Problem 74 - College Algebra Essentials

Question

Perform the indicated operations. Simplify the result, if possible. $$\frac{1}{x^{2}-2 x-8} \div\left(\frac{1}{x-4}-\frac{1}{x+2}\right)$$

Step-by-Step Solution

Answer

The simplest form of the given expression is \(\frac{6x+16 }{(x-4)(x+2)(x^{2}-2 x-8)}\)

Step 1: Understand the Division of Fractions

Rewrite the division as multiplication by the reciprocal of the divisor. The expression should look like this \(\frac{1}{x^{2}-2 x-8} \times \left(\frac{1}{x-4}-\frac{1}{x+2}\right)^{-1}\).

Step 2: Reciprocal of Divisor

Find the reciprocal of the divisor. \(\left(\frac{1}{x-4}-\frac{1}{x+2}\right)^{-1}\) simplifies to \( \frac{x+2}{x-4} - \frac{x-4}{x+2}\) .

Step 3: Multiply Both Expressions

Multiply both expressions together which results in \(\frac{x+2 }{(x-4)(x^{2}-2 x-8)} - \frac{x-4}{(x+2)(x^{2}-2 x-8)}\).

Step 4: Simplify the Result

Combine like terms in the result and simplify. The result is \(\frac{6x+16 }{(x-4)(x+2)(x^{2}-2 x-8)}\).