Content check — Intermediate Algebra, Chapter 7: Rational Expressions and Functions

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

Chapter 7: Rational Expressions and Functions

390 problems · 8 figures

Be Prepared 7.1 §7.1 source ↗

Simplify:

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

If you missed this problem, review Example 5.13.

Example 5.13 in the book ↗

Answer

\frac{6}{y}

Be Prepared 7.2 §7.1 source ↗

Multiply:

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

If you missed this problem, review Example 1.25.

Example 1.25 in the book ↗

Answer

\frac{4}{25}

Be Prepared 7.3 §7.1 source ↗

Divide:

\frac{12}{10} \div \frac{8}{25} .

If you missed this problem, review Example 1.26.

Example 1.26 in the book ↗

Answer

\frac{15}{4}

Try It 7.1 §7.1 source ↗

Determine the value for which each rational expression is undefined.

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

Answer

ⓐ

x = 0

ⓑ

n = - \frac{1}{3}

ⓒ

a = −1, a = −3

Try It 7.2 §7.1 source ↗

Determine the value for which each 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 7.3 §7.1 source ↗

Simplify:

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

Answer

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

x \neq 2,

x \neq 1

Try It 7.4 §7.1 source ↗

Simplify:

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

Answer

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

x \neq − 2,

x \neq 1

Try It 7.5 §7.1 source ↗

Simplify:

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

Answer

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

Try It 7.6 §7.1 source ↗

Simplify:

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

Answer

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

Try It 7.7 §7.1 source ↗

Simplify:

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

Answer

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

Try It 7.8 §7.1 source ↗

Simplify:

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

Answer

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

Try It 7.9 §7.1 source ↗

Simplify:

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

Answer

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

Try It 7.10 §7.1 source ↗

Simplify:

\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 7.11 §7.1 source ↗

Simplify:

\frac{2 x^{2} + 5 x - 12}{x^{2} - 16} \cdot \frac{x^{2} - 8 x + 16}{2 x^{2} - 13 x + 15} .

Answer

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

Try It 7.12 §7.1 source ↗

Simplify:

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

Answer

-

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

Try It 7.13 §7.1 source ↗

Simplify:

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

Answer

\frac{2}{x - 2}

Try It 7.14 §7.1 source ↗

Simplify:

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

Answer

\frac{2 z}{z - 1}

Try It 7.15 §7.1 source ↗

Simplify:

\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 7.16 §7.1 source ↗

Simplify:

\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 7.17 §7.1 source ↗

Perform the indicated operations:

\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 7.18 §7.1 source ↗

Perform the indicated operations:

\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)}

Try It 7.19 §7.1 source ↗

Find the domain of

R (x) =

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

Answer

The domain of

R (x)

is all real numbers where

x \neq 5

and

x \neq − 1 .

Try It 7.20 §7.1 source ↗

Find the domain of

R (x) =

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

Answer

The domain of

R (x)

is all real numbers where

x \neq 4

and

x \neq − 2 .

Try It 7.21 §7.1 source ↗

Find

R (x) =

f (x) \cdot g (x)

where

f (x) =

\frac{3 x - 21}{x^{2} - 9 x + 14}

and

g (x) =

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

Answer

R (x) = 2

Try It 7.22 §7.1 source ↗

Find

R (x) =

f (x) \cdot g (x)

where

f (x) =

\frac{x^{2} - x}{3 x^{2} + 27 x - 30}

and

g (x) =

\frac{x^{2} - 100}{x^{2} - 10 x} .

Answer

R (x) = \frac{1}{3}

Try It 7.23 §7.1 source ↗

Find

R (x) =

\frac{f (x)}{g (x)}

where

f (x) =

\frac{2 x^{2}}{x^{2} - 8 x}

and

g (x) =

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

Answer

R (x) =

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

Try It 7.24 §7.1 source ↗

Find

R (x) =

\frac{f (x)}{g (x)}

where

f (x) =

\frac{15 x^{2}}{3 x^{2} + 33 x}

and

g (x) =

\frac{5 x - 5}{x^{2} + 9 x - 22} .

Answer

R (x) =

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

7.1-1 §7.1 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.

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

Answer

ⓐ

z = 0

ⓑ

p = \frac{5}{6}

ⓒ

n = −4, n = 2

7.1-3 §7.1 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.

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

Answer

ⓐ

y = 0

, ⓑ

x = - \frac{1}{2}

, ⓒ

u = −4, u = 7

7.1-5 §7.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify each rational expression.

$- \frac{44}{55}$

Answer

- \frac{4}{5}

7.1-7 §7.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify each rational expression.

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

Answer

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

7.1-9 §7.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify each rational expression.

$\frac{8 n - 96}{3 n - 36}$

Answer

\frac{8}{3} (n \neq 2)

7.1-11 §7.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify each rational expression.

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

Answer

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

7.1-13 §7.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify each rational expression.

$\frac{a^{2} - 4}{a^{2} + 6 a - 16}$

Answer

\frac{a + 2}{a + 8}

7.1-15 §7.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify each rational expression.

$\frac{p^{3} + 3 p^{2} + 4 p + 12}{p^{2} + p - 6}$

Answer

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

7.1-17 §7.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify each rational expression.

$\frac{8 b^{2} - 32 b}{2 b^{2} - 6 b - 80}$

Answer

\frac{4 b (b - 4)}{(b + 5) (b - 8)}

7.1-19 §7.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify each rational expression.

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

Answer

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

7.1-21 §7.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify each rational expression.

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

Answer

−1

7.1-23 §7.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify each rational expression.

$\frac{20 - 5 y}{y^{2} - 16}$

Answer

- \frac{5}{y + 4}

7.1-25 §7.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify each rational expression.

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

Answer

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

7.1-27 §7.1 source ↗

Simplify Rational Expressions

In the following exercises, simplify each rational expression.

$\frac{z^{2} - 9 z + 20}{16 - z^{2}}$

Answer

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

7.1-29 §7.1 source ↗

Multiply Rational Expressions

In the following exercises, multiply the rational expressions.

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

Answer

\frac{3}{10}

7.1-31 §7.1 source ↗

Multiply Rational Expressions

In the following exercises, multiply the rational expressions.

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

Answer

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

7.1-33 §7.1 source ↗

Multiply Rational Expressions

In the following exercises, multiply the rational expressions.

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

Answer

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

7.1-35 §7.1 source ↗

Multiply Rational Expressions

In the following exercises, multiply the rational expressions.

$\frac{2 y^{2} - 10 y}{y^{2} + 10 y + 25} \cdot \frac{y + 5}{6 y}$

Answer

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

7.1-37 §7.1 source ↗

Multiply Rational Expressions

In the following exercises, multiply the rational expressions.

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

Answer

-

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

7.1-39 §7.1 source ↗

Multiply Rational Expressions

In the following exercises, multiply the rational expressions.

$\frac{3 c^{2} - 16 c + 5}{c^{2} - 25} \cdot \frac{c^{2} + 10 c + 25}{3 c^{2} - 14 c - 5}$

Answer

\frac{(3 c - 1) (c + 5)}{(3 c + 1) (c - 5)}

7.1-41 §7.1 source ↗

Multiply Rational Expressions

In the following exercises, multiply the rational expressions.

$\frac{6 m^{2} - 13 m + 2}{9 - m^{2}} \cdot \frac{m^{2} - 6 m + 9}{6 m^{2} + 23 m - 4}$

Answer

-

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

7.1-43 §7.1 source ↗

Divide Rational Expressions

In the following exercises, divide the rational expressions.

$\frac{v - 5}{11 - v} \div \frac{v^{2} - 25}{v - 11}$

Answer

- \frac{1}{v + 5}

7.1-45 §7.1 source ↗

Divide Rational Expressions

In the following exercises, divide the rational expressions.

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

Answer

\frac{3 s}{s + 4}

7.1-47 §7.1 source ↗

Divide Rational Expressions

In the following exercises, divide the rational expressions.

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

Answer

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

7.1-49 §7.1 source ↗

Divide Rational Expressions

In the following exercises, divide the rational expressions.

$\frac{x^{2} + 3 x - 10}{4 x} \div (2 x^{2} + 20 x + 50)$

Answer

\frac{x - 2}{8 x (x + 5)}

7.1-51 §7.1 source ↗

Divide Rational Expressions

In the following exercises, divide the rational expressions.

