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Elementary Algebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2nd Edition 路 1240 pages

ISBN: 9780998625713

Chapter 8: Q. 270 (page 966)

In the following exercises, simplify.
$\frac{4 - \frac{4}{b - 5}}{\frac{1}{b - 5} + \frac{b}{4}}$

Short Answer

Expert verified

The simplified form of the given expression is $\frac{16 (b - 6)}{(b - 4) (b - 1)}$ .

Step by step solution

01

Step 1. Given information.

The given expression is:

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

02

Step 2. Concept.

The following concept is required for simplification:

$\frac{A}{B} 梅 \frac{C}{D} = \frac{A}{B} 路 \frac{D}{C}$

03

Step 3. Rewrite the expression.

Simplify the numerator and denominator.

$= \frac{\frac{4 (b - 5) - 4}{b - 5}}{\frac{4 + b (b - 5)}{4 (b - 5)}} = \frac{\frac{4 b - 20 - 4}{b - 5}}{\frac{4 + b^{2} - 5 b}{4 (b - 5)}} = \frac{\frac{4 b - 24}{b - 5}}{\frac{b^{2} - 5 b + 4}{4 (b - 5)}} = \frac{4 b - 24}{b - 5} 梅 \frac{b^{2} - 5 b + 4}{4 (b - 5)}$

04

Step 4. Simplify the expression.

Rewrite as the product of first times the reciprocal of the second.

$= \frac{4 b - 24}{b - 5} 路 \frac{4 (b - 5)}{b^{2} - 5 b + 4} = \frac{4 (b - 6)}{b - 5} 路 \frac{4 (b - 5)}{b^{2} - 4 b - b + 4} = \frac{4 (b - 6)}{b - 5} 路 \frac{4 (b - 5)}{b (b - 4) - 1 (b - 4)} = \frac{4 (b - 6)}{b - 5} 路 \frac{4 (b - 5)}{(b - 4) (b - 1)} = \frac{16 (b - 6)}{(b - 4) (b - 1)}$

05

Step 5. Conclusion.

The simplified form of the given expression is $\frac{16 (b - 6)}{(b - 4) (b - 1)}$ .

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