, simplify.

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

Answer

-

\frac{n + 1 n + 8}{n + 3 n - 2}

8.4-241 §8.4 source ↗

In the following exercises, simplify.

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

Answer

\frac{42 q + 25}{90 p q}

8.4-243 §8.4 source ↗

In the following exercises, simplify.

$\frac{4}{x - 2} + \frac{3}{x + 5}$

Answer

\frac{7 (x + 2)}{(x - 2) (x + 5)}

8.4-245 §8.4 source ↗

In the following exercises, simplify.

$\frac{2 q + 7}{q + 4} - 2$

Answer

\frac{- 1}{q + 4}

8.4-247 §8.4 source ↗

In the following exercises, simplify.

$\frac{z + 2}{z - 5} - \frac{z}{z + 1}$

Answer

\frac{2 4 z + 1}{(z - 5) (z + 1)}

8.4-249 §8.4 source ↗

In the following exercises, simplify.

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

Answer

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

8.4-251 §8.4 source ↗

In the following exercises, simplify.

Decorating cupcakes Victoria can decorate an order of cupcakes for a wedding in $t$ 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

ⓐ

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

ⓑ

\frac{13}{40}

8.4-253 §8.4 source ↗

In the following exercises, simplify.

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).

ⓓ Find the correct expression for $\frac{1}{x} + \frac{1}{y} .$

Answer

Answers may vary.

Be Prepared 8.17 §8.5 source ↗

Simplify:

\frac{\frac{3}{5}}{\frac{9}{10}} .

If you missed this problem, review Example 1.72.

Example 1.72 in the book ↗

Answer

\frac{2}{3}

Be Prepared 8.18 §8.5 source ↗

Simplify:

\frac{1 - \frac{1}{3}}{4^{2} + 4 \cdot 5} .

If you missed this problem, review Example 1.74.

Example 1.74 in the book ↗

Answer

\frac{1}{54}

Try It 8.99 §8.5 source ↗

Simplify:

\frac{\frac{2}{x^{2} - 1}}{\frac{3}{x + 1}} .

Answer

\frac{2}{3 (x - 1)}

Try It 8.100 §8.5 source ↗

Simplify:

\frac{\frac{1}{x^{2} - 7 x + 12}}{\frac{2}{x - 4}} .

Answer

\frac{1}{2 (x - 3)}

Try It 8.101 §8.5 source ↗

Simplify:

\frac{\frac{1}{2} + \frac{2}{3}}{\frac{5}{6} + \frac{1}{12}} .

Answer

\frac{14}{11}

Try It 8.102 §8.5 source ↗

Simplify:

\frac{\frac{3}{4} - \frac{1}{3}}{\frac{1}{8} + \frac{5}{6}} .

Answer

\frac{10}{23}

Try It 8.103 §8.5 source ↗

Simplify:

\frac{\frac{1}{x} + \frac{1}{y}}{\frac{1}{x} - \frac{1}{y}} .

Answer

\frac{y + x}{y - x}

Try It 8.104 §8.5 source ↗

Simplify:

\frac{\frac{1}{a} + \frac{1}{b}}{\frac{1}{a^{2}} - \frac{1}{b^{2}}} .

Answer

\frac{a b}{b - a}

Try It 8.105 §8.5 source ↗

Simplify:

\frac{b - \frac{3 b}{b + 5}}{\frac{2}{b + 5} + \frac{1}{b - 5}} .

Answer

\frac{b b + 2 b - 5}{3 b - 5}

Try It 8.106 §8.5 source ↗

Simplify:

\frac{1 - \frac{3}{c + 4}}{\frac{1}{c + 4} + \frac{c}{3}} .

Answer

\frac{3}{c + 3}

Try It 8.107 §8.5 source ↗

Simplify:

\frac{\frac{1}{2} + \frac{1}{5}}{\frac{1}{10} + \frac{1}{5}} .

Answer

\frac{7}{3}

Try It 8.108 §8.5 source ↗

Simplify:

\frac{\frac{1}{4} + \frac{3}{8}}{\frac{1}{2} - \frac{5}{16}} .

Answer

\frac{10}{3}

Try It 8.109 §8.5 source ↗

Simplify:

\frac{\frac{1}{a} + \frac{1}{b}}{\frac{a}{b} + \frac{b}{a}} .

Answer

\frac{b + a}{a^{2} + b^{2}}

Try It 8.110 §8.5 source ↗

Simplify:

\frac{\frac{1}{x^{2}} - \frac{1}{y^{2}}}{\frac{1}{x} + \frac{1}{y}} .

Answer

\frac{y - x}{x y}

Try It 8.111 §8.5 source ↗

Simplify:

\frac{\frac{3}{x + 2}}{\frac{5}{x - 2} - \frac{3}{x^{2} - 4}} .

Answer

\frac{3 x - 6}{5 x + 7}

Try It 8.112 §8.5 source ↗

Simplify:

\frac{\frac{2}{x - 7} - \frac{1}{x + 7}}{\frac{6}{x + 7} - \frac{1}{x^{2} - 49}} .

Answer

\frac{x + 21}{6 x - 43}

Try It 8.113 §8.5 source ↗

Simplify:

\frac{\frac{3}{x^{2} + 7 x + 10}}{\frac{4}{x + 2} + \frac{1}{x + 5}} .

Answer

\frac{3}{5 x + 22}

Try It 8.114 §8.5 source ↗

Simplify:

\frac{\frac{4}{y + 5} + \frac{2}{y + 6}}{\frac{3 y}{y^{2} + 11 y + 30}} .

Answer

\frac{6 y + 34}{3 y}

Try It 8.115 §8.5 source ↗

Simplify:

\frac{\frac{x}{x + 3}}{1 + \frac{1}{x + 3}} .

Answer

\frac{x}{x + 4}

Try It 8.116 §8.5 source ↗

Simplify:

\frac{1 + \frac{1}{x - 1}}{\frac{3}{x + 1}} .

Answer

\frac{x (x + 1)}{3 (x - 1)}

8.5-255 §8.5 source ↗

Simplify a Complex Rational Expression by Writing It as Division

In the following exercises, simplify.

$\frac{\frac{2 a}{a + 4}}{\frac{4 a^{2}}{a^{2} - 16}}$

Answer

\frac{a - 4}{2 a}

8.5-257 §8.5 source ↗

Simplify a Complex Rational Expression by Writing It as Division

In the following exercises, simplify.

$\frac{\frac{5}{c^{2} + 5 c - 14}}{\frac{10}{c + 7}}$

Answer

\frac{1}{2 (c - 2)}

8.5-259 §8.5 source ↗

Simplify a Complex Rational Expression by Writing It as Division

In the following exercises, simplify.

$\frac{\frac{1}{2} + \frac{5}{6}}{\frac{2}{3} + \frac{7}{9}}$

Answer

\frac{12}{13}

8.5-261 §8.5 source ↗

Simplify a Complex Rational Expression by Writing It as Division

In the following exercises, simplify.

$\frac{\frac{2}{3} - \frac{1}{9}}{\frac{3}{4} + \frac{5}{6}}$

Answer

\frac{20}{57}

8.5-263 §8.5 source ↗

Simplify a Complex Rational Expression by Writing It as Division

In the following exercises, simplify.

$\frac{\frac{n}{m} + \frac{1}{n}}{\frac{1}{n} - \frac{n}{m}}$

Answer

\frac{n^{2} + m}{m - n^{2}}

8.5-265 §8.5 source ↗

Simplify a Complex Rational Expression by Writing It as Division

In the following exercises, simplify.

$\frac{\frac{1}{r} + \frac{1}{t}}{\frac{1}{r^{2}} - \frac{1}{t^{2}}}$

Answer

\frac{r t}{t - r}

8.5-267 §8.5 source ↗

Simplify a Complex Rational Expression by Writing It as Division

In the following exercises, simplify.

$\frac{x - \frac{2 x}{x + 3}}{\frac{1}{x + 3} + \frac{1}{x - 3}}$

Answer

\frac{(x + 1) (x - 3)}{2}

8.5-269 §8.5 source ↗

Simplify a Complex Rational Expression by Writing It as Division

In the following exercises, simplify.

$\frac{2 - \frac{2}{a + 3}}{\frac{1}{a + 3} + \frac{a}{2}}$

Answer

\frac{4}{a + 1}

8.5-271 §8.5 source ↗

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

$\frac{\frac{1}{3} + \frac{1}{8}}{\frac{1}{4} + \frac{1}{12}}$

Answer

\frac{11}{8}

8.5-273 §8.5 source ↗

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

$\frac{\frac{5}{6} + \frac{2}{9}}{\frac{7}{18} - \frac{1}{3}}$

Answer

19

8.5-275 §8.5 source ↗

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

$\frac{\frac{c}{d} + \frac{1}{d}}{\frac{1}{d} - \frac{d}{c}}$

Answer

\frac{c^{2} + c}{c - d^{2}}

8.5-277 §8.5 source ↗

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

$\frac{\frac{1}{p} + \frac{1}{q}}{\frac{1}{p^{2}} - \frac{1}{q^{2}}}$

Answer

\frac{p q}{q - p}

8.5-279 §8.5 source ↗

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

$\frac{\frac{2}{x + 5}}{\frac{3}{x - 5} + \frac{1}{x^{2} - 25}}$

Answer

\frac{2 x - 10}{3 x + 16}

8.5-281 §8.5 source ↗

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

$\frac{\frac{5}{z^{2} - 64} + \frac{3}{z + 8}}{\frac{1}{z + 8} + \frac{2}{z - 8}}$

Answer

\frac{3 z - 19}{3 z + 8}

8.5-283 §8.5 source ↗

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

$\frac{\frac{4}{a^{2} - 2 a - 15}}{\frac{1}{a - 5} + \frac{2}{a + 3}}$

Answer

\frac{4}{3 a - 7}

8.5-285 §8.5 source ↗

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

$\frac{\frac{5}{c + 2} - \frac{3}{c + 7}}{\frac{5 c}{c^{2} + 9 c + 14}}$

Answer

\frac{2 c + 29}{5 c}

8.5-287 §8.5 source ↗

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

$\frac{2 + \frac{1}{p - 3}}{\frac{5}{p - 3}}$

Answer

\frac{(2 p - 5)}{5}

8.5-289 §8.5 source ↗

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

$\frac{\frac{m}{m + 5}}{4 + \frac{1}{m - 5}}$

Answer

\frac{m (m - 5)}{4 m^{2} + m - 95}

8.5-291 §8.5 source ↗

Simplify

In the following exercises, use either method.