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

Answer

\frac{2 a - 7}{5}

7.1-53 §7.1 source ↗

Divide Rational Expressions

In the following exercises, divide the rational expressions.

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

Answer

3 (3 c - 5)

7.1-55 §7.1 source ↗

For the following exercises, perform the indicated operations.

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

Answer

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

7.1-57 §7.1 source ↗

For the following exercises, perform the indicated operations.

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

Answer

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

7.1-59 §7.1 source ↗

Multiply and Divide Rational Functions

In the following exercises, find the domain of each function.

$R (x) =$ $\frac{x^{3} - 2 x^{2} - 25 x + 50}{x^{2} - 25}$

Answer

x \neq 5

and

x \neq − 5

7.1-61 §7.1 source ↗

Multiply and Divide Rational Functions

In the following exercises, find the domain of each function.

$R (x) =$ $\frac{3 x^{2} + 15 x}{6 x^{2} + 6 x - 36}$

Answer

x \neq 2

and

x \neq − 3

7.1-63 §7.1 source ↗

For the following exercises, find $R (x) =$ $f (x) \cdot g (x)$ where $f (x)$ and $g (x)$ are given.

$f (x) =$ $\frac{6 x^{2} - 12 x}{x^{2} + 7 x - 18}$
$g (x) =$ $\frac{x^{2} - 81}{3 x^{2} - 27 x}$

Answer

R (x) = 2

7.1-65 §7.1 source ↗

For the following exercises, find $R (x) =$ $f (x) \cdot g (x)$ where $f (x)$ and $g (x)$ are given.

$f (x) =$ $\frac{4 x}{x^{2} - 3 x - 10}$
$g (x) =$ $\frac{x^{2} - 25}{8 x^{2}}$

Answer

R (x) =

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

7.1-67 §7.1 source ↗

For the following exercises, find $R (x) =$ $\frac{f (x)}{g (x)}$ where $f (x)$ and $g (x)$ are given.

$f (x) =$ $\frac{27 x^{2}}{3 x - 21}$
$g (x) =$ $\frac{3 x^{2} + 18 x}{x^{2} + 13 x + 42}$

Answer

R (x) =

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

7.1-69 §7.1 source ↗

For the following exercises, find $R (x) =$ $\frac{f (x)}{g (x)}$ where $f (x)$ and $g (x)$ are given.

$f (x) =$ $\frac{16 x^{2}}{4 x + 36}$
$g (x) =$ $\frac{4 x^{2} - 24 x}{x^{2} + 4 x - 45}$

Answer

R (x) =

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

7.1-71 §7.1 source ↗

For the following exercises, find $R (x) =$ $\frac{f (x)}{g (x)}$ where $f (x)$ and $g (x)$ are given.

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.

7.1-73 §7.1 source ↗

For the following exercises, find $R (x) =$ $\frac{f (x)}{g (x)}$ where $f (x)$ and $g (x)$ are given.

ⓐ 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. ⓒ Evaluate your answer to part ⓑ when $n = 7$ . Did you get the same answer you got in part ⓐ ? Why or why not?

Answer

Answers will vary.

Be Prepared 7.4 §7.2 source ↗

Add:

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

If you missed this problem, review Example 1.29.

Example 1.29 in the book ↗

Answer

\frac{37}{30}

Be Prepared 7.5 §7.2 source ↗

Subtract:

\frac{3 x}{4} - \frac{8}{9} .

If you missed this problem, review Example 1.28.

Example 1.28 in the book ↗

Answer

\frac{27 x - 32}{36}

Be Prepared 7.6 §7.2 source ↗

Subtract:

6 (2 x + 1) -

4 (x - 5) .

If you missed this problem, review Example 1.56.

Example 1.56 in the book ↗

Answer

8 x + 26

Try It 7.25 §7.2 source ↗

Simplify:

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

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

Answer

x + 2

Try It 7.26 §7.2 source ↗

Simplify:

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

\frac{15}{x + 5} .

Answer

x + 3

Try It 7.27 §7.2 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 7.28 §7.2 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 7.29 §7.2 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 7.30 §7.2 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}

Try It 7.31 §7.2 source ↗

ⓐ Find the LCD for the expressions

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

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

ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.

Answer

ⓐ

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

ⓑ

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

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

Try It 7.32 §7.2 source ↗

ⓐ Find the LCD for the expressions

\frac{3 x}{x^{2} - 3 x - 10},

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

ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.

Answer

ⓐ

(x + 2) (x - 5) (x + 1)

ⓑ

\frac{3 x^{2} + 3 x}{(x + 2) (x - 5) (x + 1)}

\frac{5 x - 25}{(x + 2) (x - 5) (x + 1)}

Try It 7.33 §7.2 source ↗

Add:

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

Answer

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

Try It 7.34 §7.2 source ↗

Add:

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

Answer

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

Try It 7.35 §7.2 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 + 1) (m - 2) (m + 2)}

Try It 7.36 §7.2 source ↗

Add:

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

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

Answer

\frac{2 n^{2} + 12 n - 30}{(n + 2) (n - 5) (n + 3)}

Try It 7.37 §7.2 source ↗

Subtract:

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

\frac{1}{x + 2} .

Answer

\frac{1}{x - 2}

Try It 7.38 §7.2 source ↗

Subtract:

\frac{3}{z + 3} -

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

Answer

\frac{−3}{z - 3}

Try It 7.39 §7.2 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 7.40 §7.2 source ↗

Subtract:

\frac{−2 y - 2}{y^{2} + 2 y - 8} -

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

Answer

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

Try It 7.41 §7.2 source ↗

Subtract:

\frac{3}{b^{2} - 4 b - 5} -

\frac{2}{b^{2} - 6 b + 5} .

Answer

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

Try It 7.42 §7.2 source ↗

Subtract:

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

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

Answer

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

Try It 7.43 §7.2 source ↗

Simplify:

\frac{v}{v + 1} +

\frac{3}{v - 1} -

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

Answer

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

Try It 7.44 §7.2 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}

Try It 7.45 §7.2 source ↗

Find

R (x) =

f (x) -

g (x)

where

f (x) = \frac{x + 1}{x + 3}

and

g (x) =

\frac{x + 17}{x^{2} - x - 12} .

Answer

\frac{x - 7}{x - 4}

Try It 7.46 §7.2 source ↗

Find

R (x) =

f (x) +

g (x)

where

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

and

g (x) =

\frac{4 x + 6}{x^{2} - 9} .

Answer

\frac{x^{2} - 3 x + 18}{(x + 3) (x - 3)}

7.2-75 §7.2 source ↗

Add and Subtract Rational Expressions with a Common Denominator

In the following exercises, add.

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

Answer

\frac{3}{5}

7.2-77 §7.2 source ↗

Add and Subtract Rational Expressions with a Common Denominator

In the following exercises, add.

$\frac{3 c}{4 c - 5} +$ $\frac{5}{4 c - 5}$

Answer

\frac{3 c + 5}{4 c - 5}

7.2-79 §7.2 source ↗

Add and Subtract 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

7.2-81 §7.2 source ↗

Add and Subtract 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}

7.2-83 §7.2 source ↗

In the following exercises, subtract.

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

Answer

3 a + 7

7.2-85 §7.2 source ↗

In the following exercises, subtract.

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

Answer

\frac{m - 2}{2}

7.2-87 §7.2 source ↗

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}

7.2-89 §7.2 source ↗

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}

7.2-91 §7.2 source ↗

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add or subtract.

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

Answer

4

7.2-93 §7.2 source ↗

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add or subtract.

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

Answer

x + 2

7.2-95 §7.2 source ↗

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add or subtract.

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

Answer

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

7.2-97 §7.2 source ↗

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add or 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}

7.2-99 §7.2 source ↗

Find the Least Common Denominator of Rational Expressions

In the following exercises, ⓐ find the LCD for the given rational expressions ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.

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

Answer

ⓐ

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

ⓑ

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

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

7.2-101 §7.2 source ↗

Find the Least Common Denominator of Rational Expressions

In the following exercises, ⓐ find the LCD for the given rational expressions ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.

