Precalculus Examples Rational Expressions and Equations

Simplify

$\frac{1}{x + 4} - \frac{8}{x - 6}$

Step 1

To write $\frac{1}{x + 4}$ as a fraction with a common denominator, multiply by $\frac{x - 6}{x - 6}$ .

$\frac{1}{x + 4} \cdot \frac{x - 6}{x - 6} - \frac{8}{x - 6}$

Step 2

To write $- \frac{8}{x - 6}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

$\frac{1}{x + 4} \cdot \frac{x - 6}{x - 6} - \frac{8}{x - 6} \cdot \frac{x + 4}{x + 4}$

Step 3

Write each expression with a common denominator of $(x + 4) (x - 6)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{1}{x + 4}$ by $\frac{x - 6}{x - 6}$ .

$\frac{x - 6}{(x + 4) (x - 6)} - \frac{8}{x - 6} \cdot \frac{x + 4}{x + 4}$

Step 3.2

Multiply $\frac{8}{x - 6}$ by $\frac{x + 4}{x + 4}$ .

$\frac{x - 6}{(x + 4) (x - 6)} - \frac{8 (x + 4)}{(x - 6) (x + 4)}$

Step 3.3

Reorder the factors of $(x - 6) (x + 4)$ .

$\frac{x - 6}{(x + 4) (x - 6)} - \frac{8 (x + 4)}{(x + 4) (x - 6)}$

Step 4

Combine the numerators over the common denominator.

$\frac{x - 6 - 8 (x + 4)}{(x + 4) (x - 6)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{x - 6 - 8 x - 8 \cdot 4}{(x + 4) (x - 6)}$

Step 5.2

Multiply $- 8$ by $4$ .

$\frac{x - 6 - 8 x - 32}{(x + 4) (x - 6)}$

Step 5.3

Subtract $8 x$ from $x$ .

$\frac{- 7 x - 6 - 32}{(x + 4) (x - 6)}$

Step 5.4

Subtract $32$ from $- 6$ .

$\frac{- 7 x - 38}{(x + 4) (x - 6)}$

Step 6

Simplify with factoring out.

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Step 6.1

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

$\frac{- (7 x) - 38}{(x + 4) (x - 6)}$

Step 6.2

Rewrite $- 38$ as $- 1 (38)$ .

$\frac{- (7 x) - 1 (38)}{(x + 4) (x - 6)}$

Step 6.3

Factor $- 1$ out of $- (7 x) - 1 (38)$ .

$\frac{- (7 x + 38)}{(x + 4) (x - 6)}$

Step 6.4

Simplify the expression.

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Step 6.4.1

Rewrite $- (7 x + 38)$ as $- 1 (7 x + 38)$ .

$\frac{- 1 (7 x + 38)}{(x + 4) (x - 6)}$

Step 6.4.2

Move the negative in front of the fraction.

$- \frac{7 x + 38}{(x + 4) (x - 6)}$

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