$\frac{\frac{3}{4} - \frac{2}{7}}{\frac{1}{2} + \frac{5}{14}}$

Answer

\frac{13}{24}

8.5-293 §8.5 source ↗

Simplify

In the following exercises, use either method.

$\frac{\frac{2}{a + 4}}{\frac{1}{a^{2} - 16}}$

Answer

2 (a - 4)

8.5-295 §8.5 source ↗

Simplify

In the following exercises, use either method.

$\frac{\frac{3}{m} + \frac{3}{n}}{\frac{1}{m^{2}} - \frac{1}{n^{2}}}$

Answer

\frac{3 m n}{n - m}

8.5-297 §8.5 source ↗

Simplify

In the following exercises, use either method.

$\frac{x - \frac{3 x}{x + 2}}{\frac{3}{x + 2} + \frac{3}{x - 2}}$

Answer

\frac{(x - 1) (x - 2)}{6}

8.5-299 §8.5 source ↗

Simplify

In the following exercises, use either method.

Electronics The resistance of a circuit formed by connecting two resistors in parallel is $\frac{1}{\frac{1}{R_{1}} + \frac{1}{R_{2}}}$ .

ⓐ Simplify the complex fraction $\frac{1}{\frac{1}{R_{1}} + \frac{1}{R_{2}}}$ .

ⓑ Find the resistance of the circuit when $R_{1} = 8$ and $R_{2} = 12$ .

Answer

ⓐ

\frac{R_{1} R_{2}}{R_{2} + R_{1}}

ⓑ

\frac{24}{5}

8.5-301 §8.5 source ↗

Simplify

In the following exercises, use either method.

In this section, you learned to simplify the complex fraction $\frac{\frac{3}{x + 2}}{\frac{x}{x^{2} - 4}}$ two ways:

rewriting it as a division problem

multiplying the numerator and denominator by the LCD

Which method do you prefer? Why?

Answer

Answers will vary.

Be Prepared 8.19 §8.6 source ↗

Solve:

\frac{1}{6} x + \frac{1}{2} = \frac{1}{3} .

If you missed this problem, review Example 2.48.

Example 2.48 in the book ↗

Answer

x = - 1

Be Prepared 8.20 §8.6 source ↗

Solve:

n^{2} - 5 n - 36 = 0 .

If you missed this problem, review Example 7.73.

Example 7.73 in the book ↗

Answer

n = 9, n = - 4

Be Prepared 8.21 §8.6 source ↗

Solve for

y

in terms of

x

5 x + 2 y = 10

for

y .

If you missed this problem, review Example 2.65.

Example 2.65 in the book ↗

Answer

y = \frac{10 - 5 x}{2}

Try It 8.117 §8.6 source ↗

Solve:

\frac{1}{y} + \frac{2}{3} = \frac{1}{5} .

Answer

- \frac{15}{7}

Try It 8.118 §8.6 source ↗

Solve:

\frac{2}{3} + \frac{1}{5} = \frac{1}{x} .

Answer

\frac{15}{13}

Try It 8.119 §8.6 source ↗

Solve:

1 - \frac{2}{a} = \frac{15}{a^{2}} .

Answer

5, −3

Try It 8.120 §8.6 source ↗

Solve:

1 - \frac{4}{b} = \frac{12}{b^{2}} .

Answer

6, −2

Try It 8.121 §8.6 source ↗

Solve:

\frac{1}{x - 1} = \frac{2}{3 x} .

Answer

−2

Try It 8.122 §8.6 source ↗

Solve:

\frac{3}{5 n + 1} = \frac{2}{3 n} .

Answer

−2

Try It 8.123 §8.6 source ↗

Solve:

\frac{2}{x + 1} +

\frac{1}{x - 1} =

\frac{1}{x^{2} - 1} .

Answer

\frac{2}{3}

Try It 8.124 §8.6 source ↗

Solve:

\frac{5}{y + 3} +

\frac{2}{y - 3} =

\frac{5}{y^{2} - 9} .

Answer

2

Try It 8.125 §8.6 source ↗

Solve:

\frac{2}{x + 5} -

\frac{1}{x - 1} =

1 .

Answer

−1, −2

Try It 8.126 §8.6 source ↗

Solve:

\frac{3}{x + 8} -

\frac{2}{x - 2} =

1 .

Answer

−2, −3

Try It 8.127 §8.6 source ↗

Solve:

\frac{x + 13}{x^{2} - 7 x + 10} =

\frac{6}{x - 5} -

\frac{4}{x - 2} .

Answer

no solution

Try It 8.128 §8.6 source ↗

Solve:

\frac{y - 14}{y^{2} + 3 y - 4} =

\frac{2}{y + 4} +

\frac{7}{y - 1} .

Answer

y = - 5

Try It 8.129 §8.6 source ↗

Solve:

\frac{x}{18} +

\frac{x + 6}{9 x} =

\frac{2}{3 x} .

Answer

−2

Try It 8.130 §8.6 source ↗

Solve:

\frac{y + 5}{5 y} +

\frac{y}{15} =

\frac{1}{y} .

Answer

−3

Try It 8.131 §8.6 source ↗

Solve:

\frac{x}{x + 4} =

\frac{32}{x^{2} - 16} +

5 .

Answer

3

Try It 8.132 §8.6 source ↗

Solve:

\frac{y}{y + 8} =

\frac{128}{y^{2} - 64} +

9 .

Answer

7

Try It 8.133 §8.6 source ↗

Solve:

\frac{y}{5 y - 10} -

\frac{5}{3 y + 6} =

\frac{2 y^{2} - 19 y + 54}{15 y^{2} - 60} .

Answer

no solution

Try It 8.134 §8.6 source ↗

Solve:

\frac{z}{2 z + 8} -

\frac{3}{4 z - 8} =

\frac{3 z^{2} - 16 z - 16}{8 z^{2} + 16 z - 64} .

Answer

no solution

Try It 8.135 §8.6 source ↗

Solve:

\frac{A}{L} = W

for

L .

Answer

L = \frac{A}{W}

Try It 8.136 §8.6 source ↗

Solve:

\frac{F}{A} = M

for

A .

Answer

A = \frac{F}{M}

Try It 8.137 §8.6 source ↗

Solve:

\frac{y - 2}{x + 1} = \frac{2}{3}

for

x .

Answer

x = \frac{3 y - 8}{2}

Try It 8.138 §8.6 source ↗

Solve:

x = \frac{y}{1 - y}

for

y .

Answer

y = \frac{x}{1 + x}

Try It 8.139 §8.6 source ↗

Solve:

\frac{1}{a} + \frac{1}{b} = c

for

a .

Answer

a = \frac{b}{c b - 1}

Try It 8.140 §8.6 source ↗

Solve:

\frac{2}{x} + \frac{1}{3} = \frac{1}{y}

for

y .

Answer

y = \frac{3 x}{6 + x}

8.6-303 §8.6 source ↗

Solve Rational Equations

In the following exercises, solve.

$\frac{1}{a} + \frac{2}{5} = \frac{1}{2}$

Answer

10

8.6-305 §8.6 source ↗

Solve Rational Equations

In the following exercises, solve.

$\frac{5}{2} - \frac{1}{c} = \frac{3}{4}$

Answer

\frac{4}{7}

8.6-307 §8.6 source ↗

Solve Rational Equations

In the following exercises, solve.

$\frac{4}{5} + \frac{1}{4} = \frac{2}{v}$

Answer

\frac{40}{21}

8.6-309 §8.6 source ↗

Solve Rational Equations

In the following exercises, solve.

$\frac{7}{9} + \frac{1}{x} = \frac{2}{3}$

Answer

−9

8.6-311 §8.6 source ↗

Solve Rational Equations

In the following exercises, solve.

$1 - \frac{2}{m} = \frac{8}{m^{2}}$

Answer

−2, 4

8.6-313 §8.6 source ↗

Solve Rational Equations

In the following exercises, solve.

$1 + \frac{9}{p} = \frac{−20}{p^{2}}$

Answer

−5, −4

8.6-315 §8.6 source ↗

Solve Rational Equations

In the following exercises, solve.

$\frac{1}{r + 3} = \frac{4}{2 r}$

Answer

−6

8.6-317 §8.6 source ↗

Solve Rational Equations

In the following exercises, solve.