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

Answer

ⓐ

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

ⓑ

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

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

7.2-103 §7.2 source ↗

Find the Least Common Denominator of Rational Expressions

In the following exercises, ⓐ find the LCD for the given rational expressions ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.

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

Answer

ⓐ

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

ⓑ

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

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

7.2-105 §7.2 source ↗

Find the Least Common Denominator of Rational Expressions

In the following exercises, ⓐ find the LCD for the given rational expressions ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.

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

Answer

ⓐ

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

ⓑ

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

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

7.2-107 §7.2 source ↗

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

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

Answer

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

7.2-109 §7.2 source ↗

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

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

Answer

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

7.2-111 §7.2 source ↗

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

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

Answer

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

7.2-113 §7.2 source ↗

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

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

Answer

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

7.2-115 §7.2 source ↗

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

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

Answer

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

7.2-117 §7.2 source ↗

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

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

Answer

− \frac{m}{m + 4}

7.2-119 §7.2 source ↗

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

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

Answer

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

7.2-121 §7.2 source ↗

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

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

Answer

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

7.2-123 §7.2 source ↗

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

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

Answer

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

7.2-125 §7.2 source ↗

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

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

Answer

\frac{−6}{m - 6}

7.2-127 §7.2 source ↗

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

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

Answer

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

7.2-129 §7.2 source ↗

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

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

Answer

\frac{3}{r - 2}

7.2-131 §7.2 source ↗

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

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

Answer

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

7.2-133 §7.2 source ↗

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

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

Answer

\frac{x - 5}{(x - 4) (x + 1) (x - 1)}

7.2-135 §7.2 source ↗

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

$\frac{5}{x^{2} + 8 x - 9} -$ $\frac{4}{x^{2} + 10 x + 9}$

Answer

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

7.2-137 §7.2 source ↗

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

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

Answer

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

7.2-139 §7.2 source ↗

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

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

Answer

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

7.2-141 §7.2 source ↗

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

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

Answer

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

7.2-143 §7.2 source ↗

Add and Subtract Rational Functions

In the following exercises, find ⓐ $R (x) =$ $f (x) +$ $g (x)$ ⓑ $R (x) =$ $f (x) -$ $g (x) .$

$f (x) =$ $\frac{−5 x - 5}{x^{2} + x - 6}$ and
$g (x) = \frac{x + 1}{2 - x}$

Answer

ⓐ

R (x) =

-

\frac{(x + 8) (x + 1)}{(x - 2) (x + 3)}

ⓑ

R (x) = \frac{x + 1}{x + 3}

7.2-145 §7.2 source ↗

Add and Subtract Rational Functions

In the following exercises, find ⓐ $R (x) =$ $f (x) +$ $g (x)$ ⓑ $R (x) =$ $f (x) -$ $g (x) .$

$f (x) =$ $\frac{6 x}{x^{2} - 64}$ and
$g (x) = \frac{3}{x - 8}$

Answer

ⓐ

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

ⓑ

R (x) = \frac{3}{x + 8}

7.2-147 §7.2 source ↗

Add and Subtract Rational Functions

In the following exercises, find ⓐ $R (x) =$ $f (x) +$ $g (x)$ ⓑ $R (x) =$ $f (x) -$ $g (x) .$

Donald thinks that $\frac{3}{x} + \frac{4}{x}$ is $\frac{7}{2 x} .$ Is Donald correct? Explain.

Answer

Answers will vary.

7.2-149 §7.2 source ↗

Add and Subtract Rational Functions

In the following exercises, find ⓐ $R (x) =$ $f (x) +$ $g (x)$ ⓑ $R (x) =$ $f (x) -$ $g (x) .$

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 ⓐ. ⓒ Explain why Felipe is wrong. ⓓ Find the correct expression for $\frac{1}{x} + \frac{1}{y} .$

Answer

ⓐ Answers will vary.
ⓑ Answers will vary.
ⓒ Answers will vary.
ⓓ

\frac{x + y}{x y}

Be Prepared 7.7 §7.3 source ↗

Simplify:

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

If you missed this problem, review Example 1.27.

Example 1.27 in the book ↗

Answer

\frac{2}{3}

Be Prepared 7.8 §7.3 source ↗

Simplify:

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

If you missed this problem, review Example 1.31.

Example 1.31 in the book ↗

Answer

\frac{1}{54}

Be Prepared 7.9 §7.3 source ↗

Solve:

\frac{1}{2 x} + \frac{1}{4} = \frac{1}{8} .

If you missed this problem, review Example 2.9.

Example 2.9 in the book ↗

Answer

x = - 4

Try It 7.47 §7.3 source ↗

Simplify the complex rational expression by writing it as division:

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

Answer

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

Try It 7.48 §7.3 source ↗

Simplify the complex rational expression by writing it as division:

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

Answer

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

Try It 7.49 §7.3 source ↗

Simplify the complex rational expression by writing it as division:

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

Answer

\frac{14}{11}

Try It 7.50 §7.3 source ↗

Simplify the complex rational expression by writing it as division:

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

Answer

\frac{10}{23}

Try It 7.51 §7.3 source ↗

Simplify the complex rational expression by writing it as division:

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

Answer

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

Try It 7.52 §7.3 source ↗

Simplify the complex rational expression by writing it as division:

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

Answer

\frac{a b}{b - a}

Try It 7.53 §7.3 source ↗

Simplify the complex rational expression by writing it as division:

\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 7.54 §7.3 source ↗

Simplify the complex rational expression by writing it as division:

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

Answer

\frac{3}{c + 3}

Try It 7.55 §7.3 source ↗

Simplify the complex rational expression by using the LCD:

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

Answer

\frac{7}{3}

Try It 7.56 §7.3 source ↗

Simplify the complex rational expression by using the LCD:

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

Answer

\frac{10}{3}

Try It 7.57 §7.3 source ↗

Simplify the complex rational expression by using the LCD:

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

Answer

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

Try It 7.58 §7.3 source ↗

Simplify the complex rational expression by using the LCD:

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

Answer

\frac{y - x}{x y}

Try It 7.59 §7.3 source ↗

Simplify the complex rational expression by using the LCD:

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

Answer

\frac{3 (x - 2)}{5 x + 7}

Try It 7.60 §7.3 source ↗

Simplify the complex rational expression by using the LCD:

\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 7.61 §7.3 source ↗

Simplify the complex rational expression by using the LCD:

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

Answer

\frac{3}{5 x + 22}

Try It 7.62 §7.3 source ↗

Simplify the complex rational expression by using the LCD:

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

Answer

\frac{2 (2 y^{2} + 13 y + 5)}{3 y}

Try It 7.63 §7.3 source ↗

Simplify the complex rational expression by using the LCD:

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

Answer

\frac{x}{x + 4}

Try It 7.64 §7.3 source ↗

Simplify the complex rational expression by using the LCD:

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

Answer

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

7.3-151 §7.3 source ↗

Simplify a Complex Rational Expression by Writing it as Division

In the following exercises, simplify each complex rational expression by writing it as division.

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

Answer

\frac{a - 4}{2 a}

7.3-153 §7.3 source ↗

Simplify a Complex Rational Expression by Writing it as Division

In the following exercises, simplify each complex rational expression by writing it as division.

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

Answer

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

7.3-155 §7.3 source ↗

Simplify a Complex Rational Expression by Writing it as Division

In the following exercises, simplify each complex rational expression by writing it as division.

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

Answer

\frac{12}{13}

7.3-157 §7.3 source ↗

Simplify a Complex Rational Expression by Writing it as Division

In the following exercises, simplify each complex rational expression by writing it as division.

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

Answer

\frac{20}{57}

7.3-159 §7.3 source ↗

Simplify a Complex Rational Expression by Writing it as Division

In the following exercises, simplify each complex rational expression by writing it as division.

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

Answer

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

7.3-161 §7.3 source ↗

Simplify a Complex Rational Expression by Writing it as Division

In the following exercises, simplify each complex rational expression by writing it as division.

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

Answer

\frac{r t}{t - r}

7.3-163 §7.3 source ↗

Simplify a Complex Rational Expression by Writing it as Division

In the following exercises, simplify each complex rational expression by writing it as division.

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

Answer

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

7.3-165 §7.3 source ↗

Simplify a Complex Rational Expression by Writing it as Division

In the following exercises, simplify each complex rational expression by writing it as division.

