Precalculus Examples

\frac{4}{x - 3} + \frac{9}{x - 6}

$\frac{4}{x - 3} + \frac{9}{x - 6}$

Step 1

To write

\frac{4}{x - 3}

$\frac{4}{x - 3}$ as a fraction with a common denominator, multiply by

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

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

\frac{4}{x - 3} \cdot \frac{x - 6}{x - 6} + \frac{9}{x - 6}

$\frac{4}{x - 3} \cdot \frac{x - 6}{x - 6} + \frac{9}{x - 6}$

Step 2

To write

\frac{9}{x - 6}

$\frac{9}{x - 6}$ as a fraction with a common denominator, multiply by

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

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

\frac{4}{x - 3} \cdot \frac{x - 6}{x - 6} + \frac{9}{x - 6} \cdot \frac{x - 3}{x - 3}

$\frac{4}{x - 3} \cdot \frac{x - 6}{x - 6} + \frac{9}{x - 6} \cdot \frac{x - 3}{x - 3}$

Step 3

Write each expression with a common denominator of

(x - 3) (x - 6)

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

1

$1$ .

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Step 3.1

Multiply

\frac{4}{x - 3}

$\frac{4}{x - 3}$ by

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

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

\frac{4 (x - 6)}{(x - 3) (x - 6)} + \frac{9}{x - 6} \cdot \frac{x - 3}{x - 3}

$\frac{4 (x - 6)}{(x - 3) (x - 6)} + \frac{9}{x - 6} \cdot \frac{x - 3}{x - 3}$

Step 3.2

Multiply

\frac{9}{x - 6}

$\frac{9}{x - 6}$ by

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

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

\frac{4 (x - 6)}{(x - 3) (x - 6)} + \frac{9 (x - 3)}{(x - 6) (x - 3)}

$\frac{4 (x - 6)}{(x - 3) (x - 6)} + \frac{9 (x - 3)}{(x - 6) (x - 3)}$

Step 3.3

Reorder the factors of

(x - 6) (x - 3)

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

\frac{4 (x - 6)}{(x - 3) (x - 6)} + \frac{9 (x - 3)}{(x - 3) (x - 6)}

$\frac{4 (x - 6)}{(x - 3) (x - 6)} + \frac{9 (x - 3)}{(x - 3) (x - 6)}$

\frac{4 (x - 6)}{(x - 3) (x - 6)} + \frac{9 (x - 3)}{(x - 3) (x - 6)}

$\frac{4 (x - 6)}{(x - 3) (x - 6)} + \frac{9 (x - 3)}{(x - 3) (x - 6)}$

Step 4

Combine the numerators over the common denominator.

\frac{4 (x - 6) + 9 (x - 3)}{(x - 3) (x - 6)}

$\frac{4 (x - 6) + 9 (x - 3)}{(x - 3) (x - 6)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

\frac{4 x + 4 \cdot - 6 + 9 (x - 3)}{(x - 3) (x - 6)}

$\frac{4 x + 4 \cdot - 6 + 9 (x - 3)}{(x - 3) (x - 6)}$

Step 5.2

Multiply

4

$4$ by

- 6

$- 6$ .

\frac{4 x - 24 + 9 (x - 3)}{(x - 3) (x - 6)}

$\frac{4 x - 24 + 9 (x - 3)}{(x - 3) (x - 6)}$

Step 5.3

Apply the distributive property.

\frac{4 x - 24 + 9 x + 9 \cdot - 3}{(x - 3) (x - 6)}

$\frac{4 x - 24 + 9 x + 9 \cdot - 3}{(x - 3) (x - 6)}$

Step 5.4

Multiply

9

$9$ by

- 3

$- 3$ .

\frac{4 x - 24 + 9 x - 27}{(x - 3) (x - 6)}

$\frac{4 x - 24 + 9 x - 27}{(x - 3) (x - 6)}$

Step 5.5

Add

4 x

$4 x$ and

9 x

$9 x$ .

\frac{13 x - 24 - 27}{(x - 3) (x - 6)}

$\frac{13 x - 24 - 27}{(x - 3) (x - 6)}$

Step 5.6

Subtract

27

$27$ from

- 24

$- 24$ .

\frac{13 x - 51}{(x - 3) (x - 6)}

$\frac{13 x - 51}{(x - 3) (x - 6)}$

\frac{13 x - 51}{(x - 3) (x - 6)}

$\frac{13 x - 51}{(x - 3) (x - 6)}$

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