$\frac{5}{3 v - 2} = \frac{7}{4 v}$

Answer

14

8.6-319 §8.6 source ↗

Solve Rational Equations

In the following exercises, solve.

$\frac{3}{x + 4} +$ $\frac{7}{x - 4} =$ $\frac{8}{x^{2} - 16}$

Answer

\frac{−4}{5}

8.6-321 §8.6 source ↗

Solve Rational Equations

In the following exercises, solve.

$\frac{8}{z - 10} +$ $\frac{7}{z + 10} =$ $\frac{5}{z^{2} - 100}$

Answer

- \frac{1}{3}

8.6-323 §8.6 source ↗

Solve Rational Equations

In the following exercises, solve.

$\frac{1}{q + 4} - \frac{2}{q - 2} = 1$

Answer

- 2, - 1

8.6-325 §8.6 source ↗

Solve Rational Equations

In the following exercises, solve.

$\frac{1}{t + 7} - \frac{5}{t - 5} = 1$

Answer

−5, −1

8.6-327 §8.6 source ↗

Solve Rational Equations

In the following exercises, solve.

$\frac{v - 10}{v^{2} - 5 v + 4} =$ $\frac{3}{v - 1} -$ $\frac{6}{v - 4}$

Answer

no solution

8.6-329 §8.6 source ↗

Solve Rational Equations

In the following exercises, solve.

$\frac{x - 10}{x^{2} + 8 x + 12} =$ $\frac{3}{x + 2} +$ $\frac{4}{x + 6}$

Answer

no solution

8.6-331 §8.6 source ↗

Solve Rational Equations

In the following exercises, solve.

$\frac{z}{16} +$ $\frac{z + 2}{4 z} =$ $\frac{1}{2 z}$

Answer

−4

8.6-333 §8.6 source ↗

Solve Rational Equations

In the following exercises, solve.

$\frac{b + 3}{3 b} +$ $\frac{b}{24} =$ $\frac{1}{b}$

Answer

−8

8.6-335 §8.6 source ↗

Solve Rational Equations

In the following exercises, solve.

$\frac{d}{d + 3} =$ $\frac{18}{d^{2} - 9} +$ $4$

Answer

2

8.6-337 §8.6 source ↗

Solve Rational Equations

In the following exercises, solve.

$\frac{n}{n + 2} =$ $\frac{8}{n^{2} - 4} +$ $3$

Answer

1

8.6-339 §8.6 source ↗

Solve Rational Equations

In the following exercises, solve.

$\frac{q}{3 q - 9} -$ $\frac{3}{4 q + 12}$
$=$ $\frac{7 q^{2} + 6 q + 63}{24 q^{2} - 216}$

Answer

no solution

8.6-341 §8.6 source ↗

Solve Rational Equations

In the following exercises, solve.

$\frac{s}{2 s + 6} - \frac{2}{5 s + 5}$
$=$ $\frac{5 s^{2} - s - 18}{10 s^{2} + 40 s + 30}$

Answer

no solution

8.6-343 §8.6 source ↗

Solve a Rational Equation for a Specific Variable

In the following exercises, solve.

$\frac{C}{r} = 2 π for r$

Answer

r = \frac{C}{2 π}

8.6-345 §8.6 source ↗

Solve a Rational Equation for a Specific Variable

In the following exercises, solve.

$\frac{V}{h} = l w for h$

Answer

h = \frac{v}{l w}

8.6-347 §8.6 source ↗

Solve a Rational Equation for a Specific Variable

In the following exercises, solve.

$\frac{v + 3}{w - 1} =$ $\frac{1}{2} for w$

Answer

w = 2 v + 7

8.6-349 §8.6 source ↗

Solve a Rational Equation for a Specific Variable

In the following exercises, solve.

$a =$ $\frac{b + 3}{c - 2} for c$

Answer

c = \frac{b + 3 + 2 a}{a}

8.6-351 §8.6 source ↗

Solve a Rational Equation for a Specific Variable

In the following exercises, solve.

$\frac{1}{p} + \frac{2}{q} = 4 for p$

Answer

p = \frac{q}{4 q - 2}

8.6-353 §8.6 source ↗

Solve a Rational Equation for a Specific Variable

In the following exercises, solve.

$\frac{2}{v} +$ $\frac{1}{5} =$ $\frac{3}{w} for w$

Answer

w = \frac{15 v}{10 + v}

8.6-355 §8.6 source ↗

Solve a Rational Equation for a Specific Variable

In the following exercises, solve.

$\frac{m + 3}{n - 2} =$ $\frac{4}{5} for n$

Answer

n = \frac{5 m + 23}{4}

8.6-357 §8.6 source ↗

Solve a Rational Equation for a Specific Variable

In the following exercises, solve.

$\frac{3}{x} -$ $\frac{5}{y} =$ $\frac{1}{4} for y$

Answer

y = \frac{20 x}{12 - x}

8.6-359 §8.6 source ↗

Solve a Rational Equation for a Specific Variable

In the following exercises, solve.

$r = \frac{s}{3 - t} for t$

Answer

t = \frac{3 r - s}{r}

8.6-361 §8.6 source ↗

Solve a Rational Equation for a Specific Variable

In the following exercises, solve.

House Painting Alain can paint a house in 4 days. Spiro would take 7 days to paint the same house. Solve the equation $\frac{1}{4} + \frac{1}{7} = \frac{1}{t}$ for t to find the number of days it would take them to paint the house if they worked together.

Answer

2 \frac{6}{11}

days

8.6-363 §8.6 source ↗

Solve a Rational Equation for a Specific Variable

In the following exercises, solve.

Why is there no solution to the equation $\frac{3}{x - 2} = \frac{5}{x - 2}$ ?

Answer

Answers will vary.

Be Prepared 8.22 §8.7 source ↗

Solve

\frac{n}{3} = 30

.
If you missed this problem, review Example 2.21.

Example 2.21 in the book ↗

Answer

n = 90

Be Prepared 8.23 §8.7 source ↗

The perimeter of a triangular window is 23 feet. The lengths of two sides are ten feet and six feet. How long is the third side?
If you missed this problem, review Example 3.35.

Example 3.35 in the book ↗

Answer

7 feet

Try It 8.141 §8.7 source ↗

Solve the proportion:

\frac{n}{84} = \frac{11}{12} .

Answer

77

Try It 8.142 §8.7 source ↗

Solve the proportion:

\frac{y}{96} = \frac{13}{12} .

Answer

104

Try It 8.143 §8.7 source ↗

Solve the proportion:

\frac{91}{b} = \frac{7}{5} .

Answer

65

Try It 8.144 §8.7 source ↗

Solve the proportion:

\frac{39}{c} = \frac{13}{8} .

Answer

24

Try It 8.145 §8.7 source ↗

Solve the proportion:

\frac{y}{y + 55} = \frac{3}{8} .

Answer

33

Try It 8.146 §8.7 source ↗

Solve the proportion:

\frac{z}{z - 84} = - \frac{1}{5} .

Answer

14

Try It 8.147 §8.7 source ↗

Solve:

\frac{v + 30}{8} =

\frac{v + 66}{12} .

Answer

42

Try It 8.148 §8.7 source ↗

Solve:

\frac{2 x + 15}{9} =

\frac{7 x + 3}{15} .

Answer

6

Try It 8.149 §8.7 source ↗

Pediatricians prescribe 5 milliliters (ml) of acetaminophen for every 25 pounds of a child’s weight. How many milliliters of acetaminophen will the doctor prescribe for Emilia, who weighs 60 pounds?

Answer

12 ml

Try It 8.150 §8.7 source ↗

For every 1 kilogram (kg) of a child’s weight, pediatricians prescribe 15 milligrams (mg) of a fever reducer. If Isabella weighs 12 kg, how many milligrams of the fever reducer will the pediatrician prescribe?

Answer

180 mg

Try It 8.151 §8.7 source ↗

At a fast-food restaurant, a 22-ounce chocolate shake has 850 calories. How many calories are in their 12-ounce chocolate shake? Round your answer to nearest whole number.

Answer

464 calories

Try It 8.152 §8.7 source ↗

Yaneli loves Starburst candies, but wants to keep her snacks to 100 calories. If the candies have 160 calories for 8 pieces, how many pieces can she have in her snack?

Answer

5 pieces

Try It 8.153 §8.7 source ↗

Yurianna is going to Europe and wants to change $800 dollars into Euros. At the current exchange rate, $1 US is equal to 0.738 Euro. How many Euros will she have for her trip?

Answer

590.4 Euros

Try It 8.154 §8.7 source ↗

Corey and Nicole are traveling to Japan and need to exchange $600 into Japanese yen. If each dollar is 94.1 yen, how many yen will they get?

Answer

56,460 yen

Try It 8.155 §8.7 source ↗

$Δ A B C$ is similar to $Δ X Y Z$ . The lengths of two sides of each triangle are given in the figure.

The above image shows two similar triangles. The smaller triangle is labeled A B C. The length of two sides is given for the smaller triangle A B C. The length from A to B is 17. The length from B to C is a. The length from C to D is 15. The larger triangle is labeled X Y Z. The length is given for two sides. The length from X to Y is 25.5. The length from Y to Z is 12. The length from Z to X is y.

Find the length of side $a$ .

Answer

Try It 8.156 §8.7 source ↗

$Δ A B C$ is similar to $Δ X Y Z$ . The lengths of two sides of each triangle are given in the figure.

Find the length of side $y$ .