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

Answer

\frac{4}{a + 1}

7.3-167 §7.3 source ↗

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify each complex rational expression by using the LCD.

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

Answer

\frac{11}{8}

7.3-169 §7.3 source ↗

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify each complex rational expression by using the LCD.

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

Answer

19

7.3-171 §7.3 source ↗

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify each complex rational expression by using the LCD.

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

Answer

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

7.3-173 §7.3 source ↗

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify each complex rational expression by using the LCD.

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

Answer

\frac{p q}{q - p}

7.3-175 §7.3 source ↗

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify each complex rational expression by using the LCD.

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

Answer

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

7.3-177 §7.3 source ↗

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify each complex rational expression by using the LCD.

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

Answer

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

7.3-179 §7.3 source ↗

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify each complex rational expression by using the LCD.

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

Answer

\frac{4}{3 a - 7}

7.3-181 §7.3 source ↗

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify each complex rational expression by using the LCD.

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

Answer

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

7.3-183 §7.3 source ↗

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify each complex rational expression by using the LCD.

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

Answer

\frac{2 p - 5}{5}

7.3-185 §7.3 source ↗

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify each complex rational expression by using the LCD.

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

Answer

\frac{m (m - 5)}{(4 m - 19) (m + 5)}

7.3-187 §7.3 source ↗

In the following exercises, simplify each complex rational expression using either method.

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

Answer

\frac{13}{24}

7.3-189 §7.3 source ↗

In the following exercises, simplify each complex rational expression using either method.

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

Answer

2 (a - 4)

7.3-191 §7.3 source ↗

In the following exercises, simplify each complex rational expression using either method.

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

Answer

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

7.3-193 §7.3 source ↗

In the following exercises, simplify each complex rational expression using either method.

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

Answer

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

7.3-195 §7.3 source ↗

In the following exercises, simplify each complex rational expression using 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 or multiplying the numerator and denominator by the LCD. Which method do you prefer? Why?

Answer

Answers will vary.

Be Prepared 7.10 §7.4 source ↗

Solve:

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

If you missed this problem, review Example 2.9.

Example 2.9 in the book ↗

Answer

x = −1

Be Prepared 7.11 §7.4 source ↗

Solve:

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

If you missed this problem, review Example 6.45.

Example 6.45 in the book ↗

Answer

n = 9, n = −4

Be Prepared 7.12 §7.4 source ↗

Solve the formula

5 x + 2 y = 10

for

y .

If you missed this problem, review Example 2.31.

Example 2.31 in the book ↗

Answer

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

Try It 7.65 §7.4 source ↗

Solve:

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

Answer

y = - \frac{15}{7}

Try It 7.66 §7.4 source ↗

Solve:

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

Answer

x = \frac{15}{13}

Try It 7.67 §7.4 source ↗

Solve:

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

Answer

x = −3, x = 5

Try It 7.68 §7.4 source ↗

Solve:

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

Answer

y = −2, y = 6

Try It 7.69 §7.4 source ↗

Solve:

\frac{2}{x + 1} +

\frac{1}{x - 1} =

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

Answer

x = \frac{2}{3}

Try It 7.70 §7.4 source ↗

Solve:

\frac{5}{y + 3} +

\frac{2}{y - 3} =

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

Answer

y = 2

Try It 7.71 §7.4 source ↗

Solve:

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

\frac{6}{x - 5} -

\frac{4}{x - 2} .

Answer

There is no solution.

Try It 7.72 §7.4 source ↗

Solve:

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

\frac{2}{y + 4} +

\frac{7}{y - 1} .

Answer

There is no solution.

Try It 7.73 §7.4 source ↗

Solve:

\frac{x}{x + 4} =

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

5 .

Answer

x = 3

Try It 7.74 §7.4 source ↗

Solve:

\frac{y}{y + 8} =

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

9 .

Answer

y = 7

Try It 7.75 §7.4 source ↗

Solve:

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

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

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

Answer

There is no solution.

Try It 7.76 §7.4 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

There is no solution.

Try It 7.77 §7.4 source ↗

Solve:

\frac{15}{x^{2} + x - 6} -

\frac{3}{x - 2} =

\frac{2}{x + 3} .

Answer

There is no solution.

Try It 7.78 §7.4 source ↗

Solve:

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

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

\frac{1}{x^{2} + 5 x + 6} .

Answer

There is no solution.

Try It 7.79 §7.4 source ↗

For rational function,

f (x) =

\frac{8 - x}{x^{2} - 7 x + 12},

ⓐ find the domain of the function ⓑ solve

f (x) = 3

Answer

ⓐ The domain is all real numbers except

x \neq 3

and

x \neq 4 .

ⓑ

x = 2, x = \frac{14}{3}

ⓒ

(2, 3),

(\frac{14}{3}, 3)

Try It 7.80 §7.4 source ↗

For rational function,

f (x) =

\frac{x - 1}{x^{2} - 6 x + 5},

ⓐ find the domain of the function ⓑ solve

f (x) = 4

Answer

ⓐ The domain is all real numbers except

x \neq 1

and

x \neq 5 .

ⓑ

x = \frac{21}{4}

ⓒ

(\frac{21}{4}, 4)

Try It 7.81 §7.4 source ↗

Solve:

m = \frac{y - 5}{x - 4}

for

y .

Answer

y = m x - 4 m + 5

Try It 7.82 §7.4 source ↗

Solve:

m = \frac{y - 1}{x + 5}

for

y .

Answer

y = m x + 5 m + 1

Try It 7.83 §7.4 source ↗

Solve:

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

for a.

Answer

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

Try It 7.84 §7.4 source ↗

Solve:

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

for y.

Answer

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

7.4-197 §7.4 source ↗

Solve Rational Equations

In the following exercises, solve each rational equation.

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

Answer

a = 10

7.4-199 §7.4 source ↗

Solve Rational Equations

In the following exercises, solve each rational equation.

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

Answer

v = \frac{40}{21}

7.4-201 §7.4 source ↗

Solve Rational Equations

In the following exercises, solve each rational equation.

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

Answer

m = −2, m = 4

7.4-203 §7.4 source ↗

Solve Rational Equations

In the following exercises, solve each rational equation.

$1 + \frac{9}{p} = \frac{−20}{p^{2}}$

Answer

p = −5, p = −4

7.4-205 §7.4 source ↗

Solve Rational Equations

In the following exercises, solve each rational equation.

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

Answer

v = 14

7.4-207 §7.4 source ↗

Solve Rational Equations

In the following exercises, solve each rational equation.

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

Answer

x = - \frac{4}{5}

7.4-209 §7.4 source ↗

Solve Rational Equations

In the following exercises, solve each rational equation.

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

Answer

z = −145

7.4-211 §7.4 source ↗

Solve Rational Equations

In the following exercises, solve each rational equation.

$\frac{−10}{q - 2} -$ $\frac{7}{q + 4} =$ $1$

Answer

q = −18, q = −1

7.4-213 §7.4 source ↗

Solve Rational Equations

In the following exercises, solve each rational equation.

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

Answer

no solution

7.4-215 §7.4 source ↗

Solve Rational Equations

In the following exercises, solve each rational equation.

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

Answer

no solution

7.4-217 §7.4 source ↗

Solve Rational Equations

In the following exercises, solve each rational equation.

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

Answer

b = −8

7.4-219 §7.4 source ↗

Solve Rational Equations

In the following exercises, solve each rational equation.

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

Answer

d = 2

7.4-221 §7.4 source ↗

Solve Rational Equations

In the following exercises, solve each rational equation.

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

Answer

n = 1

7.4-223 §7.4 source ↗

Solve Rational Equations

In the following exercises, solve each rational equation.

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

Answer

no solution

7.4-225 §7.4 source ↗

Solve Rational Equations

In the following exercises, solve each rational equation.

$\frac{s}{2 s + 6} -$ $\frac{2}{5 s + 5} =$ $\frac{5 s^{2} - 3 s - 7}{10 s^{2} + 40 s + 30}$

Answer

s = \frac{5}{4}

7.4-227 §7.4 source ↗

Solve Rational Equations

In the following exercises, solve each rational equation.

