Simplify ((x-9)/(x-1))÷((x^2-10x+9)/(x^2-8x+7))

$\frac{x - 9}{x - 1} \div \frac{x^{2} - 10 x + 9}{x^{2} - 8 x + 7}$

To divide by a fraction, multiply by its reciprocal.

$\frac{x - 9}{x - 1} \cdot \frac{x^{2} - 8 x + 7}{x^{2} - 10 x + 9}$

Factor $x^{2} - 8 x + 7$ 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 $7$ and whose sum is $- 8$ .

$- 7, - 1$

Write the factored form using these integers.

$\frac{x - 9}{x - 1} \cdot \frac{(x - 7) (x - 1)}{x^{2} - 10 x + 9}$

Factor $x^{2} - 10 x + 9$ 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 $9$ and whose sum is $- 10$ .

$- 9, - 1$

Write the factored form using these integers.

$\frac{x - 9}{x - 1} \cdot \frac{(x - 7) (x - 1)}{(x - 9) (x - 1)}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x - 9$ .

Cancel the common factor.

$\frac{x - 9}{x - 1} \cdot \frac{(x - 7) (x - 1)}{(x - 9) (x - 1)}$

Rewrite the expression.

$\frac{1}{x - 1} \cdot \frac{(x - 7) (x - 1)}{x - 1}$

Cancel the common factor of $x - 1$ .

Factor $x - 1$ out of $(x - 7) (x - 1)$ .

$\frac{1}{x - 1} \cdot \frac{(x - 1) (x - 7)}{x - 1}$

Cancel the common factor.

$\frac{1}{x - 1} \cdot \frac{(x - 1) (x - 7)}{x - 1}$

Rewrite the expression.

$\frac{x - 7}{x - 1}$

Simplify ((x-9)/(x-1))÷((x^2-10x+9)/(x^2-8x+7))

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