Answer

22.5

Try It 8.157 §8.7 source ↗

On the map, Seattle, Portland, and Boise form a triangle whose sides are shown in the figure below. If the actual distance from Seattle to Boise is 400 miles, find the distance from Seattle to Portland.

The above image is a triangle with one side labeled “Seattle, 4.5 inches”. The other side is labeled “Portland 3.5 inches”. The third side is labeled 1.5 inches. The vertex is labeled “Boise.”

Answer

150 miles

Try It 8.158 §8.7 source ↗

Using the map above, find the distance from Portland to Boise.

Answer

350 miles

Try It 8.159 §8.7 source ↗

A telephone pole casts a shadow that is 50 feet long. Nearby, an 8 foot tall traffic sign casts a shadow that is 10 feet long. How tall is the telephone pole?

Answer

40 feet

Try It 8.160 §8.7 source ↗

A pine tree casts a shadow of 80 feet next to a 30-foot tall building which casts a 40 feet shadow. How tall is the pine tree?

Answer

60 feet

8.7-365 §8.7 source ↗

Solve Proportions

In the following exercises, solve.

$\frac{x}{56} = \frac{7}{8}$

Answer

49

8.7-367 §8.7 source ↗

Solve Proportions

In the following exercises, solve.

$\frac{49}{63} = \frac{z}{9}$

Answer

7

8.7-369 §8.7 source ↗

Solve Proportions

In the following exercises, solve.

$\frac{5}{a} = \frac{65}{117}$

Answer

9

8.7-371 §8.7 source ↗

Solve Proportions

In the following exercises, solve.

$\frac{98}{154} = \frac{−7}{p}$

Answer

−11

8.7-373 §8.7 source ↗

Solve Proportions

In the following exercises, solve.

$\frac{a}{−8} = \frac{−42}{48}$

Answer

7

8.7-375 §8.7 source ↗

Solve Proportions

In the following exercises, solve.

$\frac{2.7}{j} = \frac{0.9}{0.2}$

Answer

0.6

8.7-377 §8.7 source ↗

Solve Proportions

In the following exercises, solve.

$\frac{a}{a + 12} = \frac{4}{7}$

Answer

16

8.7-379 §8.7 source ↗

Solve Proportions

In the following exercises, solve.

$\frac{c}{c - 104} = - \frac{5}{8}$

Answer

40

8.7-381 §8.7 source ↗

Solve Proportions

In the following exercises, solve.

$\frac{m + 90}{25} =$ $\frac{m + 30}{15}$

Answer

60

8.7-383 §8.7 source ↗

Solve Proportions

In the following exercises, solve.

$\frac{2 p + 4}{8} = \frac{p + 18}{6}$

Answer

30

8.7-385 §8.7 source ↗

Solve Proportions

In the following exercises, solve.

Pediatricians prescribe 5 milliliters (ml) of acetaminophen for every 25 pounds of a child’s weight. How many milliliters of acetaminophen will the doctor prescribe for Jocelyn, who weighs 45 pounds?

Answer

9 ml

8.7-387 §8.7 source ↗

Solve Proportions

In the following exercises, solve.

A veterinarian prescribed Sunny, a 65 pound dog, an antibacterial medicine in case an infection emerges after her teeth were cleaned. If the dosage is 5 mg for every pound, how much medicine was Sunny given?

Answer

325 mg

8.7-389 §8.7 source ↗

Solve Proportions

In the following exercises, solve.

A new energy drink advertises 106 calories for 8 ounces. How many calories are in 12 ounces of the drink?

Answer

159 calories

8.7-391 §8.7 source ↗

Solve Proportions

In the following exercises, solve.

A new 7 ounce lemon ice drink is advertised for having only 140 calories. How many ounces could Sally drink if she wanted to drink just 100 calories?

Answer

5 oz

8.7-393 §8.7 source ↗

Solve Proportions

In the following exercises, solve.

Janice is traveling to Canada and will change $250 US dollars into Canadian dollars. At the current exchange rate, $1 US is equal to $1.01 Canadian. How many Canadian dollars will she get for her trip?

Answer

252.5 Canadian dollars

8.7-395 §8.7 source ↗

Solve Proportions

In the following exercises, solve.

Steve changed $600 into 480 Euros. How many Euros did he receive for each US dollar?

Answer

0.80 Euros

8.7-397 §8.7 source ↗

Solve Proportions

In the following exercises, solve.

When traveling to Great Britain, Bethany exchanged her $900 into 570 British pounds. How many pounds did she receive for each American dollar?

Answer

0.63 British pounds

8.7-399 §8.7 source ↗

Solve Proportions

In the following exercises, solve.

Ronald needs a morning breakfast drink that will give him at least 390 calories. Orange juice has 130 calories in one cup. How many cups does he need to drink to reach his calorie goal?

Answer

3 cups

8.7-401 §8.7 source ↗

Solve Proportions

In the following exercises, solve.

Elizabeth is returning to the United States from Canada. She changes the remaining 300 Canadian dollars she has to $230.05 in American dollars. What was $1 worth in Canadian dollars?

Answer

1.30 Canadian dollars

8.7-403 §8.7 source ↗

Solve Proportions

In the following exercises, solve.

A golden retriever weighing 85 pounds has diarrhea. His medicine is prescribed as 1 teaspoon per 5 pounds. How much medicine should he be given?

Answer

17 tsp

8.7-405 §8.7 source ↗

Solve Proportions

In the following exercises, solve.

Karen eats $\frac{1}{2}$ cup of oatmeal that counts for 2 points on her weight loss program. Her husband, Joe, can have 3 points of oatmeal for breakfast. How much oatmeal can he have?

Answer

\frac{3}{4}

cup

8.7-407 §8.7 source ↗

Solve Similar Figure Applications

In the following exercises, $Δ A B C$ is similar to $Δ X Y Z$ . Find the length of the indicated side.

The above image shows two triangles. The larger triangle is labeled A B C. The length from A to B is 15. The length from A to C is b. The length from B to C is 9. The smaller triangle is labeled X Y Z. The length from X to Y is 10. The length from X to Z is 8. The length from Y to Z is x.

side b

Answer

12

8.7-409 §8.7 source ↗

In the following exercises, $Δ D E F$ is similar to $Δ N P Q$ .

Find the length of side d.

Answer

\frac{77}{18}

8.7-411 §8.7 source ↗

In the following two exercises, use the map shown. On the map, New York City, Chicago, and Memphis form a triangle whose sides are shown in the figure below. The actual distance from New York to Chicago is 800 miles.

The above image shows a triangle. Each angle is labeled, clockwise, “Chicago”, “New York”, and “Memphis”. The side that extends from Chicago to New York is labeled 8 inches. The side that extends from New York to Memphis is labeled 9.5 inches and the side extending from Memphis to Chicago is labeled 5.4 inches.

Find the actual distance from New York to Memphis.

Answer

950 miles

8.7-413 §8.7 source ↗

In the following two exercises, use the map shown. On the map, Atlanta, Miami, and New Orleans form a triangle whose sides are shown in the figure below. The actual distance from Atlanta to New Orleans is 420 miles.

The above image shows a triangle. Each angle is labeled, clockwise, “Atlanta”, “Miami”, and “New Orleans”. The side that extends from Atlanta to Miami is labeled 3 inches. The side that extends from Miami to New Orleans is labeled 3.4 inches and the side extending from New Orleans to Atlanta is labeled 2.1 inches.

Find the actual distance from New Orleans to Miami.

Answer

680 miles

8.7-415 §8.7 source ↗

A 2 foot tall dog casts a 3 foot shadow at the same time a cat casts a one foot shadow. How tall is the cat?

Answer

\frac{2}{3}

foot

(8 in)

8.7-417 §8.7 source ↗

The tower portion of a windmill is 212 feet tall. A six foot tall person standing next to the tower casts a seven foot shadow. How long is the windmill’s shadow?

Answer

247.3 feet

8.7-419 §8.7 source ↗

Heart Rate At the gym, Carol takes her pulse for 10 seconds and counts 19 beats.

ⓐ How many beats per minute is this?

ⓑ Has Carol met her target heart rate of 140 beats per minute?

Answer

114 beats per minute
no

8.7-421 §8.7 source ↗

Cost of a Road Trip Jesse’s car gets 30 miles per gallon of gas.

ⓐ If Las Vegas is 285 miles away, how many gallons of gas are needed to get there and then home?

ⓑ If gas is $3.09 per gallon, what is the total cost of the gas for the trip?

Answer

19 gallons
$58.71

8.7-423 §8.7 source ↗

Lawn Fertilizer Phil wants to fertilize his lawn. Each bag of fertilizer covers about 4,000 square feet of lawn. Phil’s lawn is approximately 13,500 square feet. How many bags of fertilizer will he have to buy?

Answer

4 bags

8.7-425 §8.7 source ↗

Cooking Natalia’s pasta recipe calls for 2 pounds of pasta for 1 quart of sauce. How many pounds of pasta should Natalia cook if she has 2.5 quarts of sauce?

Answer

5 pounds

8.7-427 §8.7 source ↗

Marisol solves the proportion $\frac{144}{a} = \frac{9}{4}$ by ‘cross multiplying’, so her first step looks like $4 \cdot 144 = 9 \cdot a .$ Explain how this differs from the method of solution shown in Example 8.72.

Answer

Answers will vary.

Be Prepared 8.24 §8.8 source ↗

An express bus and a local bus leave Chicago to travel to Champaign. The express bus can make the trip in 2 hours and the local bus takes 5 hours for the trip. The speed of the express bus is 42 miles per hour faster than the speed of the local bus. Find the speed of the local bus.
If you missed this problem, review Example 3.48.

