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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. 134 (page 668)

In the following exercises, perform indicated operations.
$\frac{4}{x^{2} - 6 x + 5} - \frac{3}{x^{2} - 7 x + 10}$

Short Answer

Expert verified

The required expression is $\begin{array}{rcl} \frac{1}{(x - 1) (x - 2)} \end{array}$

Step by step solution

01

Step 1. Given information

The given expression is $\frac{4}{x^{2} - 6 x + 5} - \frac{3}{x^{2} - 7 x + 10}$

02

Step 2. Find LCD

Factor the denominators of the given rational expressions.

$\begin{array}{rcl} x^{2} - 6 x + 5 & = & (x - 5) (x - 1) \\ x^{2} - 7 x + 10 & = & (x - 5) (x - 2) \end{array}$

So, the LCD is $(x - 5) (x - 1) (x - 2)$

03

Step 3. Express with LCD

Rewrite the expression in terms of LCD.

$\frac{4}{x^{2} - 6 x + 5} - \frac{3}{x^{2} - 7 x + 10} = \frac{4 x - 8}{(x - 5) (x - 1) (x - 2)} - \frac{3 x - 3}{(x - 5) (x - 1) (x - 2)}$

04

Stp 4. Simplify

As the denominator is the same for both the terms, perform subtraction in the numerator and place it over the common denominator.

$\begin{array}{rcl} \frac{4 x - 8 - 3 x + 3}{(x - 5) (x - 1) (x - 2)} & = & \frac{x - 5}{(x - 5) (x - 1) (x - 2)} \\ = & \frac{1}{(x - 1) (x - 2)} \end{array}$

So, the required expression is $\begin{array}{rcl} \frac{1}{(x - 1) (x - 2)} \end{array}$ .

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