Algebra Examples

$\frac{\frac{5}{x - 3} - \frac{1}{x + 3}}{\frac{3}{x^{2} - 9}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$(\frac{5}{x - 3} - \frac{1}{x + 3}) \frac{x^{2} - 9}{3}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $9$ as $3^{2}$ .

$(\frac{5}{x - 3} - \frac{1}{x + 3}) \frac{x^{2} - 3^{2}}{3}$

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 $b = 3$ .

$(\frac{5}{x - 3} - \frac{1}{x + 3}) \frac{(x + 3) (x - 3)}{3}$

Step 3

To write $\frac{5}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$(\frac{5}{x - 3} \cdot \frac{x + 3}{x + 3} - \frac{1}{x + 3}) \frac{(x + 3) (x - 3)}{3}$

Step 4

To write $- \frac{1}{x + 3}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$(\frac{5}{x - 3} \cdot \frac{x + 3}{x + 3} - \frac{1}{x + 3} \cdot \frac{x - 3}{x - 3}) \frac{(x + 3) (x - 3)}{3}$

Step 5

Write each expression with a common denominator of $(x - 3) (x + 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.1

Multiply $\frac{5}{x - 3}$ by $\frac{x + 3}{x + 3}$ .

$(\frac{5 (x + 3)}{(x - 3) (x + 3)} - \frac{1}{x + 3} \cdot \frac{x - 3}{x - 3}) \frac{(x + 3) (x - 3)}{3}$

Step 5.2

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

$(\frac{5 (x + 3)}{(x - 3) (x + 3)} - \frac{x - 3}{(x + 3) (x - 3)}) \frac{(x + 3) (x - 3)}{3}$

Step 5.3

Reorder the factors of $(x - 3) (x + 3)$ .

$(\frac{5 (x + 3)}{(x + 3) (x - 3)} - \frac{x - 3}{(x + 3) (x - 3)}) \frac{(x + 3) (x - 3)}{3}$

Step 6

Combine the numerators over the common denominator.

$\frac{5 (x + 3) - (x - 3)}{(x + 3) (x - 3)} \cdot \frac{(x + 3) (x - 3)}{3}$

Step 7

Simplify the numerator.

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Step 7.1

Apply the distributive property.

$\frac{5 x + 5 \cdot 3 - (x - 3)}{(x + 3) (x - 3)} \cdot \frac{(x + 3) (x - 3)}{3}$

Step 7.2

Multiply $5$ by $3$ .

$\frac{5 x + 15 - (x - 3)}{(x + 3) (x - 3)} \cdot \frac{(x + 3) (x - 3)}{3}$

Step 7.3

Apply the distributive property.

$\frac{5 x + 15 - x - - 3}{(x + 3) (x - 3)} \cdot \frac{(x + 3) (x - 3)}{3}$

Step 7.4

Multiply $- 1$ by $- 3$ .

$\frac{5 x + 15 - x + 3}{(x + 3) (x - 3)} \cdot \frac{(x + 3) (x - 3)}{3}$

Step 7.5

Subtract $x$ from $5 x$ .

$\frac{4 x + 15 + 3}{(x + 3) (x - 3)} \cdot \frac{(x + 3) (x - 3)}{3}$

Step 7.6

Add $15$ and $3$ .

$\frac{4 x + 18}{(x + 3) (x - 3)} \cdot \frac{(x + 3) (x - 3)}{3}$

Step 7.7

Factor $2$ out of $4 x + 18$ .

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Step 7.7.1

Factor $2$ out of $4 x$ .

$\frac{2 (2 x) + 18}{(x + 3) (x - 3)} \cdot \frac{(x + 3) (x - 3)}{3}$

Step 7.7.2

Factor $2$ out of $18$ .

$\frac{2 (2 x) + 2 (9)}{(x + 3) (x - 3)} \cdot \frac{(x + 3) (x - 3)}{3}$

Step 7.7.3

Factor $2$ out of $2 (2 x) + 2 (9)$ .

$\frac{2 (2 x + 9)}{(x + 3) (x - 3)} \cdot \frac{(x + 3) (x - 3)}{3}$

Step 8

Simplify terms.

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Step 8.1

Cancel the common factor of $(x + 3) (x - 3)$ .

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Step 8.1.1

Cancel the common factor.

$\frac{2 (2 x + 9)}{(x + 3) (x - 3)} \cdot \frac{(x + 3) (x - 3)}{3}$

Step 8.1.2

Rewrite the expression.

$2 (2 x + 9) \frac{1}{3}$

Step 8.2

Combine $\frac{1}{3}$ and $2$ .

$\frac{2}{3} (2 x + 9)$

Step 8.3

Apply the distributive property.

$\frac{2}{3} (2 x) + \frac{2}{3} \cdot 9$

Step 9

Multiply $\frac{2}{3} (2 x)$ .

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Step 9.1

Combine $2$ and $\frac{2}{3}$ .

$\frac{2 \cdot 2}{3} x + \frac{2}{3} \cdot 9$

Step 9.2

Multiply $2$ by $2$ .

$\frac{4}{3} x + \frac{2}{3} \cdot 9$

Step 9.3

Combine $\frac{4}{3}$ and $x$ .

$\frac{4 x}{3} + \frac{2}{3} \cdot 9$

Step 10

Cancel the common factor of $3$ .

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Step 10.1

Factor $3$ out of $9$ .

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

Step 10.2

Cancel the common factor.

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

Step 10.3

Rewrite the expression.

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

Step 11

Multiply $2$ by $3$ .

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

Step 12

To write $6$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 13

Simplify terms.

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Step 13.1

Combine $6$ and $\frac{3}{3}$ .

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

Step 13.2

Combine the numerators over the common denominator.

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

Step 14

Simplify the numerator.

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Step 14.1

Factor $2$ out of $4 x + 6 \cdot 3$ .

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Step 14.1.1

Factor $2$ out of $4 x$ .

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

Step 14.1.2

Factor $2$ out of $6 \cdot 3$ .

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

Step 14.1.3

Factor $2$ out of $2 (2 x) + 2 (3 \cdot 3)$ .

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

Step 14.2

Multiply $3$ by $3$ .

$\frac{2 (2 x + 9)}{3}$