Example 3.48 in the book ↗

Answer

28 mph

Be Prepared 8.25 §8.8 source ↗

Solve

\frac{1}{3} x + \frac{1}{4} x = \frac{5}{6}

.
If you missed this problem, review Example 3.49.

Example 3.49 in the book ↗

Answer

x = \frac{10}{7}

Be Prepared 8.26 §8.8 source ↗

Solve:

18 t^{2} -

30 =

−33 t

.
If you missed this problem, review Example 7.79.

Example 7.79 in the book ↗

Answer

t = - \frac{5}{2}, t = \frac{2}{3}

Try It 8.161 §8.8 source ↗

Link has an electric bike which runs at a constant speed. That speed will be reduced by the amount of any headwind and increased by the amount of any tailwind. Link can ride his bike 20 miles into a 3 mph headwind in the same amount of time he can ride 30 miles with a 3 mph tailwind. What is Link’s biking speed?

Answer

15 mph

Try It 8.162 §8.8 source ↗

Judy can pilot her powerboat 5 miles into a 7 mph wind in the same amount of time she can cover 12 miles with a 7 mph tailwind. What is the speed of Judy’s boat without a wind?

Answer

17 mph

Try It 8.163 §8.8 source ↗

Dennis went cross-country skiing for 6 hours on Saturday. He skied 20 miles uphill and then 20 miles back downhill, returning to his starting point. His uphill speed was 5 mph slower than his downhill speed. What was Dennis’ speed going uphill and his speed going downhill?

Answer

5 mph uphill and 10 mph downhill

Try It 8.164 §8.8 source ↗

Tony drove 4 hours to his home, driving 208 miles on the interstate and 40 miles on country roads. If he drove 15 mph faster on the interstate than on the country roads, what was his rate on the country roads?

Answer

50 mph

Try It 8.165 §8.8 source ↗

Kayla rode her bike 75 miles home from college one weekend and then rode the bus back to college. It took her 2 hours less to ride back to college on the bus than it took her to ride home on her bike, and the average speed of the bus was 10 miles per hour faster than Kayla’s biking speed. Find Kayla’s biking speed.

Answer

15 mph

Try It 8.166 §8.8 source ↗

Victoria jogs 12 miles to the park along a flat trail and then returns by jogging on a 20 mile hilly trail. She jogs 1 mile per hour slower on the hilly trail than on the flat trail, and her return trip takes her two hours longer. Find her rate of jogging on the flat trail.

Answer

6 mph

Try It 8.167 §8.8 source ↗

One gardener can mow a golf course in 4 hours, while another gardener can mow the same golf course in 6 hours. How long would it take if the two gardeners worked together to mow the golf course?

Answer

2 hours and 24 minutes

Try It 8.168 §8.8 source ↗

Carrie can weed the garden in 7 hours, while her mother can do it in 3. How long will it take the two of them working together?

Answer

2 hours and 6 minutes

Try It 8.169 §8.8 source ↗

Two hoses can fill a swimming pool in 10 hours. It would take one hose 26 hours to fill the pool by itself. How long would it take for the other hose, working alone, to fill the pool?

Answer

16.25 hours

Try It 8.170 §8.8 source ↗

Cara and Cindy, working together, can rake the yard in 4 hours. Working alone, it takes Cindy 6 hours to rake the yard. How long would it take Cara to rake the yard alone?

Answer

12 hours

8.8-429 §8.8 source ↗

Solve Uniform Motion Applications

In the following exercises, solve uniform motion applications

Mary takes a sightseeing tour on a helicopter that can fly 450 miles against a 35 mph headwind in the same amount of time it can travel 702 miles with a 35 mph tailwind. Find the speed of the helicopter.

Answer

160 mph

8.8-431 §8.8 source ↗

Solve Uniform Motion Applications

In the following exercises, solve uniform motion applications

A boat travels 140 miles downstream in the same time as it travels 92 miles upstream. The speed of the current is 6mph. What is the speed of the boat?

Answer

29 mph

8.8-433 §8.8 source ↗

Solve Uniform Motion Applications

In the following exercises, solve uniform motion applications

Jane spent 2 hours exploring a mountain with a dirt bike. When she rode the 40 miles uphill, she went 5 mph slower than when she reached the peak and rode for 12 miles along the summit. What was her rate along the summit?

Answer

30 mph

8.8-435 §8.8 source ↗

Solve Uniform Motion Applications

In the following exercises, solve uniform motion applications

Bill wanted to try out different water craft. He went 62 miles downstream in a motor boat and 27 miles downstream on a jet ski. His speed on the jet ski was 10 mph faster than in the motor boat. Bill spent a total of 4 hours on the water. What was his rate of speed in the motor boat?

Answer

20 mph

8.8-437 §8.8 source ↗

Solve Uniform Motion Applications

In the following exercises, solve uniform motion applications

Chester rode his bike uphill 24 miles and then back downhill at 2 mph faster than his uphill. If it took him 2 hours longer to ride uphill than downhill, l, what was his uphill rate?

Answer

4 mph

8.8-439 §8.8 source ↗

Solve Uniform Motion Applications

In the following exercises, solve uniform motion applications

Hudson travels 1080 miles in a jet and then 240 miles by car to get to a business meeting. The jet goes 300 mph faster than the rate of the car, and the car ride takes 1 hour longer than the jet. What is the speed of the car?

Answer

60 mph

8.8-441 §8.8 source ↗

Solve Uniform Motion Applications

In the following exercises, solve uniform motion applications

John can fly his airplane 2800 miles with a wind speed of 50 mph in the same time he can travel 2400 miles against the wind. If the speed of the wind is 50 mph, find the speed of his airplane.

Answer

650 mph

8.8-443 §8.8 source ↗

Solve Uniform Motion Applications

In the following exercises, solve uniform motion applications

Hazel needs to get to her granddaughter’s house by taking an airplane and a rental car. She travels 900 miles by plane and 250 miles by car. The plane travels 250 mph faster than the car. If she drives the rental car for 2 hours more than she rode the plane, find the speed of the car.

Answer

50 mph

8.8-445 §8.8 source ↗

Solve Uniform Motion Applications

In the following exercises, solve uniform motion applications

When driving the 9 hour trip home, Sharon drove 390 miles on the interstate and 150 miles on country roads. Her speed on the interstate was 15 more than on country roads. What was her speed on country roads?

Answer

50 mph

8.8-447 §8.8 source ↗

Solve Work Applications

In the following exercises, solve work applications.

Mike, an experienced bricklayer, can build a wall in 3 hours, while his son, who is learning, can do the job in 6 hours. How long does it take for them to build a wall together?

Answer

2 hours

8.8-449 §8.8 source ↗

Solve Work Applications

In the following exercises, solve work applications.

Mary can clean her apartment in 6 hours while her roommate can clean the apartment in 5 hours. If they work together, how long would it take them to clean the apartment?

Answer

2 hours and 44 minutes

8.8-451 §8.8 source ↗

Solve Work Applications

In the following exercises, solve work applications.

Leeson can proofread a newspaper copy in 4 hours. If Ryan helps, they can do the job in 3 hours. How long would it take for Ryan to do his job alone?

Answer

12 hours

8.8-453 §8.8 source ↗

Solve Work Applications

In the following exercises, solve work applications.

Josephine can correct her students’ test papers in 5 hours, but if her teacher’s assistant helps, it would take them 3 hours. How long would it take the assistant to do it alone?

Answer

7 hours and 30 minutes

8.8-455 §8.8 source ↗

Solve Work Applications

In the following exercises, solve work applications.

Jackson can remove the shingles off of a house in 7 hours, while Martin can remove the shingles in 5 hours. How long will it take them to remove the shingles if they work together?

Answer

2 hours and 55 minutes

8.8-457 §8.8 source ↗

Solve Work Applications

In the following exercises, solve work applications.

Ronald can shovel the driveway in 4 hours, but if his brother Donald helps it would take 2 hours. How long would it take Donald to shovel the driveway alone?

Answer

4 hours

8.8-459 §8.8 source ↗

Solve Work Applications

In the following exercises, solve work applications.

Dana enjoys taking her dog for a walk, but sometimes her dog gets away and she has to run after him. Dana walked her dog for 7 miles but then had to run for 1 mile, spending a total time of 2.5 hours with her dog. Her running speed was 3 mph faster than her walking speed. Find her walking speed.

Answer

3 mph

Be Prepared 8.27 §8.9 source ↗

Find the multiplicative inverse of

−8

.
If you missed this problem, review Example 1.126.

Example 1.126 in the book ↗

Answer

- \frac{1}{8}

Be Prepared 8.28 §8.9 source ↗

Solve for

n

45 = 20 n

.
If you missed this problem, review Example 2.13.

Example 2.13 in the book ↗

Answer

n = 2.25

Be Prepared 8.29 §8.9 source ↗

Evaluate

5 x^{2}

when

x = 10

.
If you missed this problem, review Example 1.20.

Example 1.20 in the book ↗

Answer

500

Try It 8.171 §8.9 source ↗

y

varies directly as

x

and

y =

3,

when x =

10 .

find the equation that relates x and y.

Answer

y = \frac{3}{10} x

Try It 8.172 §8.9 source ↗

y

varies directly as

x

and

y = 12 when x = 4

find the equation that relates x and y.

