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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 7: Q. 164 (page 680)

In the following exercises, simplify each complex rational expression by writing it as division.
$\frac{y - \frac{2 y}{y - 4}}{\frac{2}{y - 4} + \frac{2}{y + 4}}$

Short Answer

Expert verified

The simplified expression is $\frac{(y - 6) (y + 4)}{4}$ .

Step by step solution

01

Step 1. Given Information.

We have:

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

02

Step 2. Simplify the numerator.

The numerator is $y - \frac{2 y}{y - 4}$ .

Apply the fraction rule: $a - \frac{b}{c} = \frac{a c - b}{c}$

$= \frac{y (y - 4) - 2 y}{y - 4} = \frac{y^{2} - 6 y}{y - 4}$

03

Step 3. Simplify the denominator.

The denominator is $\frac{2}{y - 4} + \frac{2}{y + 4}$ .

The least common divisor is $(y - 4) (y + 4)$ .

Adjust the fractions based on the LCM:

$= \frac{2 (y + 4)}{(y - 4) (y + 4)} + \frac{2 (y - 4)}{(y - 4) (y + 4)} = \frac{2 (y + 4) + 2 (y - 4)}{(y - 4) (y + 4)} = \frac{4 y}{(y - 4) (y + 4)}$

04

Step 4. Substitute the simplified numerator and denominator into the original expression.

We have:

$\frac{y - \frac{2 y}{y - 4}}{\frac{2}{y - 4} + \frac{2}{y + 4}} = \frac{\frac{y^{2} - 6 y}{y - 4}}{\frac{4 y}{(y - 4) (y + 4)}}$

Apply fraction rule: $\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a 路鈥� d}{b 路鈥� c}$

$= \frac{(y^{2} - 6 y) (y - 4) (y + 4)}{(y - 4) 路鈥� 4 y} = \frac{(y - 6) (y + 4)}{4}$

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Most popular questions from this chapter

In the following exercises, divide the rational expressions.

$\frac{\frac{3 b^{2} + 2 b - 8}{12 b + 18}}{\frac{3 b^{2} + 2 b - 8}{2 b^{2} - 7 b - 15}}$

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

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

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Determine the values for which the rational expression is undefined.

$\frac{x - 3}{x^{2} - x - 30}$

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