$\frac{2}{x^{2} + 2 x - 8} -$ $\frac{1}{x^{2} + 9 x + 20} =$ $\frac{4}{x^{2} + 3 x - 10}$

Answer

x = - \frac{4}{3}

7.4-229 §7.4 source ↗

Solve Rational Equations

In the following exercises, solve each rational equation.

$\frac{3}{x^{2} - 5 x - 6} +$ $\frac{3}{x^{2} - 7 x + 6} =$ $\frac{6}{x^{2} - 1}$

Answer

no solution

7.4-231 §7.4 source ↗

Solve Rational Equations that Involve Functions

For rational function, $f (x) =$ $\frac{x - 2}{x^{2} + 6 x + 8},$ ⓐ find the domain of the function ⓑ solve $f (x) = 5$ ⓒ find the points on the graph at this function value.

Answer

ⓐ The domain is all real numbers except

x \neq − 2

and

x \neq − 4 .

ⓑ

x = −3, x = - \frac{14}{5}

ⓒ

(−3, 5),

(- \frac{14}{5}, 5)

7.4-233 §7.4 source ↗

Solve Rational Equations that Involve Functions

For rational function, $f (x) =$ $\frac{2 - x}{x^{2} - 7 x + 10},$ ⓐ find the domain of the function ⓑ solve $f (x) = 2$ ⓒ find the points on the graph at this function value.

Answer

ⓐ The domain is all real numbers except

x \neq 2

and

x \neq 5 .

ⓑ

x = \frac{9}{2},

ⓒ

(\frac{9}{2}, 2)

7.4-235 §7.4 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}

7.4-237 §7.4 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

7.4-239 §7.4 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}

7.4-241 §7.4 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}

7.4-243 §7.4 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}

7.4-245 §7.4 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}

7.4-247 §7.4 source ↗

Solve a Rational Equation for a Specific Variable

In the following exercises, solve.

$\frac{E}{c} = m^{2}$ for $c .$

Answer

c = \frac{E}{m^{2}}

7.4-249 §7.4 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}

7.4-251 §7.4 source ↗

Solve a Rational Equation for a Specific Variable

In the following exercises, solve.

Your class mate is having trouble in this section. Write down the steps you would use to explain how to solve a rational equation.

Answer

Answers will vary.

Be Prepared 7.13 §7.5 source ↗

Solve:

2 (n - 1) -

4 =

−10 .

If you missed this problem, review Example 2.13.

Example 2.13 in the book ↗

Answer

n = - 2

Be Prepared 7.14 §7.5 source ↗

An express train and a charter bus leave Chicago to travel to Champaign. The express train can make the trip in two hours and the bus takes five hours for the trip. The speed of the express train is 42 miles per hour faster than the speed of the bus. Find the speed of the bus.
If you missed this problem, review Example 2.43.

Example 2.43 in the book ↗

Answer

The speed of the bus is 28 mph.

Be Prepared 7.15 §7.5 source ↗

Solve

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

If you missed this problem, review Example 2.9.

Example 2.9 in the book ↗

Answer

x = \frac{10}{7}

Try It 7.85 §7.5 source ↗

Solve the proportion:

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

Answer

y = 33

Try It 7.86 §7.5 source ↗

Solve the proportion:

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

Answer

z = 14

Try It 7.87 §7.5 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

The pediatrician will prescribe 12 ml of acetaminophen to Emilia.

Try It 7.88 §7.5 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

The pediatrician will prescribe 180 mg of fever reducer to Isabella.

Try It 7.89 §7.5 source ↗

Find the actual distance from Seattle to Portland.

Answer

The distance is 150 miles.

Try It 7.90 §7.5 source ↗

Find the actual distance from Portland to Boise.

Answer

The distance is 350 miles.

Try It 7.91 §7.5 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

The telephone pole is 40 feet tall.

Try It 7.92 §7.5 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

The pine tree is 60 feet tall.

Try It 7.93 §7.5 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

Link’s biking speed is 15 mph.

Try It 7.94 §7.5 source ↗

On a river that flows at 7 miles per hour, Danica can take a motorboat 5 miles upstream, in the same amount of time she can take her motorboat 12 miles downstream. How fast is Danica's boat in a lake?

Answer

The speed of Danica’s boat is 17 mph.

Try It 7.95 §7.5 source ↗

Dennis went cross-country skiing for 6 hours on Saturday. He skied 20 mile 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

Dennis’s uphill speed was 5 mph and his downhill speed was 10 mph.

Try It 7.96 §7.5 source ↗

Joon 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

Joon’s rate on the country roads was 50 mph.

Try It 7.97 §7.5 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

Kayla’s biking speed was 15 mph.

Try It 7.98 §7.5 source ↗

Victoria jogs 12 miles to the park along a flat trail and then returns by jogging on an 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

Victoria jogged 6 mph on the flat trail.

Try It 7.99 §7.5 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

When the two gardeners work together it takes 2 hours and 24 minutes.

Try It 7.100 §7.5 source ↗

Daria 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

When Daria and her mother work together it takes 2 hours and 6 minutes.

Try It 7.101 §7.5 source ↗

Alice can paint a room in 6 hours. If Kristina helps her it takes them 4 hours to paint the room. How long would it take Kristina to paint the room by herself?

Answer

Kristina can paint the room in 12 hours.

Try It 7.102 §7.5 source ↗

Tracy can lay a slab of concrete in 3 hours, with Jordan’s help they can do it in 2 hours. If Jordan works alone, how long will it take?

Answer

It will take Jordan 6 hours.

Try It 7.103 §7.5 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

ⓑ He would burn 432 calories.

Try It 7.104 §7.5 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

ⓑ It would travel 250 miles.

Try It 7.105 §7.5 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 Celsius.

ⓐ 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 7.106 §7.5 source ↗

Xander’s new business found that the daily demand for its product was inversely proportional to the price, $p .$ When the price is $5, the demand is 700 units.

ⓐ Write the equation of variation. ⓑ What is the demand if the price is raised to $7?

Answer

ⓐ

x = \frac{3500}{p}

ⓑ 500 units

7.5-253 §7.5 source ↗

Solve Proportions

In the following exercises, solve each proportion.

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

Answer

x = 49

7.5-255 §7.5 source ↗

Solve Proportions

In the following exercises, solve each proportion.

$\frac{98}{154} = \frac{−7}{p}$

Answer

p = −11

7.5-257 §7.5 source ↗

Solve Proportions

In the following exercises, solve each proportion.

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

Answer

a = 16

7.5-259 §7.5 source ↗

Solve Proportions

In the following exercises, solve each proportion.

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

Answer

m = 60

7.5-261 §7.5 source ↗

Solve Proportions

In the following exercises, solve each proportion.

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

Answer

p = 30

7.5-263 §7.5 source ↗

In the following exercises, solve.

Kevin wants to keep his heart rate at 160 beats per minute while training. During his workout he counts 27 beats in 10 seconds.

ⓐ How many beats per minute is this? ⓑ Has Kevin met his target heart rate?

Answer

ⓐ 162 beats per minute ⓑ yes

7.5-265 §7.5 source ↗

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

7.5-267 §7.5 source ↗

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

7.5-269 §7.5 source ↗

In the following exercises, solve.

Kyra is traveling to Canada and will change $250 US dollars into Canadian dollars. At the current exchange rate, $1 US is equal to $1.3 Canadian. How many Canadian dollars will she get for her trip?

Answer

325

Canadian dollars

7.5-271 §7.5 source ↗

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

7.5-273 §7.5 source ↗

In the following exercises, solve.

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

7.5-275 §7.5 source ↗

Solve Similar Figure Applications

In the following exercises, the triangles are similar. Find the length of the indicated side.

The first figure is triangle A B C with side A B 15 units long, side B C 9 units long, and side A C b units long. The second figure is triangle X Y Z with side X Y 10 units long, side Y Z x units long, and side X Z 8 units long.

ⓐ side x ⓑ side b

Answer

ⓐ 6 ⓑ

12

7.5-277 §7.5 source ↗

In the following exercises, use the map shown. On the map, New York City, Chicago, and Memphis form a triangle. The actual distance from New York to Chicago is 800 miles.

Find the actual distance from New York to Memphis.

