Algebra Examples

$\frac{\frac{x - 3}{x^{2} - 16}}{1 + \frac{1}{x - 4}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x - 3}{x^{2} - 16} \cdot \frac{1}{1 + \frac{1}{x - 4}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $16$ as $4^{2}$ .

$\frac{x - 3}{x^{2} - 4^{2}} \cdot \frac{1}{1 + \frac{1}{x - 4}}$

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 $b = 4$ .

$\frac{x - 3}{(x + 4) (x - 4)} \cdot \frac{1}{1 + \frac{1}{x - 4}}$

Step 3

Simplify the denominator.

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Step 3.1

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

$\frac{x - 3}{(x + 4) (x - 4)} \cdot \frac{1}{\frac{x - 4}{x - 4} + \frac{1}{x - 4}}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{x - 3}{(x + 4) (x - 4)} \cdot \frac{1}{\frac{x - 4 + 1}{x - 4}}$

Step 3.3

Add $- 4$ and $1$ .

$\frac{x - 3}{(x + 4) (x - 4)} \cdot \frac{1}{\frac{x - 3}{x - 4}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{x - 3}{(x + 4) (x - 4)} (1 \frac{x - 4}{x - 3})$

Step 5

Multiply $\frac{x - 4}{x - 3}$ by $1$ .

$\frac{x - 3}{(x + 4) (x - 4)} \cdot \frac{x - 4}{x - 3}$

Step 6

Cancel the common factor of $x - 3$ .

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Step 6.1

Cancel the common factor.

$\frac{x - 3}{(x + 4) (x - 4)} \cdot \frac{x - 4}{x - 3}$

Step 6.2

Rewrite the expression.

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

Step 7

Cancel the common factor of $x - 4$ .

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Step 7.1

Factor $x - 4$ out of $(x + 4) (x - 4)$ .

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

$\frac{1}{x + 4}$