Simplify ((x-2)/(x-1))÷((x^2-4)/(x^2+x-2))

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Factor $x^{2} + x - 2$ 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 $- 2$ and whose sum is $1$ .

$- 1, 2$

Write the factored form using these integers.

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

Step 3

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

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

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 $b = 2$ .

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

Step 4

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x - 2$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 1$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{x + 2}{x + 2}$

Cancel the common factor of $x + 2$ .

Cancel the common factor.

$\frac{x + 2}{x + 2}$

Rewrite the expression.

$1$

Simplify ((x-2)/(x-1))÷((x^2-4)/(x^2+x-2))

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