Precalculus Examples

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

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

Step 1

To write

\frac{6}{x + 6}

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

\frac{x - 3}{x - 3}

$\frac{x - 3}{x - 3}$ .

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

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

Step 2

To write

\frac{5}{x - 3}

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

\frac{x + 6}{x + 6}

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

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

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

Step 3

Write each expression with a common denominator of

(x + 6) (x - 3)

$(x + 6) (x - 3)$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 3.1

Multiply

\frac{6}{x + 6}

$\frac{6}{x + 6}$ by

\frac{x - 3}{x - 3}

$\frac{x - 3}{x - 3}$ .

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

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

Step 3.2

Multiply

\frac{5}{x - 3}

$\frac{5}{x - 3}$ by

\frac{x + 6}{x + 6}

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

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

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

Step 3.3

Reorder the factors of

(x - 3) (x + 6)

$(x - 3) (x + 6)$ .

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

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

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

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

Step 4

Combine the numerators over the common denominator.

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

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

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

Step 5.2

Multiply

6

$6$ by

- 3

$- 3$ .

\frac{6 x - 18 + 5 (x + 6)}{(x + 6) (x - 3)}

$\frac{6 x - 18 + 5 (x + 6)}{(x + 6) (x - 3)}$

Step 5.3

Apply the distributive property.

\frac{6 x - 18 + 5 x + 5 \cdot 6}{(x + 6) (x - 3)}

$\frac{6 x - 18 + 5 x + 5 \cdot 6}{(x + 6) (x - 3)}$

Step 5.4

Multiply

5

$5$ by

6

$6$ .

\frac{6 x - 18 + 5 x + 30}{(x + 6) (x - 3)}

$\frac{6 x - 18 + 5 x + 30}{(x + 6) (x - 3)}$

Step 5.5

Add

6 x

$6 x$ and

5 x

$5 x$ .

\frac{11 x - 18 + 30}{(x + 6) (x - 3)}

$\frac{11 x - 18 + 30}{(x + 6) (x - 3)}$

Step 5.6

Add

- 18

$- 18$ and

30

$30$ .

\frac{11 x + 12}{(x + 6) (x - 3)}

$\frac{11 x + 12}{(x + 6) (x - 3)}$

\frac{11 x + 12}{(x + 6) (x - 3)}

$\frac{11 x + 12}{(x + 6) (x - 3)}$

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