Algebra Examples

\frac{2}{x + 4} - \frac{2}{x - 4}

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

Step 1

To write

\frac{2}{x + 4}

$\frac{2}{x + 4}$ as a fraction with a common denominator, multiply by

\frac{x - 4}{x - 4}

$\frac{x - 4}{x - 4}$ .

\frac{2}{x + 4} \cdot \frac{x - 4}{x - 4} - \frac{2}{x - 4}

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

Step 2

To write

- \frac{2}{x - 4}

$- \frac{2}{x - 4}$ as a fraction with a common denominator, multiply by

\frac{x + 4}{x + 4}

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

\frac{2}{x + 4} \cdot \frac{x - 4}{x - 4} - \frac{2}{x - 4} \cdot \frac{x + 4}{x + 4}

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

Step 3

Write each expression with a common denominator of

(x + 4) (x - 4)

$(x + 4) (x - 4)$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 3.1

Multiply

\frac{2}{x + 4}

$\frac{2}{x + 4}$ by

\frac{x - 4}{x - 4}

$\frac{x - 4}{x - 4}$ .

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

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

Step 3.2

Multiply

\frac{2}{x - 4}

$\frac{2}{x - 4}$ by

\frac{x + 4}{x + 4}

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

\frac{2 (x - 4)}{(x + 4) (x - 4)} - \frac{2 (x + 4)}{(x - 4) (x + 4)}

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

Step 3.3

Reorder the factors of

(x - 4) (x + 4)

$(x - 4) (x + 4)$ .

\frac{2 (x - 4)}{(x + 4) (x - 4)} - \frac{2 (x + 4)}{(x + 4) (x - 4)}

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

\frac{2 (x - 4)}{(x + 4) (x - 4)} - \frac{2 (x + 4)}{(x + 4) (x - 4)}

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

Step 4

Combine the numerators over the common denominator.

\frac{2 (x - 4) - 2 (x + 4)}{(x + 4) (x - 4)}

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

Step 5

Simplify the numerator.

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Step 5.1

Factor

2

$2$ out of

2 (x - 4) - 2 (x + 4)

$2 (x - 4) - 2 (x + 4)$ .

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Step 5.1.1

Factor

2

$2$ out of

- 2 (x + 4)

$- 2 (x + 4)$ .

\frac{2 (x - 4) + 2 (- (x + 4))}{(x + 4) (x - 4)}

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

Step 5.1.2

Factor

2

$2$ out of

2 (x - 4) + 2 (- (x + 4))

$2 (x - 4) + 2 (- (x + 4))$ .

\frac{2 (x - 4 - (x + 4))}{(x + 4) (x - 4)}

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

\frac{2 (x - 4 - (x + 4))}{(x + 4) (x - 4)}

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Multiply

$- 1$ by

$4$ .

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

Step 5.4

Subtract

$x$ from

$x$ .

$\frac{2 (0 - 4 - 4)}{(x + 4) (x - 4)}$

Step 5.5

Subtract

$4$ from

$0$ .

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

Step 5.6

Subtract

$4$ from

$- 4$ .

$\frac{2 \cdot - 8}{(x + 4) (x - 4)}$

Step 6

Simplify the expression.

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Step 6.1

Multiply

$2$ by

$- 8$ .

$\frac{- 16}{(x + 4) (x - 4)}$

Step 6.2

Move the negative in front of the fraction.

$- \frac{16}{(x + 4) (x - 4)}$

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