Answer

y = 3 x

Try It 8.173 §8.9 source ↗

The number of calories, c, burned varies directly with the amount of time, t, spent exercising. Arnold burned 312 calories in 65 minutes exercising.

ⓐ Write the equation that relates c and t.

ⓑ How many calories would he burn if he exercises for 90 minutes?

Answer

ⓐ

c = 4.8 t

ⓑ 432 calories

Try It 8.174 §8.9 source ↗

The distance a moving body travels, d, varies directly with time, t, it moves. A train travels 100 miles in 2 hours

ⓐ Write the equation that relates d and t. ⓑ How many miles would it travel in 5 hours?

Answer

ⓐ

d = 50 t

ⓑ 250 miles

Try It 8.175 §8.9 source ↗

The distance that Brad travels varies directly with the time spent traveling. Brad travelled 660 miles in 12 hours,

ⓐ Write the equation that relates the number of miles travelled to the time.

ⓑ How many miles could Brad travel in 4 hours?

Answer

ⓐ

m = 55 h

ⓑ 220 miles

Try It 8.176 §8.9 source ↗

The weight of a liquid varies directly as its volume. A liquid that weighs 24 pounds has a volume of 4 gallons.

ⓐ Write the equation that relates the weight to the volume.

ⓑ If a liquid has volume 13 gallons, what is its weight?

Answer

ⓐ

w = 6 v

ⓑ 78 pounds

Try It 8.177 §8.9 source ↗

The distance an object falls is directly proportional to the square of the time it falls. A ball falls 144 feet in 3 seconds.

ⓐ Write the equation that relates the distance to the time.

ⓑ How far will an object fall in 4 seconds?

Answer

ⓐ

d = 16 t^{2}

ⓑ 256 feet

Try It 8.178 §8.9 source ↗

The area of a circle varies directly as the square of the radius. A circular pizza with a radius of 6 inches has an area of 113.04 square inches.

ⓐ Write the equation that relates the area to the radius.

ⓑ What is the area of a pizza with a radius of 9 inches?

Answer

ⓐ

A = 3.14 r^{2}

ⓑ 254.34 square inches

Try It 8.179 §8.9 source ↗

p

varies inversely with

q

and

p = 30

when

q = 12

find the equation that relates

p

and

q .

Answer

p = \frac{360}{q}

Try It 8.180 §8.9 source ↗

y

varies inversely with

x

and

y = 8

when

x = 2

find the equation that relates

x

and

y

Answer

y = \frac{16}{x}

Try It 8.181 §8.9 source ↗

A car’s value varies inversely with its age. Elena bought a two-year-old car for $20,000.

ⓐ Write the equation of variation. ⓑ What will be the value of Elena’s car when it is 5 years old?

Answer

ⓐ

v = \frac{40,000}{a}

ⓑ $8,000

Try It 8.182 §8.9 source ↗

The time required to empty a pool varies inversely as the rate of pumping. It took Lucy 2.5 hours to empty her pool using a pump that was rated at 400 gpm (gallons per minute).

ⓐ Write the equation of variation.

ⓑ How long will it take her to empty the pool using a pump rated at 500 gpm?

Answer

ⓐ

t = \frac{1000}{r}

ⓑ 2 hours

Try It 8.183 §8.9 source ↗

The number of hours it takes for ice to melt varies inversely with the air temperature. Suppose a block of ice melts in 2 hours when the temperature is 65 degrees.

ⓐ Write the equation of variation.

ⓑ How many hours would it take for the same block of ice to melt if the temperature was 78 degrees?

Answer

ⓐ

h = \frac{130}{t}

ⓑ

1 \frac{2}{3}

hours

Try It 8.184 §8.9 source ↗

The force needed to break a board varies inversely with its length. Richard uses 24 pounds of pressure to break a 2-foot long board.

ⓐ Write the equation of variation.

ⓑ How many pounds of pressure is needed to break a 5-foot long board?

Answer

ⓐ

F = \frac{48}{L}

ⓑ 9.6 pounds

8.9-463 §8.9 source ↗

Solve Direct Variation Problems

In the following exercises, solve.

If $y$ varies directly as $x$ and $y =$ $14,$ $when x =$ $3$ , find the equation that relates $x and y$ .

Answer

y = \frac{14}{3} x

8.9-465 §8.9 source ↗

Solve Direct Variation Problems

In the following exercises, solve.

If $v$ varies directly as $w$ and $v =$ $24,$ $when w =$ $8$ , find the equation that relates $v and w .$

Answer

v = 3 w

8.9-467 §8.9 source ↗

Solve Direct Variation Problems

In the following exercises, solve.

If $p$ varies directly as $q$ and $p =$ $9.6,$ $when q =$ $3$ , find the equation that relates $p and q .$

Answer

p = 3.2 q

8.9-469 §8.9 source ↗

Solve Direct Variation Problems

In the following exercises, solve.

If $a$ varies directly as $b$ and $a =$ $6,$ $when b =$ $\frac{1}{3}$ , find the equation that relates $a and b .$

Answer

a = 18 b

8.9-471 §8.9 source ↗

Solve Direct Variation Problems

In the following exercises, solve.

The amount of money Sally earns, P, varies directly with the number, n, of necklaces she sells. When Sally sells 15 necklaces she earns $150.

ⓐ Write the equation that relates P and n.

ⓑ How much money would she earn if she sold 4 necklaces?

Answer

ⓐ

P = 10 n

ⓑ

$ 40

8.9-473 §8.9 source ↗

Solve Direct Variation Problems

In the following exercises, solve.

Terri needs to make some pies for a fundraiser. The number of apples, a, varies directly with number of pies, p. It takes nine apples to make two pies.

ⓐ Write the equation that relates a and p.

ⓑ How many apples would Terri need for six pies?

Answer

ⓐ

a = 4.5 p

ⓑ 27 apples

8.9-475 §8.9 source ↗

Solve Direct Variation Problems

In the following exercises, solve.

The price of gas that Jesse purchased varies directly to how many gallons he purchased. He purchased 10 gallons of gas for $39.80.

ⓐ Write the equation that relates the price to the number of gallons.

ⓑ How much will it cost Jesse for 15 gallons of gas?

Answer

ⓐ

p = 3.98 g

ⓑ

$ 59.70

8.9-477 §8.9 source ↗

Solve Direct Variation Problems

In the following exercises, solve.

The mass of a liquid varies directly with its volume. A liquid with mass 16 kilograms has a volume of 2 liters.

ⓐ Write the equation that relates the mass to the volume.

ⓑ What is the volume of this liquid if its mass is 128 kilograms?

Answer

ⓐ

m = 8 v

ⓑ

16 liters

8.9-479 §8.9 source ↗

Solve Direct Variation Problems

In the following exercises, solve.

The distance an object falls varies directly to the square of the time it falls. A ball falls 45 feet in 3 seconds.

ⓐ Write the equation that relates the distance to the time.

ⓑ How far will the ball fall in 7 seconds?

Answer

ⓐ

d = 5 t^{2}

ⓑ 245 feet

8.9-481 §8.9 source ↗

Solve Direct Variation Problems

In the following exercises, solve.

The area of a circle varies directly as the square of the radius. A circular pizza with a radius of 6 inches has an area of 113.04 square inches.

ⓐ Write the equation that relates the area to the radius.

ⓑ What is the area of a personal pizza with a radius 4 inches?

Answer

ⓐ

A = 3.14 r^{2}

ⓑ

50.24 sq. in.

8.9-483 §8.9 source ↗

Solve Inverse Variation Problems

In the following exercises, solve.

If $y$ varies inversely with $x$ and $y = 5$ when $x = 4$ find the equation that relates $x$ and $y .$

Answer

y = \frac{20}{x}

8.9-485 §8.9 source ↗

Solve Inverse Variation Problems

In the following exercises, solve.

If $v$ varies inversely with $w$ and $v = 6$ when $w = \frac{1}{2}$ find the equation that relates $v$ and $w .$

Answer

v = \frac{3}{w}

8.9-487 §8.9 source ↗

Write an inverse variation equation to solve the following problems.

The fuel consumption (mpg) of a car varies inversely with its weight. A Toyota Corolla weighs 2800 pounds and gets 33 mpg on the highway.

ⓐ Write the equation that relates the mpg to the car’s weight.

ⓑ What would the fuel consumption be for a Toyota Sequoia that weighs 5500 pounds?

Answer

ⓐ

g = \frac{92,400}{w}

ⓑ 16.8 mpg

8.9-489 §8.9 source ↗

Write an inverse variation equation to solve the following problems.

The time required to empty a tank varies inversely as the rate of pumping. It took Janet 5 hours to pump her flooded basement using a pump that was rated at 200 gpm (gallons per minute),

ⓐ Write the equation that relates the number of hours to the pump rate.

ⓑ How long would it take Janet to pump her basement if she used a pump rated at 400 gpm?

Answer

ⓐ

t = \frac{1000}{r}

ⓑ

2.5 hours

8.9-491 §8.9 source ↗

Write an inverse variation equation to solve the following problems.

On a string instrument, the length of a string varies inversely as the frequency of its vibrations. An 11-inch string on a violin has a frequency of 400 cycles per second.

ⓐ Write the equation that relates the string length to its frequency.

ⓑ What is the frequency of a 10-inch string?

Answer

ⓐ

L = \frac{4,400}{f}

ⓑ 440 cycles per second

8.9-493 §8.9 source ↗

Write an inverse variation equation to solve the following problems.

