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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 7: Q145E. (page 669)

In the following exercise, find
(a) $R (x) = f (x) + g (x)$
(b) $R (x) = f (x) - g (x)$ :
$f (x) = \frac{6 x}{x^{2} - 64} g (x) = \frac{3}{x - 8}$

Short Answer

Expert verified

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

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

Step by step solution

01

Step 1. Given information.

The following information is given:

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

02

Step 1. Finding (a) Rx=f(x)+g(x).

$R (x) = f (x) + g (x)$

Substituting the values

$R (x) = \frac{6 x}{x^{2} - 64} + \frac{3}{x - 8}$

Factorise the denominators:

$R (x) = \frac{6 x}{(x - 8) (x + 8)} + \frac{3}{(x - 8)}$

Find out the LCD:

$(x - 8) (x + 8)$

Rewrite the expression with a denominator equal to the LCD:

$R (x) = \frac{6 x}{(x - 8) (x + 8)} + \frac{3 (x + 8)}{(x - 8) (x + 8)}$

Simplify:

$R (x) = \frac{6 x + 3 x + 24}{(x - 8) (x + 8)} R (x) = \frac{9 x + 24}{(x - 8) (x + 8)} R (x) = \frac{3 (3 x + 8)}{(x - 8) (x + 8)}$

03

Step 3. Finding (b) Rx=fx-gx.

$R (x) = f (x) - g (x)$

By substituting the values and factorising:

$R (x) = \frac{6 x}{(x - 8) (x + 8)} - \frac{3}{(x - 8)}$

Finding the LCD:

$(x - 8) (x + 8)$

Rewriting the expression of $R (x)$ :

$R (x) = \frac{6 x}{(x - 8) (x + 8)} - \frac{3 (x + 8)}{(x - 8) (x + 8)}$

Simplifying:

$R (x) = \frac{6 x - 3 x - 24}{(x - 8) (x + 8)} R (x) = \frac{3 x - 24}{(x - 8) (x + 8)} R (x) = \frac{3 (x - 8)}{(x - 8) (x + 8)}$

By cancelling the common factors from the numerator and the denominator:

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

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