Algebra Examples

$\frac{\frac{3}{x - 2} - \frac{4}{x + 2}}{\frac{7}{x^{2} - 4}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$(\frac{3}{x - 2} - \frac{4}{x + 2}) \frac{x^{2} - 4}{7}$

Step 2

Simplify the numerator.

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Step 2.1

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

$(\frac{3}{x - 2} - \frac{4}{x + 2}) \frac{x^{2} - 2^{2}}{7}$

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 $b = 2$ .

$(\frac{3}{x - 2} - \frac{4}{x + 2}) \frac{(x + 2) (x - 2)}{7}$

Step 3

To write $\frac{3}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$(\frac{3}{x - 2} \cdot \frac{x + 2}{x + 2} - \frac{4}{x + 2}) \frac{(x + 2) (x - 2)}{7}$

Step 4

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

$(\frac{3}{x - 2} \cdot \frac{x + 2}{x + 2} - \frac{4}{x + 2} \cdot \frac{x - 2}{x - 2}) \frac{(x + 2) (x - 2)}{7}$

Step 5

Write each expression with a common denominator of $(x - 2) (x + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.1

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

$(\frac{3 (x + 2)}{(x - 2) (x + 2)} - \frac{4}{x + 2} \cdot \frac{x - 2}{x - 2}) \frac{(x + 2) (x - 2)}{7}$

Step 5.2

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

$(\frac{3 (x + 2)}{(x - 2) (x + 2)} - \frac{4 (x - 2)}{(x + 2) (x - 2)}) \frac{(x + 2) (x - 2)}{7}$

Step 5.3

Reorder the factors of $(x - 2) (x + 2)$ .

$(\frac{3 (x + 2)}{(x + 2) (x - 2)} - \frac{4 (x - 2)}{(x + 2) (x - 2)}) \frac{(x + 2) (x - 2)}{7}$

Step 6

Combine the numerators over the common denominator.

$\frac{3 (x + 2) - 4 (x - 2)}{(x + 2) (x - 2)} \cdot \frac{(x + 2) (x - 2)}{7}$

Step 7

Simplify the numerator.

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Step 7.1

Apply the distributive property.

$\frac{3 x + 3 \cdot 2 - 4 (x - 2)}{(x + 2) (x - 2)} \cdot \frac{(x + 2) (x - 2)}{7}$

Step 7.2

Multiply $3$ by $2$ .

$\frac{3 x + 6 - 4 (x - 2)}{(x + 2) (x - 2)} \cdot \frac{(x + 2) (x - 2)}{7}$

Step 7.3

Apply the distributive property.

$\frac{3 x + 6 - 4 x - 4 \cdot - 2}{(x + 2) (x - 2)} \cdot \frac{(x + 2) (x - 2)}{7}$

Step 7.4

Multiply $- 4$ by $- 2$ .

$\frac{3 x + 6 - 4 x + 8}{(x + 2) (x - 2)} \cdot \frac{(x + 2) (x - 2)}{7}$

Step 7.5

Subtract $4 x$ from $3 x$ .

$\frac{- x + 6 + 8}{(x + 2) (x - 2)} \cdot \frac{(x + 2) (x - 2)}{7}$

Step 7.6

Add $6$ and $8$ .

$\frac{- x + 14}{(x + 2) (x - 2)} \cdot \frac{(x + 2) (x - 2)}{7}$

Step 8

Simplify terms.

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Step 8.1

Cancel the common factor of $(x + 2) (x - 2)$ .

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Step 8.1.1

Cancel the common factor.

$\frac{- x + 14}{(x + 2) (x - 2)} \cdot \frac{(x + 2) (x - 2)}{7}$

Step 8.1.2

Rewrite the expression.

$(- x + 14) \frac{1}{7}$

Step 8.2

Apply the distributive property.

$- x \frac{1}{7} + 14 (\frac{1}{7})$

Step 8.3

Combine $\frac{1}{7}$ and $x$ .

$- \frac{x}{7} + 14 (\frac{1}{7})$

Step 8.4

Cancel the common factor of $7$ .

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Step 8.4.1

Factor $7$ out of $14$ .

$- \frac{x}{7} + 7 (2) \frac{1}{7}$

Step 8.4.2

Cancel the common factor.

$- \frac{x}{7} + 7 \cdot 2 \frac{1}{7}$

Step 8.4.3

Rewrite the expression.

$- \frac{x}{7} + 2$

Step 9

To write $2$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$- \frac{x}{7} + 2 \cdot \frac{7}{7}$

Step 10

Combine $2$ and $\frac{7}{7}$ .

$- \frac{x}{7} + \frac{2 \cdot 7}{7}$

Step 11

Combine the numerators over the common denominator.

$\frac{- x + 2 \cdot 7}{7}$

Step 12

Multiply $2$ by $7$ .

$\frac{- x + 14}{7}$

Step 13

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

$\frac{- (x) + 14}{7}$

Step 14

Rewrite $14$ as $- 1 (- 14)$ .

$\frac{- (x) - 1 (- 14)}{7}$

Step 15

Factor $- 1$ out of $- (x) - 1 (- 14)$ .

$\frac{- (x - 14)}{7}$

Step 16

Simplify the expression.

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Step 16.1

Rewrite $- (x - 14)$ as $- 1 (x - 14)$ .

$\frac{- 1 (x - 14)}{7}$

Step 16.2

Move the negative in front of the fraction.

$- \frac{x - 14}{7}$