Algebra Examples

$\frac{- \frac{6}{x + 4} + \frac{2}{x - 5}}{- \frac{3}{x + 4}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$(- \frac{6}{x + 4} + \frac{2}{x - 5}) (- \frac{x + 4}{3})$

Step 2

To write $- \frac{6}{x + 4}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$(- \frac{6}{x + 4} \cdot \frac{x - 5}{x - 5} + \frac{2}{x - 5}) (- \frac{x + 4}{3})$

Step 3

To write $\frac{2}{x - 5}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

$(- \frac{6}{x + 4} \cdot \frac{x - 5}{x - 5} + \frac{2}{x - 5} \cdot \frac{x + 4}{x + 4}) (- \frac{x + 4}{3})$

Step 4

Write each expression with a common denominator of $(x + 4) (x - 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{6}{x + 4}$ by $\frac{x - 5}{x - 5}$ .

$(- \frac{6 (x - 5)}{(x + 4) (x - 5)} + \frac{2}{x - 5} \cdot \frac{x + 4}{x + 4}) (- \frac{x + 4}{3})$

Step 4.2

Multiply $\frac{2}{x - 5}$ by $\frac{x + 4}{x + 4}$ .

$(- \frac{6 (x - 5)}{(x + 4) (x - 5)} + \frac{2 (x + 4)}{(x - 5) (x + 4)}) (- \frac{x + 4}{3})$

Step 4.3

Reorder the factors of $(x - 5) (x + 4)$ .

$(- \frac{6 (x - 5)}{(x + 4) (x - 5)} + \frac{2 (x + 4)}{(x + 4) (x - 5)}) (- \frac{x + 4}{3})$

Step 5

Simplify terms.

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Step 5.1

Combine the numerators over the common denominator.

$\frac{- 6 (x - 5) + 2 (x + 4)}{(x + 4) (x - 5)} (- \frac{x + 4}{3})$

Step 5.2

Rewrite using the commutative property of multiplication.

$- \frac{- 6 (x - 5) + 2 (x + 4)}{(x + 4) (x - 5)} \cdot \frac{x + 4}{3}$

Step 5.3

Cancel the common factor of $x + 4$ .

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Step 5.3.1

Move the leading negative in $- \frac{- 6 (x - 5) + 2 (x + 4)}{(x + 4) (x - 5)}$ into the numerator.

$\frac{- (- 6 (x - 5) + 2 (x + 4))}{(x + 4) (x - 5)} \cdot \frac{x + 4}{3}$

Step 5.3.2

Cancel the common factor.

$\frac{- (- 6 (x - 5) + 2 (x + 4))}{(x + 4) (x - 5)} \cdot \frac{x + 4}{3}$

Step 5.3.3

Rewrite the expression.

$\frac{- (- 6 (x - 5) + 2 (x + 4))}{x - 5} \cdot \frac{1}{3}$

Step 5.4

Multiply $\frac{- (- 6 (x - 5) + 2 (x + 4))}{x - 5}$ by $\frac{1}{3}$ .

$\frac{- (- 6 (x - 5) + 2 (x + 4))}{(x - 5) \cdot 3}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $2$ out of $- 6 (x - 5) + 2 (x + 4)$ .

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Step 6.1.1

Factor $2$ out of $- 6 (x - 5)$ .

$\frac{- (2 (- 3 (x - 5)) + 2 (x + 4))}{(x - 5) \cdot 3}$

Step 6.1.2

Factor $2$ out of $2 (- 3 (x - 5)) + 2 (x + 4)$ .

$\frac{- (2 (- 3 (x - 5) + x + 4))}{(x - 5) \cdot 3}$

Step 6.2

Apply the distributive property.

$\frac{- (2 (- 3 x - 3 \cdot - 5 + x + 4))}{(x - 5) \cdot 3}$

Step 6.3

Multiply $- 3$ by $- 5$ .

$\frac{- (2 (- 3 x + 15 + x + 4))}{(x - 5) \cdot 3}$

Step 6.4

Add $- 3 x$ and $x$ .

$\frac{- (2 (- 2 x + 15 + 4))}{(x - 5) \cdot 3}$

Step 6.5

Add $15$ and $4$ .

$\frac{- 1 \cdot 2 (- 2 x + 19)}{(x - 5) \cdot 3}$

Step 6.6

Multiply $- 1$ by $2$ .

$\frac{- 2 (- 2 x + 19)}{(x - 5) \cdot 3}$

Step 7

Simplify with factoring out.

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Step 7.1

Move $3$ to the left of $x - 5$ .

$\frac{- 2 (- 2 x + 19)}{3 \cdot (x - 5)}$

Step 7.2

Move the negative in front of the fraction.

$- \frac{2 (- 2 x + 19)}{3 \cdot (x - 5)}$

Step 7.3

Factor $- 1$ out of $- 2 x$ .

$- \frac{2 (- (2 x) + 19)}{3 (x - 5)}$

Step 7.4

Rewrite $19$ as $- 1 (- 19)$ .

$- \frac{2 (- (2 x) - 1 (- 19))}{3 (x - 5)}$

Step 7.5

Factor $- 1$ out of $- (2 x) - 1 (- 19)$ .

$- \frac{2 (- (2 x - 19))}{3 (x - 5)}$

Step 7.6

Simplify the expression.

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Step 7.6.1

Rewrite $- (2 x - 19)$ as $- 1 (2 x - 19)$ .

$- \frac{2 (- 1 (2 x - 19))}{3 (x - 5)}$

Step 7.6.2

Move the negative in front of the fraction.

$- - \frac{2 (2 x - 19)}{3 (x - 5)}$

Step 7.6.3

Multiply $- 1$ by $- 1$ .

$1 \frac{2 (2 x - 19)}{3 (x - 5)}$

Step 7.6.4

Multiply $\frac{2 (2 x - 19)}{3 (x - 5)}$ by $1$ .

$\frac{2 (2 x - 19)}{3 (x - 5)}$