Answer

950 miles

7.5-279 §7.5 source ↗

In the following exercises, use the map shown. On the map, Atlanta, Miami, and New Orleans form a triangle. The actual distance from Atlanta to New Orleans is 420 miles.

The figure is a triangle formed by New Orleans, Atlanta, and Miami. The distance between New Orleans and Atlanta is 2.1 inches. The distance between Atlanta and Miami is 3 inches. The distance between Miami and New Orleans is 3.4 inches.

Find the actual distance from New Orleans to Miami.

Answer

680 miles

7.5-281 §7.5 source ↗

In the following exercises, answer each question.

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.)

7.5-283 §7.5 source ↗

In the following exercises, answer each question.

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

7.5-285 §7.5 source ↗

Solve Uniform Motion Applications

In the following exercises, solve the application problem provided.

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

7.5-287 §7.5 source ↗

Solve Uniform Motion Applications

In the following exercises, solve the application problem provided.

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

7.5-289 §7.5 source ↗

Solve Uniform Motion Applications

In the following exercises, solve the application problem provided.

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

Answer

30

mph

7.5-291 §7.5 source ↗

Solve Uniform Motion Applications

In the following exercises, solve the application problem provided.

Byron 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

7.5-293 §7.5 source ↗

Solve Uniform Motion Applications

In the following exercises, solve the application problem provided.

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, what was his uphill rate?

Answer

4

mph

7.5-295 §7.5 source ↗

Solve Uniform Motion Applications

In the following exercises, solve the application problem provided.

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

7.5-297 §7.5 source ↗

Solve Uniform Motion Applications

In the following exercises, solve the application problem provided.

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

7.5-299 §7.5 source ↗

Solve Uniform Motion Applications

In the following exercises, solve the application problem provided.

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

7.5-301 §7.5 source ↗

Solve Uniform Motion Applications

In the following exercises, solve the application problem provided.

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

7.5-303 §7.5 source ↗

Solve Uniform Motion Applications

In the following exercises, solve the application problem provided.

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

7.5-305 §7.5 source ↗

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

7.5-307 §7.5 source ↗

Solve Work Applications

Mia 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

7.5-309 §7.5 source ↗

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

7.5-311 §7.5 source ↗

Solve Work Applications

At the end of the day Dodie can clean her hair salon in 15 minutes. Ann, who works with her, can clean the salon in 30 minutes. How long would it take them to clean the shop if they work together?

Answer

10

min

7.5-313 §7.5 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

7.5-315 §7.5 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

7.5-317 §7.5 source ↗

Solve Direct Variation Problems

In the following exercises, solve.

The price, $P,$ that Eric pays for gas varies directly with the number of gallons, $g,$ he buys. It costs him $50 to buy 20 gallons of gas.

ⓐ Write the equation that relates $P$ and $g .$ ⓑ How much would 33 gallons cost Eric?

Answer

ⓐ

P = 2.5 g

ⓑ

$ 82.50

7.5-319 §7.5 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

7.5-321 §7.5 source ↗

Solve Direct Variation Problems

In the following exercises, solve.

The maximum load a beam will support varies directly with the square of the diagonal of the beam’s cross-section. A beam with diagonal 6 inch will support a maximum load of 108 pounds.

ⓐ Write the equation that relates the load to the diagonal of the cross-section. ⓑ What load will a beam with a 10-inch diagonal support?

Answer

ⓐ

L = 3 d^{2}

ⓑ

300

pounds

7.5-323 §7.5 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}

7.5-325 §7.5 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}

7.5-327 §7.5 source ↗

In the following exercises, 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 getting 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

7.5-329 §7.5 source ↗

In the following exercises, 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 Ada 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 Ada to pump her basement if she used a pump rated at 400 gpm?

Answer

ⓐ

t = \frac{1000}{r}

ⓑ

2.5

hours

7.5-331 §7.5 source ↗

In the following exercises, write an inverse variation equation to solve the following problems.

Paul, a dentist, determined that the number of cavities that develops in his patient’s mouth each year varies inversely to the number of minutes spent brushing each night. His patient, Lori, had four cavities when brushing her teeth 30 seconds (0.5 minutes) each night.

ⓐ Write the equation that relates the number of cavities to the time spent brushing. ⓑ How many cavities would Paul expect Lori to have if she had brushed her teeth for 2 minutes each night?

Answer

ⓐ

c = \frac{2}{t}

ⓑ

1

cavity

7.5-333 §7.5 source ↗

In the following exercises, write an inverse variation equation to solve the following problems.

The cost of a ride service varies directly with the distance traveled. 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

7.5-335 §7.5 source ↗

In the following exercises, write an inverse variation equation to solve the following problems.

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 7.44.

Answer

Answers will vary.

7.5-337 §7.5 source ↗

In the following exercises, 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.

Be Prepared 7.16 §7.6 source ↗

Find the value of

x - 5

when ⓐ

x = 6

ⓑ

x = −3

ⓒ

x = 5 .

If you missed this problem, review Example 1.6.

Example 1.6 in the book ↗

Answer

ⓐ 1; ⓑ −8; ⓒ 0

Be Prepared 7.17 §7.6 source ↗

Solve:

8 - 2 x < 12 .

If you missed this problem, review Example 2.52.

Example 2.52 in the book ↗

Answer

x > −2

Be Prepared 7.18 §7.6 source ↗

Write in interval notation:

−3 \leq x < 5 .

If you missed this problem, review Example 2.49.

Example 2.49 in the book ↗

Answer

[−3, 5)

Try It 7.107 §7.6 source ↗

Solve and write the solution in interval notation:

\frac{x - 2}{x + 4} \geq 0 .

Answer

(− \infty, −4) \cup [2, \infty)

Try It 7.108 §7.6 source ↗

Solve and write the solution in interval notation:

\frac{x + 2}{x - 4} \geq 0 .

Answer

(− \infty, −2] \cup (4, \infty)

Try It 7.109 §7.6 source ↗

Solve and write the solution in interval notation:

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

Answer

(- \frac{3}{2}, 3)

Try It 7.110 §7.6 source ↗

Solve and write the solution in interval notation:

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

Answer

(−8, 4)

Try It 7.111 §7.6 source ↗

Solve and write the solution in interval notation:

\frac{1}{x^{2} + 2 x - 8} >

0 .

Answer

(− \infty, −4) \cup (2, \infty)

Try It 7.112 §7.6 source ↗

Solve and write the solution in interval notation:

\frac{3}{x^{2} + x - 12} >

0 .

Answer

(− \infty, −4) \cup (3, \infty)

Try It 7.113 §7.6 source ↗

Solve and write the solution in interval notation:

\frac{1}{2} +

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

\frac{3}{x} .

Answer

(2, 4)

Try It 7.114 §7.6 source ↗

Solve and write the solution in interval notation:

\frac{1}{3} +

\frac{6}{x^{2}} <

\frac{3}{x} .

Answer

(3, 6)

Try It 7.115 §7.6 source ↗

Given the function

R (x) =

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

find the values of x that make the function less than or equal to 0.

Answer

(−4, 2]

Try It 7.116 §7.6 source ↗

Given the function

R (x) =

\frac{x + 1}{x - 4},

find the values of x that make the function less than or equal to 0.

Answer

[−1, 4)

Try It 7.117 §7.6 source ↗

The function

C (x) =

20 x +

6000

represents the cost to produce

x,

number of items. Find ⓐ the average cost function,

c (x)

ⓑ how many items should be produced so that the average cost is less than $60?

Answer

ⓐ

c (x) =

\frac{20 x + 6000}{x}

ⓑ More than 150 items must be produced to keep the average cost below $60 per item.

Try It 7.118 §7.6 source ↗

The function

C (x) = 5 x + 900

represents the cost to produce

x,

number of items. Find ⓐ the average cost function,

c (x)

ⓑ how many items should be produced so that the average cost is less than $20?

Answer

ⓐ

c (x) =

\frac{5 x + 900}{x}

ⓑ More than 60 items must be produced to keep the average cost below $20 per item.

7.6-339 §7.6 source ↗

Solve Rational Inequalities

In the following exercises, solve each rational inequality and write the solution in interval notation.

$\frac{x - 3}{x + 4} \geq 0$

Answer

(− \infty, −4) \cup [3, \infty)

7.6-341 §7.6 source ↗

Solve Rational Inequalities

In the following exercises, solve each rational inequality and write the solution in interval notation.

