Simplify ((x^2)/(x^2+7x+6))÷((9x-9)/(x^2-1))

$\frac{x^{2}}{x^{2} + 7 x + 6} \div \frac{9 x - 9}{x^{2} - 1}$

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2}}{x^{2} + 7 x + 6} \cdot \frac{x^{2} - 1}{9 x - 9}$

Factor $x^{2} + 7 x + 6$ 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 $6$ and whose sum is $7$ .

$1, 6$

Write the factored form using these integers.

$\frac{x^{2}}{(x + 1) (x + 6)} \cdot \frac{x^{2} - 1}{9 x - 9}$

Simplify the numerator.

Rewrite $1$ as $1^{2}$ .

$\frac{x^{2}}{(x + 1) (x + 6)} \cdot \frac{x^{2} - 1^{2}}{9 x - 9}$

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 $b = 1$ .

$\frac{x^{2}}{(x + 1) (x + 6)} \cdot \frac{(x + 1) (x - 1)}{9 x - 9}$

Simplify terms.

Factor $9$ out of $9 x - 9$ .

Factor $9$ out of $9 x$ .

$\frac{x^{2}}{(x + 1) (x + 6)} \cdot \frac{(x + 1) (x - 1)}{9 (x) - 9}$

Factor $9$ out of $- 9$ .

$\frac{x^{2}}{(x + 1) (x + 6)} \cdot \frac{(x + 1) (x - 1)}{9 (x) + 9 (- 1)}$

Factor $9$ out of $9 (x) + 9 (- 1)$ .

$\frac{x^{2}}{(x + 1) (x + 6)} \cdot \frac{(x + 1) (x - 1)}{9 (x - 1)}$

Combine.

$\frac{x^{2} ((x + 1) (x - 1))}{(x + 1) (x + 6) (9 (x - 1))}$

Cancel the common factor of $x + 1$ .

Cancel the common factor.

$\frac{x^{2} ((x + 1) (x - 1))}{(x + 1) (x + 6) (9 (x - 1))}$

Rewrite the expression.

$\frac{x^{2} (x - 1)}{(x + 6) (9 (x - 1))}$

Cancel the common factor of $x - 1$ .

Cancel the common factor.

$\frac{x^{2} (x - 1)}{(x + 6) (9 (x - 1))}$

Rewrite the expression.

$\frac{x^{2}}{(x + 6) \cdot (9)}$

Move $9$ to the left of $x + 6$ .

$\frac{x^{2}}{9 (x + 6)}$

Simplify ((x^2)/(x^2+7x+6))÷((9x-9)/(x^2-1))

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