Simplify ((x^2+9x-22)/(x^2-121))÷(2-x)

$\frac{x^{2} + 9 x - 22}{x^{2} - 121} \div (2 - x)$

Rewrite the division as a fraction.

$\frac{\frac{x^{2} + 9 x - 22}{x^{2} - 121}}{2 - x}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2} + 9 x - 22}{x^{2} - 121} \cdot \frac{1}{2 - x}$

Factor $x^{2} + 9 x - 22$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 22$ and whose sum is $9$ .

$- 2, 11$

Write the factored form using these integers.

$\frac{(x - 2) (x + 11)}{x^{2} - 121} \cdot \frac{1}{2 - x}$

Simplify the denominator.

Rewrite $121$ as $11^{2}$ .

$\frac{(x - 2) (x + 11)}{x^{2} - 11^{2}} \cdot \frac{1}{2 - x}$

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 $b = 11$ .

$\frac{(x - 2) (x + 11)}{(x + 11) (x - 11)} \cdot \frac{1}{2 - x}$

Simplify terms.

Cancel the common factor of $x + 11$ .

Cancel the common factor.

$\frac{(x - 2) (x + 11)}{(x + 11) (x - 11)} \cdot \frac{1}{2 - x}$

Rewrite the expression.

$\frac{x - 2}{x - 11} \cdot \frac{1}{2 - x}$

Multiply $\frac{x - 2}{x - 11}$ by $\frac{1}{2 - x}$ .

$\frac{x - 2}{(x - 11) (2 - x)}$

Cancel the common factor of $x - 2$ and $2 - x$ .

Factor $- 1$ out of $x$ .

$\frac{- 1 (- x) - 2}{(x - 11) (2 - x)}$

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{- 1 (- x) - 1 (2)}{(x - 11) (2 - x)}$

Factor $- 1$ out of $- 1 (- x) - 1 (2)$ .

$\frac{- 1 (- x + 2)}{(x - 11) (2 - x)}$

Reorder terms.

$\frac{- 1 (- x + 2)}{(x - 11) (- x + 2)}$

Cancel the common factor.

$\frac{- 1 (- x + 2)}{(x - 11) (- x + 2)}$

Rewrite the expression.

$\frac{- 1}{x - 11}$

Move the negative in front of the fraction.

$- \frac{1}{x - 11}$

Simplify ((x^2+9x-22)/(x^2-121))÷(2-x)

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