Precalculus Examples

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 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 $b = 2$ .

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

Step 3

Simplify the denominator.

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Step 3.1

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

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 3.3

Add $- 2$ and $1$ .

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

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

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

Step 6

Cancel the common factor of $x - 1$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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

Step 7

Cancel the common factor of $x - 2$ .

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Step 7.1

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

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

$\frac{1}{x + 2}$