Precalculus Examples

$\frac{\frac{x}{x + 2} - \frac{4}{x + 2}}{x - 3 - \frac{6}{x + 2}}$

Step 1

Multiply the numerator and denominator of the fraction by $x + 2$ .

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Step 1.1

Multiply $\frac{\frac{x}{x + 2} - \frac{4}{x + 2}}{x - 3 - \frac{6}{x + 2}}$ by $\frac{x + 2}{x + 2}$ .

$\frac{x + 2}{x + 2} \cdot \frac{\frac{x}{x + 2} - \frac{4}{x + 2}}{x - 3 - \frac{6}{x + 2}}$

Step 1.2

Combine.

$\frac{(x + 2) (\frac{x}{x + 2} - \frac{4}{x + 2})}{(x + 2) (x - 3 - \frac{6}{x + 2})}$

Step 2

Apply the distributive property.

$\frac{(x + 2) \frac{x}{x + 2} + (x + 2) (- \frac{4}{x + 2})}{(x + 2) x + (x + 2) \cdot - 3 + (x + 2) (- \frac{6}{x + 2})}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x + 2$ .

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Step 3.1.1

Cancel the common factor.

$\frac{(x + 2) \frac{x}{x + 2} + (x + 2) (- \frac{4}{x + 2})}{(x + 2) x + (x + 2) \cdot - 3 + (x + 2) (- \frac{6}{x + 2})}$

Step 3.1.2

Rewrite the expression.

$\frac{x + (x + 2) (- \frac{4}{x + 2})}{(x + 2) x + (x + 2) \cdot - 3 + (x + 2) (- \frac{6}{x + 2})}$

Step 3.2

Cancel the common factor of $x + 2$ .

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Step 3.2.1

Move the leading negative in $- \frac{4}{x + 2}$ into the numerator.

$\frac{x + (x + 2) \frac{- 4}{x + 2}}{(x + 2) x + (x + 2) \cdot - 3 + (x + 2) (- \frac{6}{x + 2})}$

Step 3.2.2

Cancel the common factor.

$\frac{x + (x + 2) \frac{- 4}{x + 2}}{(x + 2) x + (x + 2) \cdot - 3 + (x + 2) (- \frac{6}{x + 2})}$

Step 3.2.3

Rewrite the expression.

$\frac{x - 4}{(x + 2) x + (x + 2) \cdot - 3 + (x + 2) (- \frac{6}{x + 2})}$

Step 3.3

Cancel the common factor of $x + 2$ .

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Step 3.3.1

Move the leading negative in $- \frac{6}{x + 2}$ into the numerator.

$\frac{x - 4}{(x + 2) x + (x + 2) \cdot - 3 + (x + 2) \frac{- 6}{x + 2}}$

Step 3.3.2

Cancel the common factor.

$\frac{x - 4}{(x + 2) x + (x + 2) \cdot - 3 + (x + 2) \frac{- 6}{x + 2}}$

Step 3.3.3

Rewrite the expression.

$\frac{x - 4}{(x + 2) x + (x + 2) \cdot - 3 - 6}$

Step 4

Simplify the denominator.

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Step 4.1

Apply the distributive property.

$\frac{x - 4}{x \cdot x + 2 x + (x + 2) \cdot - 3 - 6}$

Step 4.2

Multiply $x$ by $x$ .

$\frac{x - 4}{x^{2} + 2 x + (x + 2) \cdot - 3 - 6}$

Step 4.3

Apply the distributive property.

$\frac{x - 4}{x^{2} + 2 x + x \cdot - 3 + 2 \cdot - 3 - 6}$

Step 4.4

Move $- 3$ to the left of $x$ .

$\frac{x - 4}{x^{2} + 2 x - 3 \cdot x + 2 \cdot - 3 - 6}$

Step 4.5

Multiply $2$ by $- 3$ .

$\frac{x - 4}{x^{2} + 2 x - 3 \cdot x - 6 - 6}$

Step 4.6

Subtract $3 x$ from $2 x$ .

$\frac{x - 4}{x^{2} - x - 6 - 6}$

Step 4.7

Subtract $6$ from $- 6$ .

$\frac{x - 4}{x^{2} - x - 12}$

Step 4.8

Factor $x^{2} - x - 12$ using the AC method.

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Step 4.8.1

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 $- 12$ and whose sum is $- 1$ .

$- 4, 3$

Step 4.8.2

Write the factored form using these integers.

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

Step 5

Cancel the common factor of $x - 4$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

$\frac{1}{x + 3}$