Examples

\frac{2}{x + 7} - \frac{4}{x + 9}

$\frac{2}{x + 7} - \frac{4}{x + 9}$

Step 1

To write

\frac{2}{x + 7}

$\frac{2}{x + 7}$ as a fraction with a common denominator, multiply by

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

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

\frac{2}{x + 7} \cdot \frac{x + 9}{x + 9} - \frac{4}{x + 9}

$\frac{2}{x + 7} \cdot \frac{x + 9}{x + 9} - \frac{4}{x + 9}$

Step 2

To write

- \frac{4}{x + 9}

$- \frac{4}{x + 9}$ as a fraction with a common denominator, multiply by

\frac{x + 7}{x + 7}

$\frac{x + 7}{x + 7}$ .

\frac{2}{x + 7} \cdot \frac{x + 9}{x + 9} - \frac{4}{x + 9} \cdot \frac{x + 7}{x + 7}

$\frac{2}{x + 7} \cdot \frac{x + 9}{x + 9} - \frac{4}{x + 9} \cdot \frac{x + 7}{x + 7}$

Step 3

Write each expression with a common denominator of

(x + 7) (x + 9)

$(x + 7) (x + 9)$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 3.1

Multiply

\frac{2}{x + 7}

$\frac{2}{x + 7}$ by

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

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

\frac{2 (x + 9)}{(x + 7) (x + 9)} - \frac{4}{x + 9} \cdot \frac{x + 7}{x + 7}

$\frac{2 (x + 9)}{(x + 7) (x + 9)} - \frac{4}{x + 9} \cdot \frac{x + 7}{x + 7}$

Step 3.2

Multiply

\frac{4}{x + 9}

$\frac{4}{x + 9}$ by

\frac{x + 7}{x + 7}

$\frac{x + 7}{x + 7}$ .

\frac{2 (x + 9)}{(x + 7) (x + 9)} - \frac{4 (x + 7)}{(x + 9) (x + 7)}

$\frac{2 (x + 9)}{(x + 7) (x + 9)} - \frac{4 (x + 7)}{(x + 9) (x + 7)}$

Step 3.3

Reorder the factors of

(x + 9) (x + 7)

$(x + 9) (x + 7)$ .

\frac{2 (x + 9)}{(x + 7) (x + 9)} - \frac{4 (x + 7)}{(x + 7) (x + 9)}

$\frac{2 (x + 9)}{(x + 7) (x + 9)} - \frac{4 (x + 7)}{(x + 7) (x + 9)}$

\frac{2 (x + 9)}{(x + 7) (x + 9)} - \frac{4 (x + 7)}{(x + 7) (x + 9)}

$\frac{2 (x + 9)}{(x + 7) (x + 9)} - \frac{4 (x + 7)}{(x + 7) (x + 9)}$

Step 4

Combine the numerators over the common denominator.

\frac{2 (x + 9) - 4 (x + 7)}{(x + 7) (x + 9)}

$\frac{2 (x + 9) - 4 (x + 7)}{(x + 7) (x + 9)}$

Step 5

Simplify the numerator.

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Step 5.1

Factor

2

$2$ out of

2 (x + 9) - 4 (x + 7)

$2 (x + 9) - 4 (x + 7)$ .

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Step 5.1.1

Factor

2

$2$ out of

- 4 (x + 7)

$- 4 (x + 7)$ .

\frac{2 (x + 9) + 2 (- 2 (x + 7))}{(x + 7) (x + 9)}

$\frac{2 (x + 9) + 2 (- 2 (x + 7))}{(x + 7) (x + 9)}$

Step 5.1.2

Factor

2

$2$ out of

2 (x + 9) + 2 (- 2 (x + 7))

$2 (x + 9) + 2 (- 2 (x + 7))$ .

\frac{2 (x + 9 - 2 (x + 7))}{(x + 7) (x + 9)}

$\frac{2 (x + 9 - 2 (x + 7))}{(x + 7) (x + 9)}$

\frac{2 (x + 9 - 2 (x + 7))}{(x + 7) (x + 9)}

$\frac{2 (x + 9 - 2 (x + 7))}{(x + 7) (x + 9)}$

Step 5.2

Apply the distributive property.

\frac{2 (x + 9 - 2 x - 2 \cdot 7)}{(x + 7) (x + 9)}

$\frac{2 (x + 9 - 2 x - 2 \cdot 7)}{(x + 7) (x + 9)}$

Step 5.3

Multiply

- 2

$- 2$ by

7

$7$ .

\frac{2 (x + 9 - 2 x - 14)}{(x + 7) (x + 9)}

$\frac{2 (x + 9 - 2 x - 14)}{(x + 7) (x + 9)}$

Step 5.4

Subtract

2 x

$2 x$ from

x

$x$ .

\frac{2 (- x + 9 - 14)}{(x + 7) (x + 9)}

$\frac{2 (- x + 9 - 14)}{(x + 7) (x + 9)}$

Step 5.5

Subtract

14

$14$ from

9

$9$ .

\frac{2 (- x - 5)}{(x + 7) (x + 9)}

$\frac{2 (- x - 5)}{(x + 7) (x + 9)}$

\frac{2 (- x - 5)}{(x + 7) (x + 9)}

$\frac{2 (- x - 5)}{(x + 7) (x + 9)}$

Step 6

Simplify with factoring out.

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Step 6.1

Factor

- 1

$- 1$ out of

- x

$- x$ .

\frac{2 (- (x) - 5)}{(x + 7) (x + 9)}

$\frac{2 (- (x) - 5)}{(x + 7) (x + 9)}$

Step 6.2

Rewrite

- 5

$- 5$ as

- 1 (5)

$- 1 (5)$ .

\frac{2 (- (x) - 1 (5))}{(x + 7) (x + 9)}

$\frac{2 (- (x) - 1 (5))}{(x + 7) (x + 9)}$

Step 6.3

Factor

- 1

$- 1$ out of

- (x) - 1 (5)

$- (x) - 1 (5)$ .

\frac{2 (- (x + 5))}{(x + 7) (x + 9)}

$\frac{2 (- (x + 5))}{(x + 7) (x + 9)}$

Step 6.4

Simplify the expression.

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Step 6.4.1

Rewrite

- (x + 5)

$- (x + 5)$ as

- 1 (x + 5)

$- 1 (x + 5)$ .

\frac{2 (- 1 (x + 5))}{(x + 7) (x + 9)}

$\frac{2 (- 1 (x + 5))}{(x + 7) (x + 9)}$

Step 6.4.2

Move the negative in front of the fraction.

- \frac{2 (x + 5)}{(x + 7) (x + 9)}

$- \frac{2 (x + 5)}{(x + 7) (x + 9)}$

- \frac{2 (x + 5)}{(x + 7) (x + 9)}

$- \frac{2 (x + 5)}{(x + 7) (x + 9)}$

- \frac{2 (x + 5)}{(x + 7) (x + 9)}

$- \frac{2 (x + 5)}{(x + 7) (x + 9)}$

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