Algebra Examples

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

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

Step 1

To write

\frac{9}{x + 9}

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

\frac{x + 3}{x + 3}

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

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

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

Step 2

To write

- \frac{3}{x + 3}

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

\frac{x + 9}{x + 9}

$\frac{x + 9}{x + 9}$ .

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

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

Step 3

Write each expression with a common denominator of

(x + 9) (x + 3)

$(x + 9) (x + 3)$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 3.1

Multiply

\frac{9}{x + 9}

$\frac{9}{x + 9}$ by

\frac{x + 3}{x + 3}

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

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

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

Step 3.2

Multiply

\frac{3}{x + 3}

$\frac{3}{x + 3}$ by

\frac{x + 9}{x + 9}

$\frac{x + 9}{x + 9}$ .

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

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

Step 3.3

Reorder the factors of

(x + 9) (x + 3)

$(x + 9) (x + 3)$ .

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

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

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

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

Step 4

Combine the numerators over the common denominator.

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

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

Step 5

Simplify the numerator.

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Step 5.1

Factor

3

$3$ out of

9 (x + 3) - 3 (x + 9)

$9 (x + 3) - 3 (x + 9)$ .

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Step 5.1.1

Factor

3

$3$ out of

9 (x + 3)

$9 (x + 3)$ .

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

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

Step 5.1.2

Factor

3

$3$ out of

- 3 (x + 9)

$- 3 (x + 9)$ .

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

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

Step 5.1.3

Factor

3

$3$ out of

3 (3 (x + 3)) + 3 (- (x + 9))

$3 (3 (x + 3)) + 3 (- (x + 9))$ .

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

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

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

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

Step 5.2

Apply the distributive property.

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

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

Step 5.3

Multiply

3

$3$ by

3

$3$ .

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

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

Step 5.4

Apply the distributive property.

\frac{3 (3 x + 9 - x - 1 \cdot 9)}{(x + 3) (x + 9)}

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

Step 5.5

Multiply

- 1

$- 1$ by

9

$9$ .

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

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

Step 5.6

Subtract

x

$x$ from

3 x

$3 x$ .

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

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

Step 5.7

Subtract

9

$9$ from

9

$9$ .

\frac{3 (2 x + 0)}{(x + 3) (x + 9)}

$\frac{3 (2 x + 0)}{(x + 3) (x + 9)}$

Step 5.8

Add

2 x

$2 x$ and

0

$0$ .

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

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

Step 5.9

Multiply

3

$3$ by

2

$2$ .

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

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

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

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

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