Examples

$\frac{8}{x + 5} - \frac{6}{x + 9}$

Step 1

To write $\frac{8}{x + 5}$ as a fraction with a common denominator, multiply by $\frac{x + 9}{x + 9}$ .

$\frac{8}{x + 5} \cdot \frac{x + 9}{x + 9} - \frac{6}{x + 9}$

Step 2

To write $- \frac{6}{x + 9}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$\frac{8}{x + 5} \cdot \frac{x + 9}{x + 9} - \frac{6}{x + 9} \cdot \frac{x + 5}{x + 5}$

Step 3

Write each expression with a common denominator of $(x + 5) (x + 9)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{8}{x + 5}$ by $\frac{x + 9}{x + 9}$ .

$\frac{8 (x + 9)}{(x + 5) (x + 9)} - \frac{6}{x + 9} \cdot \frac{x + 5}{x + 5}$

Step 3.2

Multiply $\frac{6}{x + 9}$ by $\frac{x + 5}{x + 5}$ .

$\frac{8 (x + 9)}{(x + 5) (x + 9)} - \frac{6 (x + 5)}{(x + 9) (x + 5)}$

Step 3.3

Reorder the factors of $(x + 9) (x + 5)$ .

$\frac{8 (x + 9)}{(x + 5) (x + 9)} - \frac{6 (x + 5)}{(x + 5) (x + 9)}$

Step 4

Combine the numerators over the common denominator.

$\frac{8 (x + 9) - 6 (x + 5)}{(x + 5) (x + 9)}$

Step 5

Simplify the numerator.

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Step 5.1

Factor $2$ out of $8 (x + 9) - 6 (x + 5)$ .

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Step 5.1.1

Factor $2$ out of $8 (x + 9)$ .

$\frac{2 (4 (x + 9)) - 6 (x + 5)}{(x + 5) (x + 9)}$

Step 5.1.2

Factor $2$ out of $- 6 (x + 5)$ .

$\frac{2 (4 (x + 9)) + 2 (- 3 (x + 5))}{(x + 5) (x + 9)}$

Step 5.1.3

Factor $2$ out of $2 (4 (x + 9)) + 2 (- 3 (x + 5))$ .

$\frac{2 (4 (x + 9) - 3 (x + 5))}{(x + 5) (x + 9)}$

Step 5.2

Apply the distributive property.

$\frac{2 (4 x + 4 \cdot 9 - 3 (x + 5))}{(x + 5) (x + 9)}$

Step 5.3

Multiply $4$ by $9$ .

$\frac{2 (4 x + 36 - 3 (x + 5))}{(x + 5) (x + 9)}$

Step 5.4

Apply the distributive property.

$\frac{2 (4 x + 36 - 3 x - 3 \cdot 5)}{(x + 5) (x + 9)}$

Step 5.5

Multiply $- 3$ by $5$ .

$\frac{2 (4 x + 36 - 3 x - 15)}{(x + 5) (x + 9)}$

Step 5.6

Subtract $3 x$ from $4 x$ .

$\frac{2 (x + 36 - 15)}{(x + 5) (x + 9)}$

Step 5.7

Subtract $15$ from $36$ .

$\frac{2 (x + 21)}{(x + 5) (x + 9)}$

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