Simplify ((x-4)/(x^2-25))/(1+1/(x-5))

$\frac{\frac{x - 4}{x^{2} - 25}}{1 + \frac{1}{x - 5}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x - 4}{x^{2} - 25} \cdot \frac{1}{1 + \frac{1}{x - 5}}$

Step 2

Simplify the denominator.

Rewrite $25$ as $5^{2}$ .

$\frac{x - 4}{x^{2} - 5^{2}} \cdot \frac{1}{1 + \frac{1}{x - 5}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 5$ .

$\frac{x - 4}{(x + 5) (x - 5)} \cdot \frac{1}{1 + \frac{1}{x - 5}}$

Step 3

Simplify the denominator.

Write $1$ as a fraction with a common denominator.

$\frac{x - 4}{(x + 5) (x - 5)} \cdot \frac{1}{\frac{x - 5}{x - 5} + \frac{1}{x - 5}}$

Combine the numerators over the common denominator.

$\frac{x - 4}{(x + 5) (x - 5)} \cdot \frac{1}{\frac{x - 5 + 1}{x - 5}}$

Add $- 5$ and $1$ .

$\frac{x - 4}{(x + 5) (x - 5)} \cdot \frac{1}{\frac{x - 4}{x - 5}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{x - 4}{(x + 5) (x - 5)} (1 \frac{x - 5}{x - 4})$

Step 5

Multiply $\frac{x - 5}{x - 4}$ by $1$ .

$\frac{x - 4}{(x + 5) (x - 5)} \cdot \frac{x - 5}{x - 4}$

Step 6

Cancel the common factor of $x - 4$ .

Cancel the common factor.

$\frac{x - 4}{(x + 5) (x - 5)} \cdot \frac{x - 5}{x - 4}$

Rewrite the expression.

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

Step 7

Cancel the common factor of $x - 5$ .

Factor $x - 5$ out of $(x + 5) (x - 5)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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