The number of tickets for a sports fundraiser varies inversely to the price of each ticket. Brianna can buy 25 tickets at $5each.

ⓐ Write the equation that relates the number of tickets to the price of each ticket.

ⓑ How many tickets could Brianna buy if the price of each ticket was $2.50?

Answer

ⓐ

t = \frac{125}{p}

ⓑ 50 tickets

8.9-495 §8.9 source ↗

Write an inverse variation equation to solve the following problems.

If $y$ varies directly as $x$ and $y =$ $5,$ $when x =$ $3 .$ , find the equation that relates $x and y$ .

Answer

y = \frac{5}{3} x

8.9-497 §8.9 source ↗

Write an inverse variation equation to solve the following problems.

If $p$ varies inversely with $q$ and $p = 5$ when $q = 6$ , find the equation that relates $p$ and $q .$

Answer

p = \frac{30}{q}

8.9-499 §8.9 source ↗

Write an inverse variation equation to solve the following problems.

If $p$ varies directly as $q$ and $p =$ $10,$ $when q =$ $2 .$ find the equation that relates $p and q$ .

Answer

p = 5 q

8.9-501 §8.9 source ↗

Write an inverse variation equation to solve the following problems.

The force needed to break a board varies inversely with its length. If Tom uses 20 pounds of pressure to break a 1.5-foot long board, how many pounds of pressure would he need to use to break a 6 foot long board?

Answer

5 pounds

8.9-503 §8.9 source ↗

Write an inverse variation equation to solve the following problems.

The length a spring stretches varies directly with a weight placed at the end of the spring. When Meredith placed a 6-pound cantaloupe on a hanging scale, the spring stretched 2 inches. How far would the spring stretch if the cantaloupe weighed 9 pounds?

Answer

3 inches

8.9-505 §8.9 source ↗

Write an inverse variation equation to solve the following problems.

The fuel consumption (mpg) of a car varies inversely with its weight. A Ford Focus weighs 3000 pounds and gets 28.7 mpg on the highway. What would the fuel consumption be for a Ford Expedition that weighs 5,500 pounds? Round to the nearest tenth.

Answer

15.6 mpg

8.9-507 §8.9 source ↗

Write an inverse variation equation to solve the following problems.

The distance an object falls varies directly to the square of the time it falls. If an object falls 52.8 feet in 4 seconds, how far will it fall in 9 seconds?

Answer

267.3 feet

8.9-509 §8.9 source ↗

Write an inverse variation equation to solve the following problems.

Ride Service It costs $35 for a ride from the city center to the airport, 14 miles away.

ⓐ Write the equation that relates the cost, c, with the number of miles, m.

ⓑ What would it cost to travel 22 miles with this service?

Answer

ⓐ

c = 2.5 m

ⓑ $55

8.9-511 §8.9 source ↗

Write an inverse variation equation to solve the following problems.

In your own words, explain the difference between direct variation and inverse variation.

Answer

Answers will vary.

8.R-513 §8.R source ↗

Determine the Values for Which a Rational Expression is Undefined

In the following exercises, determine the values for which the rational expression is undefined.

$\frac{2 a + 1}{3 a - 2}$

Answer

a \neq \frac{2}{3}

8.R-515 §8.R source ↗

Determine the Values for Which a Rational Expression is Undefined

In the following exercises, determine the values for which the rational expression is undefined.

$\frac{3 x y^{2}}{5 y}$

Answer

y \neq 0

8.R-517 §8.R source ↗

Evaluate Rational Expressions

In the following exercises, evaluate the rational expressions for the given values.

$\frac{4 p - 1}{p^{2} + 5} when p =$ $−1$

Answer

- \frac{5}{6}

8.R-519 §8.R source ↗

Evaluate Rational Expressions

In the following exercises, evaluate the rational expressions for the given values.

$\frac{y^{2} - 8}{y^{2} - y - 2} when y =$ $1$

Answer

\frac{7}{2}

8.R-521 §8.R source ↗

Simplify Rational Expressions

In the following exercises, simplify.

$\frac{10}{24}$

Answer

\frac{5}{12}

8.R-523 §8.R source ↗

Simplify Rational Expressions

In the following exercises, simplify.

$\frac{14 a - 14}{a - 1}$

Answer

14

8.R-525 §8.R source ↗

Simplify Rational Expressions with Opposite Factors

In the following exercises, simplify.

$\frac{c^{2} - c - 2}{4 - c^{2}}$

Answer

- \frac{c + 1}{c + 2}

8.R-527 §8.R source ↗

Simplify Rational Expressions with Opposite Factors

In the following exercises, simplify.

$\frac{7 v - 35}{25 - v^{2}}$

Answer

- \frac{7}{5 + v}

8.R-529 §8.R source ↗

Multiply Rational Expressions

In the following exercises, multiply.

$\frac{3}{8} \cdot \frac{2}{15}$

Answer

\frac{1}{20}

8.R-531 §8.R source ↗

Multiply Rational Expressions

In the following exercises, multiply.

$\frac{3 a^{2} + 21 a}{a^{2} + 6 a - 7} \cdot \frac{a - 1}{a b}$

Answer

\frac{3}{b}

8.R-533 §8.R source ↗

Divide Rational Expressions

In the following exercises, divide.

$\frac{t^{2} - 4 t - 12}{t^{2} + 8 t + 12} \div \frac{t^{2} - 36}{6 t}$

Answer

\frac{6 t}{(t + 6)^{2}}

8.R-535 §8.R source ↗

Divide Rational Expressions

In the following exercises, divide.

$\frac{11 + w}{w - 9} \div \frac{121 - w^{2}}{9 - w}$

Answer

\frac{−1}{11 - w}

8.R-537 §8.R source ↗

Divide Rational Expressions

In the following exercises, divide.

$\frac{\frac{c^{2} - 64}{3 c^{2} + 26 c + 16}}{\frac{c^{2} - 4 c - 32}{15 c + 10}}$

Answer

\frac{5}{c + 4}

8.R-539 §8.R source ↗

Add Rational Expressions with a Common Denominator

In the following exercises, add.

$\frac{3}{5} + \frac{2}{5}$

Answer

1

8.R-541 §8.R source ↗

Add Rational Expressions with a Common Denominator

In the following exercises, add.

$\frac{p^{2} + 10 p}{p + 5} +$ $\frac{25}{p + 5}$

Answer

p + 5

8.R-543 §8.R source ↗

Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

$\frac{d^{2}}{d + 4} -$ $\frac{3 d + 28}{d + 4}$

Answer

d - 7

8.R-545 §8.R source ↗

Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

$\frac{4 q^{2} - q + 3}{q^{2} + 6 q + 5} -$ $\frac{3 q^{2} + q + 6}{q^{2} + 6 q + 5}$

Answer

\frac{q - 3}{q + 5}

8.R-547 §8.R source ↗

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add and subtract.

$\frac{18 w}{6 w - 1} +$ $\frac{3 w - 2}{1 - 6 w}$

Answer

\frac{15 w + 2}{6 w - 1}

8.R-549 §8.R source ↗

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add and subtract.

$\frac{2 b^{2} + 3 b - 15}{b^{2} - 49} -$ $\frac{b^{2} + 16 b - 1}{49 - b^{2}}$

Answer

\frac{3 b^{2} + 19 b - 16}{b - 7 b + 7}

8.R-551 §8.R source ↗

Find the Least Common Denominator of Rational Expressions

In the following exercises, find the LCD.

$\frac{4}{m^{2} - 3 m - 10},$ $\frac{2 m}{m^{2} - m - 20}$

Answer

(m + 2) (m - 5) (m + 4)

8.R-553 §8.R source ↗

Find the Least Common Denominator of Rational Expressions

In the following exercises, find the LCD.

$\frac{5}{3 p^{2} + 19 p + 6},$ $\frac{2 p}{3 p^{2} + 25 p + 8}$

Answer

(3 p + 1) (p + 6) (p + 8)

8.R-555 §8.R source ↗

Find Equivalent Rational Expressions

In the following exercises, rewrite as equivalent rational expressions with the given denominator.

Rewrite as equivalent rational expressions with denominator $(n - 2) (n - 2) (n + 2)$ :

\frac{6}{n^{2} - 4 n + 4}, \frac{2 n}{n^{2} - 4} .

Answer

\frac{6 n + 12}{(n - 2) (n - 2) (n + 2)},

\frac{2 n^{2} - 4 n}{(n - 2) (n - 2) (n + 2)}

8.R-557 §8.R source ↗

Add Rational Expressions with Different Denominators

In the following exercises, add.

$\frac{2}{3} + \frac{3}{5}$

Answer

\frac{19}{15}

8.R-559 §8.R source ↗

Add Rational Expressions with Different Denominators

In the following exercises, add.

$\frac{2}{c - 2} + \frac{9}{c + 3}$

Answer

\frac{11 c - 12}{(c - 2) (c + 3)}

8.R-561 §8.R source ↗

Add Rational Expressions with Different Denominators

In the following exercises, add.

$\frac{2 x}{x^{2} + 10 x + 24} +$ $\frac{3 x}{x^{2} + 8 x + 16}$

Answer

\frac{5 x^{2} + 26 x}{(x + 4) (x + 4) (x + 6)}

8.R-563 §8.R source ↗

Subtract Rational Expressions with Different Denominators

In the following exercises, subtract and add.

