Simplify ((x-2)/(x^2-9))/(1+1/(x-3))

$\frac{\frac{x - 2}{x^{2} - 9}}{1 + \frac{1}{x - 3}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x - 2}{x^{2} - 9} \cdot \frac{1}{1 + \frac{1}{x - 3}}$

Step 2

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

$\frac{x - 2}{x^{2} - 3^{2}} \cdot \frac{1}{1 + \frac{1}{x - 3}}$

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 $b = 3$ .

$\frac{x - 2}{(x + 3) (x - 3)} \cdot \frac{1}{1 + \frac{1}{x - 3}}$

Step 3

Simplify the denominator.

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

$\frac{x - 2}{(x + 3) (x - 3)} \cdot \frac{1}{\frac{x - 3}{x - 3} + \frac{1}{x - 3}}$

Combine the numerators over the common denominator.

$\frac{x - 2}{(x + 3) (x - 3)} \cdot \frac{1}{\frac{x - 3 + 1}{x - 3}}$

Add $- 3$ and $1$ .

$\frac{x - 2}{(x + 3) (x - 3)} \cdot \frac{1}{\frac{x - 2}{x - 3}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{x - 2}{(x + 3) (x - 3)} (1 \frac{x - 3}{x - 2})$

Step 5

Multiply $\frac{x - 3}{x - 2}$ by $1$ .

$\frac{x - 2}{(x + 3) (x - 3)} \cdot \frac{x - 3}{x - 2}$

Step 6

Cancel the common factor of $x - 2$ .

Cancel the common factor.

$\frac{x - 2}{(x + 3) (x - 3)} \cdot \frac{x - 3}{x - 2}$

Rewrite the expression.

$\frac{1}{(x + 3) (x - 3)} (x - 3)$

Step 7

Cancel the common factor of $x - 3$ .

Factor $x - 3$ out of $(x + 3) (x - 3)$ .

$\frac{1}{(x - 3) (x + 3)} (x - 3)$

Cancel the common factor.

$\frac{1}{(x - 3) (x + 3)} (x - 3)$

Rewrite the expression.

$\frac{1}{x + 3}$