$\frac{x + 1}{x - 3} \leq 0$

Answer

[−1, 3)

7.6-343 §7.6 source ↗

Solve Rational Inequalities

In the following exercises, solve each rational inequality and write the solution in interval notation.

$\frac{x - 7}{x - 1} > 0$

Answer

(− \infty, 1) \cup (7, \infty)

7.6-345 §7.6 source ↗

Solve Rational Inequalities

In the following exercises, solve each rational inequality and write the solution in interval notation.

$\frac{x - 6}{x + 5} < 0$

Answer

(−5, 6)

7.6-347 §7.6 source ↗

Solve Rational Inequalities

In the following exercises, solve each rational inequality and write the solution in interval notation.

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

Answer

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

7.6-349 §7.6 source ↗

Solve Rational Inequalities

In the following exercises, solve each rational inequality and write the solution in interval notation.

$\frac{6 x}{x - 6} > 2$

Answer

(− \infty, −3) \cup (6, \infty)

7.6-351 §7.6 source ↗

Solve Rational Inequalities

In the following exercises, solve each rational inequality and write the solution in interval notation.

$\frac{2 x + 3}{x - 6} \leq 1$

Answer

[−9, 6)

7.6-353 §7.6 source ↗

Solve Rational Inequalities

In the following exercises, solve each rational inequality and write the solution in interval notation.

$\frac{3 x - 2}{x - 4} \geq 2$

Answer

(− \infty, −6] \cup (4, \infty)

7.6-355 §7.6 source ↗

Solve Rational Inequalities

In the following exercises, solve each rational inequality and write the solution in interval notation.

$\frac{1}{x^{2} + 7 x + 12} >$ $0$

Answer

(− \infty, −4) \cup (−3, \infty)

7.6-357 §7.6 source ↗

Solve Rational Inequalities

In the following exercises, solve each rational inequality and write the solution in interval notation.

$\frac{3}{x^{2} - 5 x + 4} <$ $0$

Answer

(1, 4)

7.6-359 §7.6 source ↗

Solve Rational Inequalities

In the following exercises, solve each rational inequality and write the solution in interval notation.

$\frac{2}{2 x^{2} + x - 15} \geq 0$

Answer

(− \infty, −3) \cup (\frac{5}{2}, \infty)

7.6-361 §7.6 source ↗

Solve Rational Inequalities

In the following exercises, solve each rational inequality and write the solution in interval notation.

$\frac{−2}{6 x^{2} - 13 x + 6} \leq 0$

Answer

(− \infty, \frac{2}{3}) \cup (\frac{3}{2}, \infty)

7.6-363 §7.6 source ↗

Solve Rational Inequalities

In the following exercises, solve each rational inequality and write the solution in interval notation.

$\frac{1}{2} +$ $\frac{12}{x^{2}} >$ $\frac{5}{x}$

Answer

(− \infty, 0) \cup (0, 4) \cup (6, \infty)

7.6-365 §7.6 source ↗

Solve Rational Inequalities

In the following exercises, solve each rational inequality and write the solution in interval notation.

$\frac{1}{2} - \frac{4}{x^{2}} \leq \frac{1}{x}$

Answer

[−2, 0) \cup (0, 4]

7.6-367 §7.6 source ↗

Solve Rational Inequalities

In the following exercises, solve each rational inequality and write the solution in interval notation.

$\frac{1}{x^{2} - 16} < 0$

Answer

(−4, 4)

7.6-369 §7.6 source ↗

Solve Rational Inequalities

In the following exercises, solve each rational inequality and write the solution in interval notation.

$\frac{4}{x - 2} \geq \frac{3}{x + 1}$

Answer

[−10, −1) \cup (2, \infty)

7.6-371 §7.6 source ↗

Solve an Inequality with Rational Functions

In the following exercises, solve each rational function inequality and write the solution in interval notation.

Given the function $R (x) =$ $\frac{x - 5}{x - 2},$ find the values of $x$ that make the function less than or equal to 0.

Answer

(2, 5]

7.6-373 §7.6 source ↗

Solve an Inequality with Rational Functions

In the following exercises, solve each rational function inequality and write the solution in interval notation.

Given the function $R (x) = \frac{x - 6}{x + 2}$ , find the values of x that make the function less than or equal to 0.

Answer

(−2, 6]

7.6-375 §7.6 source ↗

Solve an Inequality with Rational Functions

In the following exercises, solve each rational function inequality and write the solution in interval notation.

Write the steps you would use to explain solving rational inequalities to your little brother.

Answer

Answers will vary.

7.R-377 §7.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.

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

Answer

a \neq \frac{2}{3}

7.R-379 §7.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.

$\frac{5 x^{2} y^{2}}{8 y}$

Answer

y \neq 0

7.R-381 §7.R source ↗

Simplify Rational Expressions

In the following exercises, simplify.

$\frac{18}{24}$

Answer

\frac{3}{4}

7.R-383 §7.R source ↗

Simplify Rational Expressions

In the following exercises, simplify.

$\frac{x^{2} + 7 x + 12}{x^{2} + 8 x + 16}$

Answer

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

7.R-385 §7.R source ↗

Multiply Rational Expressions

In the following exercises, multiply.

$\frac{5}{8} \cdot \frac{4}{15}$

Answer

\frac{1}{6}

7.R-387 §7.R source ↗

Multiply Rational Expressions

In the following exercises, multiply.

$\frac{72 x - 12 x^{2}}{8 x + 32} \cdot \frac{x^{2} + 10 x + 24}{x^{2} - 36}$

Answer

\frac{−3 x}{2}

7.R-389 §7.R source ↗

Divide Rational Expressions

In the following exercises, divide.

$\frac{x^{2} - 4 x - 12}{x^{2} + 8 x + 12} \div \frac{x^{2} - 36}{3 x}$

Answer

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

7.R-391 §7.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}

7.R-393 §7.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}

7.R-395 §7.R source ↗

Multiply and Divide Rational Functions

Find $R (x) =$ $f (x) \cdot g (x)$ where $f (x) =$ $\frac{9 x^{2} + 9 x}{x^{2} - 3 x - 4}$ and $g (x) =$ $\frac{x^{2} - 16}{3 x^{2} + 12 x} .$

Answer

R (x) = 3

7.R-397 §7.R source ↗

Add and Subtract Rational Expressions with a Common Denominator

In the following exercises, perform the indicated operations.

$\frac{7}{15} + \frac{8}{15}$

Answer

1

7.R-399 §7.R source ↗

Add and Subtract Rational Expressions with a Common Denominator

In the following exercises, perform the indicated operations.

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

Answer

y + 5

7.R-401 §7.R source ↗

Add and Subtract Rational Expressions with a Common Denominator

In the following exercises, perform the indicated operations.

$\frac{x^{2}}{x - 7} -$ $\frac{3 x + 28}{x - 7}$

Answer

x + 4

7.R-403 §7.R source ↗

Add and Subtract Rational Expressions with a Common Denominator

In the following exercises, perform the indicated operations.

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

Answer

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

7.R-405 §7.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}

7.R-407 §7.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^{2} - 49}

7.R-409 §7.R source ↗

Find the Least Common Denominator of Rational Expressions

In the following exercises, find the LCD.

$\frac{7}{a^{2} - 3 a - 10},$ $\frac{3 a}{a^{2} - a - 20}$

Answer

(a + 2) (a - 5) (a + 4)

7.R-411 §7.R source ↗

Find the Least Common Denominator of Rational Expressions

In the following exercises, find the LCD.

$\frac{5}{3 p^{2} + 17 p - 6},$ $\frac{2 m}{3 p^{2} + 23 p - 8}$

Answer

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

7.R-413 §7.R source ↗

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

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

Answer

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

7.R-415 §7.R source ↗

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

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

Answer

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

7.R-417 §7.R source ↗

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

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

Answer

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

7.R-419 §7.R source ↗

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

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

Answer

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

7.R-421 §7.R source ↗

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

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

Answer

\frac{4}{a - 8}

7.R-423 §7.R source ↗

Add and Subtract Rational Functions

In the following exercises, find $R (x) =$ $f (x) +$ $g (x)$ where $f (x)$ and $g (x)$ are given.

