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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. 418 (page 736)

In the following exercises, perform the indicated operations .
$\frac{- 3 w - 15}{w^{2} + w - 20} - \frac{w + 2}{4 - w}$

Short Answer

Expert verified

Simplified expression is $\frac{w - 1}{w - 4}$

Step by step solution

01

Step 1. Given Information

Expression :

\frac{- 3 w - 15}{w^{2} + w - 20} - \frac{w + 2}{4 - w}

it can be rewritten as

$\frac{- (3 w + 15)}{w^{2} + w - 20} - \frac{w + 2}{- (w - 4)} = \frac{- (3 w + 15)}{w^{2} + w - 20} + \frac{w + 2}{w - 4}$

02

Step 2. Find LCD

Denominator of first fraction is $w^{2} + w - 20$

factorize this denominator

$w^{2} + 5 w - 4 w - 20 = w (w + 5) - 4 (w + 5) = (w + 5) (w - 4)$

Denominator of the second fraction

It can be written as $((w - 4))$

here $(w - 4)$ is common between both denominator

So the LCD is $(w + 5) (w - 4)$

03

Step 3. Rewrite each rational expression as an equivalent rational expression with the LCD

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

04

Step 4.Simplify the numerators.

$\frac{- 3 w - 15}{(w + 5) (w - 4)} + \frac{(w^{2} + 5 w + 2 w + 10)}{(w + 5) (w - 4)} = \frac{- 3 w - 15}{(w + 5) (w - 4)} + \frac{w^{2} + 7 w + 10}{(w + 5) (w - 4)}$

05

Step 5. Add numerators

$\frac{- 3 w - 15 + w^{2} + 7 w + 10}{(w + 5) (w - 4)}$

06

Step 6. Simplify numerator

$\frac{w^{2} + 4 w - 5}{(w + 5) (w - 4)}$

07

Step 7. Factorize numerator and remove common factor

$\frac{w^{2} + 5 w - w - 5}{(w + 5) (w - 4)} = \frac{w (w + 5) - 1 (w + 5)}{(w + 5) (w - 4)} = \frac{(w + 5) (w - 1)}{(w + 5) (w - 4)} = \frac{w - 1}{w - 4}$

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