Problem 74 - College Algebra

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 simplified form of the given expression is \(\frac{1}{6}\).

Step 1: Express division as multiplication

The expression can be written as multiplication by finding the reciprocal of the divisor, \((\frac{1}{x-4}-\frac{1}{x+2})^{-1}\). So, our expression becomes \(\frac{1}{x^{2}-2 x-8} * (\frac{1}{x-4}-\frac{1}{x+2})^{-1}\).

Step 2: Factorise the denominators in the second fraction

The second fraction can be written on a common denominator. Thus, \(\frac{1}{x-4}-\frac{1}{x+2}= \frac{x+2-x+4}{(x-4)(x+2)}=\frac{6}{x^{2}-2 x-8}\). Thus, our expression can be further written as \(\frac{1}{x^{2}-2 x-8} * (\frac{6}{x^{2}-2 x-8})^{-1}\).

Step 3: Multiply the fractions

Only thing left to do is to multiply the fractions, which simplifies to \(\frac{1}{x^{2}-2 x-8} * \frac{x^{2}-2 x-8}{6} = \frac{1}{6}\).