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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. 7.48 (page 671)

Simplify the complex rational expression by writing it as division:
$\frac{\frac{1}{x^{2} - 7 x + 12}}{\frac{2}{x - 4}}$

Short Answer

Expert verified

The simplified answer is $\frac{1}{2 (x - 3)}$

Step by step solution

01

Step 1. Given information.

The following rational expression is given:

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

02

Step 2. Simplify the rational expression by division.

Rewrite the complex fraction as division:

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

Rewrite as the product of the first and the reciprocal of the second:

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

Now,

$x^{2} - 7 x + 12 = x^{2} - 4 x - 3 x + 12 x^{2} - 7 x + 12 = x (x - 4) - 3 (x - 4) x^{2} - 7 x + 12 = (x - 4) (x - 3)$

Factor:

$\frac{1}{(x - 3) (x - 4)} . \frac{x - 4}{2}$

Multiply:

$\frac{(x - 4)}{2 (x - 3) (x - 4)}$

Remove common factors from the numerator and the denominator:

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

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Most popular questions from this chapter

In the following expression, perform the indicated operation:

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

For the following exercises, perform the indicated operations. $\frac{10 m^{2} + 80 m}{3 m - 9} . \frac{m^{2} + 4 m - 21}{m^{2} - 9 m + 20} 梅 \frac{5 m^{2} + 10 m}{2 m - 10}$

Simplify the complex rational expression by writing it as division:

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

For the following exercises, find $R (x) = f (x) g (x)$ where $f (x) a n d g (x)$ are given.

$f (x) = \frac{2 x^{2} + 8 x}{x^{2} - 9 x + 20} a n d g (x) = \frac{x - 5}{x^{2}}$

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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