Question

question

divide: \frac{\frac{1}{x + 5} + \frac{1}{x - 5}}{\frac{1}{x + 5} - \frac{1}{x - 5}}.

provide your answer below:

Explanation:

Identify the complex fraction structure

The given expression is a complex rational expression of the form:

$$ \frac{\frac{1}{x+5} + \frac{1}{x-5}}{\frac{1}{x+5} - \frac{1}{x-5}} $$

We will simplify the numerator and the denominator separately.

Simplify the numerator

Using the Adding Rational Expressions knowledge point, we find a common denominator for the terms in the numerator, which is \((x+5)(x-5)\):

$$ \frac{1}{x+5} + \frac{1}{x-5} = \frac{1(x-5) + 1(x+5)}{(x+5)(x-5)} $$

$$ = \frac{x - 5 + x + 5}{(x+5)(x-5)} = \frac{2x}{(x+5)(x-5)} $$

Simplify the denominator

Using the Subtracting Rational Expressions knowledge point, we find a common denominator for the terms in the denominator, which is also \((x+5)(x-5)\):

$$ \frac{1}{x+5} - \frac{1}{x-5} = \frac{1(x-5) - 1(x+5)}{(x+5)(x-5)} $$

$$ = \frac{x - 5 - x - 5}{(x+5)(x-5)} = \frac{-10}{(x+5)(x-5)} $$

Divide the simplified expressions

Using the Dividing Rational Expressions knowledge point, we divide the simplified numerator by the simplified denominator:

$$ \frac{\frac{2x}{(x+5)(x-5)}}{\frac{-10}{(x+5)(x-5)}} = \frac{2x}{(x+5)(x-5)} \cdot \frac{(x+5)(x-5)}{-10} $$

Simplify the final quotient

We cancel the common factor \((x+5)(x-5)\) from the numerator and denominator:

$$ = \frac{2x}{-10} = -\frac{x}{5} $$

This holds true for \(x
eq \pm 5\).

Answer:

Divide: <blank>\(-\frac{x}{5}\)</blank>