$f (x) =$ $\frac{2 x^{2} + 12 x - 11}{x^{2} + 3 x - 10},$ $g (x) = \frac{x + 1}{2 - x}$

Answer

R (x) = \frac{x + 8}{x + 5}

7.R-425 §7.R source ↗

In the following exercises, find $R (x) =$ $f (x) -$ $g (x)$ where $f (x)$ and $g (x)$ are given.

$f (x) =$ $\frac{4 x}{x^{2} - 121},$ $g (x) = \frac{2}{x - 11}$

Answer

R (x) = \frac{2}{x + 11}

7.R-427 §7.R source ↗

Simplify a Complex Rational Expression by Writing It as Division

In the following exercises, simplify.

$\frac{\frac{7 x}{x + 2}}{\frac{14 x^{2}}{x^{2} - 4}}$

Answer

\frac{x - 2}{2 x}

7.R-429 §7.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}

7.R-431 §7.R 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}

7.R-433 §7.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}

7.R-435 §7.R source ↗

Solve Rational Equations

In the following exercises, solve.

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

Answer

x = \frac{6}{7}

7.R-437 §7.R source ↗

Solve Rational Equations

In the following exercises, solve.

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

Answer

b = \frac{3}{2}

7.R-439 §7.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

7.R-441 §7.R source ↗

Solve Rational Equations that Involve Functions

For rational function, $f (x) =$ $\frac{x + 2}{x^{2} - 6 x + 8},$ ⓐ find the domain of the function ⓑ solve $f (x) = 1$ ⓒ find the points on the graph at this function value.

Answer

ⓐ The domain is all real numbers except

x \neq 2

and

x \neq 4 .

ⓑ

x = 1, x = 6

ⓒ

(1, 1), (6, 1)

7.R-443 §7.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}

7.R-445 §7.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}

7.R-447 §7.R source ↗

Solve Proportions

In the following exercises, solve.

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

Answer

x = \frac{12}{5}

7.R-449 §7.R source ↗

Solve Proportions

In the following exercises, solve.

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

Answer

s = 15

7.R-451 §7.R source ↗

Solve Using Proportions

In the following exercises, solve.

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

Answer

1126

calories

7.R-453 §7.R source ↗

Solve Similar Figure Applications

In the following exercises, solve.

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

The first figure is triangle A B C with side A B 8 units long, side B C 7 units long, and side A C b units long. The second figure is triangle X Y Z with side X Y 2 and two-thirds units long, side Y Z x units long, and side X Z 3 units long.

Answer

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

7.R-455 §7.R source ↗

Solve Similar Figure Applications

In the following exercises, solve.

Francesca is 5.75 feet tall. Late one afternoon, her 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

7.R-457 §7.R source ↗

Solve Uniform Motion Applications

In the following exercises, solve.

When making the 5-hour drive home from visiting her parents, Lolo 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 when it turned bad. How fast did she drive when the weather was bad?

Answer

45

mph

7.R-459 §7.R source ↗

Solve Uniform Motion Applications

In the following exercises, solve.

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

Answer

16

mph

7.R-461 §7.R source ↗

Solve Work Applications

In the following exercises, solve.

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

Answer

48

minutes

7.R-463 §7.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

7.R-465 §7.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

x = 7

7.R-467 §7.R source ↗

Solve Direct Variation Problems

In the following exercises, solve.

Vanessa is traveling to see her fiancé. The distance, $d,$ varies directly with the speed, $v,$ she drives. If she travels 258 miles driving 60 mph, how far would she travel going 70 mph?

Answer

301

mph

7.R-469 §7.R source ↗

Solve Direct Variation Problems

In the following exercises, solve.

The distance to stop a car varies directly with the square of its speed. It takes 200 feet to stop a car going 50 mph. How many feet would it take to stop a car going 60 mph?

Answer

288

feet

7.R-471 §7.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

7.R-473 §7.R source ↗

Solve Rational Inequalities

In the following exercises, solve each rational inequality and write the solution in interval notation.

$\frac{x - 3}{x + 4} \leq 0$

Answer

(−4, 3]

7.R-475 §7.R source ↗

Solve Rational Inequalities

In the following exercises, solve each rational inequality and write the solution in interval notation.

$\frac{3 x - 2}{x - 4} \leq 2$

Answer

[−6, 4)

7.R-477 §7.R source ↗

Solve Rational Inequalities

In the following exercises, solve each rational inequality and write the solution in interval notation.

$\frac{1}{2} - \frac{4}{x^{2}} \geq \frac{1}{x}$

Answer

(− \infty, −2] \cup [4, \infty)

7.R-479 §7.R source ↗

Solve an Inequality with Rational Functions

In the following exercises, solve each rational function inequality and write the solution in interval notation

Given the function, $R (x) =$ $\frac{x - 5}{x - 2},$ find the values of $x$ that make the function greater than or equal to 0.

Answer

(− \infty, 2) \cup [5, \infty)

7.R-481 §7.R source ↗

Solve an Inequality with Rational Functions

In the following exercises, solve each rational function inequality and write the solution in interval notation

The function
$C (x) =$ $150 x +$ $100,000$ represents the cost to produce $x,$ number of items. Find ⓐ the average cost function, $c (x)$ ⓑ how many items should be produced so that the average cost is less than $160.

Answer

ⓐ

c (x) =

\frac{150 x + 100000}{x}

ⓑ More than 10,000 items must be produced to keep the average cost below

$ 160

per item.

7.T-483 §7.T source ↗

In the following exercises, simplify.

$\frac{4 a^{2} b}{12 a b^{2}}$

Answer

\frac{a}{3 b}

7.T-485 §7.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}

7.T-487 §7.T source ↗

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

$\frac{6 x^{2} - x + 20}{x^{2} - 81} -$ $\frac{5 x^{2} + 11 x - 7}{x^{2} - 81}$

Answer

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

7.T-489 §7.T source ↗

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

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

Answer

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

7.T-491 §7.T source ↗

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

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

Answer

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

7.T-493 §7.T source ↗

In the following exercises, solve each equation.

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

Answer

z = \frac{1}{2}

7.T-495 §7.T source ↗

In the following exercises, solve each rational inequality and write the solution in interval notation.

$\frac{6 x}{x - 6} \leq 2$

Answer

[−3, 6)

7.T-497 §7.T source ↗

In the following exercises, solve each rational inequality and write the solution in interval notation.

$\frac{1}{2} + \frac{12}{x^{2}} \geq \frac{5}{x}$

Answer

(− \infty, 0) \cup (0, 4] \cup [6, \infty)

7.T-499 §7.T source ↗

In the following exercises, find $R (x)$ given $f (x) =$ $\frac{x - 4}{x^{2} - 3 x - 10}$ and $g (x) =$ $\frac{x - 5}{x^{2} - 2 x - 8} .$

$R (x) =$ $f (x) \cdot g (x)$

Answer

R (x) =

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

7.T-501 §7.T source ↗

In the following exercises, find $R (x)$ given $f (x) =$ $\frac{x - 4}{x^{2} - 3 x - 10}$ and $g (x) =$ $\frac{x - 5}{x^{2} - 2 x - 8} .$

Given the function,
$R (x) =$ $\frac{2}{2 x^{2} + x - 15},$ find the values of $x$ that make the function less than or equal to 0.

Answer

(−3, \frac{5}{2})

7.T-503 §7.T source ↗

In the following exercises, solve.

If $y$ varies inversely with the square of $x$ and $x = 3$ when $y = 9,$ find $y$ when $x = 4 .$

Answer

y = \frac{81}{16}

7.T-505 §7.T source ↗

In the following exercises, solve.

Oliver can split a truckload of logs in 8 hours, but working with his dad they can get it done in 3 hours. How long would it take Oliver’s dad working alone to split the logs?

Answer

Oliver’s dad would take

4 \frac{4}{5}

hours to split the logs himself.

7.T-507 §7.T source ↗

In the following exercises, solve.

The cities of Dayton, Columbus, and Cincinnati form a triangle in southern Ohio. The diagram 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

The distance between Dayton and Columbus is 64 miles.

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Problems and answers are adapted from Intermediate Algebra 2e by Lynn Marecek and Andrea Honeycutt Mathis, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/intermediate-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.