$\frac{3 v}{v + 2} -$ $\frac{v + 2}{v + 8}$

Answer

\frac{2 (v^{2} + 10 v - 2)}{(v + 2) (v + 8)}

8.R-565 §8.R source ↗

Subtract Rational Expressions with Different Denominators

In the following exercises, subtract and add.

$\frac{7 m + 3}{m + 2} - 5$

Answer

\frac{2 m - 7}{m + 2}

8.R-567 §8.R source ↗

Subtract Rational Expressions with Different Denominators

In the following exercises, subtract and add.

$\frac{8 d}{d^{2} - 64} -$ $\frac{4}{d + 8}$

Answer

\frac{4}{d - 8}

8.R-569 §8.R source ↗

Simplify a Complex Rational Expression by Writing it as Division

In the following exercises, simplify.

$\frac{\frac{5 a}{a + 2}}{\frac{10 a^{2}}{a^{2} - 4}}$

Answer

\frac{a - 2}{2 a}

8.R-571 §8.R source ↗

Simplify a Complex Rational Expression by Writing it as Division

In the following exercises, simplify.

$\frac{x - \frac{3 x}{x + 5}}{\frac{1}{x + 5} + \frac{1}{x - 5}}$

Answer

\frac{(x + 2) (x - 5)}{2}

8.R-573 §8.R source ↗

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

$\frac{6 + \frac{2}{q - 4}}{\frac{5}{q + 4}}$

Answer

\frac{2 3 q - 11 q + 4}{5 (q - 4)}

8.R-575 §8.R source ↗

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

$\frac{\frac{2}{z^{2} - 49} + \frac{1}{z + 7}}{\frac{9}{z + 7} + \frac{12}{z - 7}}$

Answer

\frac{z - 5}{21 z + 21}

8.R-577 §8.R source ↗

Solve Rational Equations

In the following exercises, solve.

$\frac{1}{2} + \frac{2}{3} = \frac{1}{x}$

Answer

\frac{6}{7}

8.R-579 §8.R source ↗

Solve Rational Equations

In the following exercises, solve.

$\frac{1}{b - 2} +$ $\frac{1}{b + 2} =$ $\frac{3}{b^{2} - 4}$

Answer

\frac{3}{2}

8.R-581 §8.R source ↗

Solve Rational Equations

In the following exercises, solve.

$\frac{v - 15}{v^{2} - 9 v + 18} =$ $\frac{4}{v - 3} +$ $\frac{2}{v - 6}$

Answer

no solution

8.R-583 §8.R source ↗

Solve a Rational Equation for a Specific Variable

In the following exercises, solve for the indicated variable.

$\frac{V}{l} = h w for l$

Answer

l = \frac{V}{h w}

8.R-585 §8.R source ↗

Solve a Rational Equation for a Specific Variable

In the following exercises, solve for the indicated variable.

$x =$ $\frac{y + 5}{z - 7} for z$

Answer

z = \frac{y + 5 + 7 x}{x}

8.R-587 §8.R source ↗

Solve Proportions

In the following exercises, solve.

$\frac{x}{4} = \frac{3}{5}$

Answer

\frac{12}{5}

8.R-589 §8.R source ↗

Solve Proportions

In the following exercises, solve.

$\frac{s}{s + 20} = \frac{3}{7}$

Answer

15

8.R-591 §8.R source ↗

In the following exercises, solve using proportions.

Rachael had a 21 ounce strawberry shake that has 739 calories. How many calories are there in a 32 ounce shake?

Answer

1126 calories

8.R-593 §8.R source ↗

Solve Similar Figure Applications

In the following exercises, solve.

∆ABC is similar to ∆XYZ. The lengths of two sides of each triangle are given in the figure. Find the lengths of the third sides.

This image shows two triangles. The large triangle is labeled A B C. The length from A to B is labeled 8. The length from B to C is labeled 7. The length from C to A is labeled b. The smaller triangle is triangle x y z. The length from x to y is labeled 2 and two-thirds. The length from y to z is labeled x. The length from x to z is labeled 3.

Answer

b = 9; x = 2 \frac{1}{3}

8.R-595 §8.R source ↗

Solve Similar Figure Applications

In the following exercises, solve.

Tony is 5.75 feet tall. Late one afternoon, his shadow was 8 feet long. At the same time, the shadow of a nearby tree was 32 feet long. Find the height of the tree.

Answer

23 feet

8.R-597 §8.R source ↗

Solve Uniform Motion Applications

In the following exercises, solve.

When making the 5-hour drive home from visiting her parents, Lisa ran into bad weather. She was able to drive 176 miles while the weather was good, but then driving 10 mph slower, went 81 miles in the bad weather. How fast did she drive when the weather was bad?

Answer

45 mph

8.R-599 §8.R source ↗

Solve Uniform Motion Applications

In the following exercises, solve.

John can ride his bicycle 8 mph faster than Luke can ride his bike. It takes Luke 3 hours longer than John to ride 48 miles. How fast can John ride his bike?

Answer

16 mph

8.R-601 §8.R source ↗

Solve Work Applications

In the following exercises, solve.

Jerry can frame a room in 1 hour, while Jake takes 4 hours. How long could they frame a room working together?

Answer

\frac{4}{5} hour

8.R-603 §8.R source ↗

Solve Work Applications

In the following exercises, solve.

Jeffrey can paint a house in 6 days, but if he gets a helper he can do it in 4 days. How long would it take the helper to paint the house alone?

Answer

12 days

8.R-605 §8.R source ↗

Solve Direct Variation Problems

In the following exercises, solve.

If $y$ varies directly as $x$ , when $y = 9$ and $x = 3$ , find $x$ when $y = 21 .$

Answer

7

8.R-607 §8.R source ↗

Solve Direct Variation Problems

In the following exercises, solve.

If $m$ varies inversely with the square of $n$ , when $m = 4$ and $n = 6$ find $m$ when $n = 2 .$

Answer

36

8.R-609 §8.R source ↗

Solve Direct Variation Problems

In the following exercises, solve.

If the cost of a pizza varies directly with its diameter, and if an 8” diameter pizza costs $12, how much would a 6” diameter pizza cost?

Answer

$ 9

8.R-611 §8.R source ↗

Solve Inverse Variation Problems

In the following exercises, solve.

The number of tickets for a music fundraiser varies inversely with the price of the tickets. If Madelyn has just enough money to purchase 12 tickets for $6 each, how many tickets can Madelyn afford to buy if the price increased to $8?

Answer

9 tickets

8.T-613 §8.T source ↗

In the following exercises, simplify.

$\frac{3 a^{2} b}{6 a b^{2}}$

Answer

\frac{a}{2 b}

8.T-615 §8.T source ↗

In the following exercises, perform the indicated operation and simplify.

$\frac{4 x}{x + 2} \cdot \frac{x^{2} + 5 x + 6}{12 x^{2}}$

Answer

\frac{x + 3}{3 x}

8.T-617 §8.T source ↗

In the following exercises, perform the indicated operation and simplify.

$\frac{4}{p q} + \frac{5}{p}$

Answer

\frac{4 + 5 q}{p q}

8.T-619 §8.T source ↗

In the following exercises, perform the indicated operation and simplify.

$\frac{\frac{2}{3} + \frac{3}{5}}{\frac{2}{5}}$

Answer

\frac{19}{6}

8.T-621 §8.T source ↗

In the following exercises, solve each equation.

$\frac{1}{2} + \frac{2}{7} = \frac{1}{x}$

Answer

\frac{14}{11}

8.T-623 §8.T source ↗

In the following exercises, solve each equation.

$\frac{1}{z - 5} +$ $\frac{1}{z + 5} =$ $\frac{1}{z^{2} - 25}$

Answer

\frac{1}{2}

8.T-625 §8.T source ↗

In the following exercises, solve each equation.

$\frac{2}{r - 2} = \frac{3}{r - 1}$

Answer

8.T-627 §8.T source ↗

In the following exercises, solve.

If $y$ varies inversely with $x$ and $x = 6$ when $y = 20,$ find $y$ when $x = 2 .$

Answer

8.T-629 §8.T source ↗

In the following exercises, solve.

The recommended erythromycin dosage for dogs, is 5 mg for every pound the dog weighs. If Daisy weighs 25 pounds, how many milligrams of erythromycin should her veterinarian prescribe?

Answer

125 mg

8.T-631 §8.T source ↗

In the following exercises, solve.

Kurt can ride his bike for 30 miles with the wind in the same amount of time that he can go 21 miles against the wind. If the wind’s speed is 6 mph, what is Kurt’s speed on his bike?

Answer

34 mph

8.T-633 §8.T source ↗

In the following exercises, solve.

An experienced window washer can wash all the windows in Mike’s house in 2 hours, while a new trainee can wash all the windows in 7 hours. How long would it take them working together?

Answer

1 \frac{5}{9} hours

8.T-635 §8.T source ↗

In the following exercises, solve.

The price that Tyler pays for gas varies directly with the number of gallons he buys. If 24 gallons cost him $59.76, what would 30 gallons cost?

Answer

$ 74.70

8.T-637 §8.T source ↗

In the following exercises, solve.

The cities of Dayton, Columbus, and Cincinnati form a triangle in southern Ohio, as shown on the figure below, that gives the map distances between these cities in inches.

The actual distance from Dayton to Cincinnati is 48 miles. What is the actual distance between Dayton and Columbus?

Answer

64 miles

Loading chapter 9…

Problems and answers are adapted from Elementary Algebra 2e by Lynn Marecek and MaryAnne Anthony-Smith, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/elementary-algebra-2e/pages/1-introduction. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.