Examples

\frac{4}{x + 5} + \frac{3}{x - 6}

$\frac{4}{x + 5} + \frac{3}{x - 6}$

Step 1

To write

\frac{4}{x + 5}

$\frac{4}{x + 5}$ as a fraction with a common denominator, multiply by

\frac{x - 6}{x - 6}

$\frac{x - 6}{x - 6}$ .

\frac{4}{x + 5} \cdot \frac{x - 6}{x - 6} + \frac{3}{x - 6}

$\frac{4}{x + 5} \cdot \frac{x - 6}{x - 6} + \frac{3}{x - 6}$

Step 2

To write

\frac{3}{x - 6}

$\frac{3}{x - 6}$ as a fraction with a common denominator, multiply by

\frac{x + 5}{x + 5}

$\frac{x + 5}{x + 5}$ .

\frac{4}{x + 5} \cdot \frac{x - 6}{x - 6} + \frac{3}{x - 6} \cdot \frac{x + 5}{x + 5}

$\frac{4}{x + 5} \cdot \frac{x - 6}{x - 6} + \frac{3}{x - 6} \cdot \frac{x + 5}{x + 5}$

Step 3

Write each expression with a common denominator of

(x + 5) (x - 6)

$(x + 5) (x - 6)$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 3.1

Multiply

\frac{4}{x + 5}

$\frac{4}{x + 5}$ by

\frac{x - 6}{x - 6}

$\frac{x - 6}{x - 6}$ .

\frac{4 (x - 6)}{(x + 5) (x - 6)} + \frac{3}{x - 6} \cdot \frac{x + 5}{x + 5}

$\frac{4 (x - 6)}{(x + 5) (x - 6)} + \frac{3}{x - 6} \cdot \frac{x + 5}{x + 5}$

Step 3.2

Multiply

\frac{3}{x - 6}

$\frac{3}{x - 6}$ by

\frac{x + 5}{x + 5}

$\frac{x + 5}{x + 5}$ .

\frac{4 (x - 6)}{(x + 5) (x - 6)} + \frac{3 (x + 5)}{(x - 6) (x + 5)}

$\frac{4 (x - 6)}{(x + 5) (x - 6)} + \frac{3 (x + 5)}{(x - 6) (x + 5)}$

Step 3.3

Reorder the factors of

(x - 6) (x + 5)

$(x - 6) (x + 5)$ .

\frac{4 (x - 6)}{(x + 5) (x - 6)} + \frac{3 (x + 5)}{(x + 5) (x - 6)}

$\frac{4 (x - 6)}{(x + 5) (x - 6)} + \frac{3 (x + 5)}{(x + 5) (x - 6)}$

\frac{4 (x - 6)}{(x + 5) (x - 6)} + \frac{3 (x + 5)}{(x + 5) (x - 6)}

$\frac{4 (x - 6)}{(x + 5) (x - 6)} + \frac{3 (x + 5)}{(x + 5) (x - 6)}$

Step 4

Combine the numerators over the common denominator.

\frac{4 (x - 6) + 3 (x + 5)}{(x + 5) (x - 6)}

$\frac{4 (x - 6) + 3 (x + 5)}{(x + 5) (x - 6)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

\frac{4 x + 4 \cdot - 6 + 3 (x + 5)}{(x + 5) (x - 6)}

$\frac{4 x + 4 \cdot - 6 + 3 (x + 5)}{(x + 5) (x - 6)}$

Step 5.2

Multiply

4

$4$ by

- 6

$- 6$ .

\frac{4 x - 24 + 3 (x + 5)}{(x + 5) (x - 6)}

$\frac{4 x - 24 + 3 (x + 5)}{(x + 5) (x - 6)}$

Step 5.3

Apply the distributive property.

\frac{4 x - 24 + 3 x + 3 \cdot 5}{(x + 5) (x - 6)}

$\frac{4 x - 24 + 3 x + 3 \cdot 5}{(x + 5) (x - 6)}$

Step 5.4

Multiply

3

$3$ by

5

$5$ .

\frac{4 x - 24 + 3 x + 15}{(x + 5) (x - 6)}

$\frac{4 x - 24 + 3 x + 15}{(x + 5) (x - 6)}$

Step 5.5

Add

4 x

$4 x$ and

3 x

$3 x$ .

\frac{7 x - 24 + 15}{(x + 5) (x - 6)}

$\frac{7 x - 24 + 15}{(x + 5) (x - 6)}$

Step 5.6

Add

- 24

$- 24$ and

15

$15$ .

\frac{7 x - 9}{(x + 5) (x - 6)}

$\frac{7 x - 9}{(x + 5) (x - 6)}$

\frac{7 x - 9}{(x + 5) (x - 6)}

$\frac{7 x - 9}{(x + 5) (x - 6)}$

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