Algebra Examples

$\frac{- \frac{7}{x + 7} - \frac{2}{x + 6}}{- \frac{6}{x + 7}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$(- \frac{7}{x + 7} - \frac{2}{x + 6}) (- \frac{x + 7}{6})$

Step 2

To write $- \frac{7}{x + 7}$ as a fraction with a common denominator, multiply by $\frac{x + 6}{x + 6}$ .

$(- \frac{7}{x + 7} \cdot \frac{x + 6}{x + 6} - \frac{2}{x + 6}) (- \frac{x + 7}{6})$

Step 3

To write $- \frac{2}{x + 6}$ as a fraction with a common denominator, multiply by $\frac{x + 7}{x + 7}$ .

$(- \frac{7}{x + 7} \cdot \frac{x + 6}{x + 6} - \frac{2}{x + 6} \cdot \frac{x + 7}{x + 7}) (- \frac{x + 7}{6})$

Step 4

Write each expression with a common denominator of $(x + 7) (x + 6)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{7}{x + 7}$ by $\frac{x + 6}{x + 6}$ .

$(- \frac{7 (x + 6)}{(x + 7) (x + 6)} - \frac{2}{x + 6} \cdot \frac{x + 7}{x + 7}) (- \frac{x + 7}{6})$

Step 4.2

Multiply $\frac{2}{x + 6}$ by $\frac{x + 7}{x + 7}$ .

$(- \frac{7 (x + 6)}{(x + 7) (x + 6)} - \frac{2 (x + 7)}{(x + 6) (x + 7)}) (- \frac{x + 7}{6})$

Step 4.3

Reorder the factors of $(x + 7) (x + 6)$ .

$(- \frac{7 (x + 6)}{(x + 6) (x + 7)} - \frac{2 (x + 7)}{(x + 6) (x + 7)}) (- \frac{x + 7}{6})$

Step 5

Simplify terms.

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Step 5.1

Combine the numerators over the common denominator.

$\frac{- 7 (x + 6) - 2 (x + 7)}{(x + 6) (x + 7)} (- \frac{x + 7}{6})$

Step 5.2

Rewrite using the commutative property of multiplication.

$- \frac{- 7 (x + 6) - 2 (x + 7)}{(x + 6) (x + 7)} \cdot \frac{x + 7}{6}$

Step 5.3

Cancel the common factor of $x + 7$ .

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Step 5.3.1

Move the leading negative in $- \frac{- 7 (x + 6) - 2 (x + 7)}{(x + 6) (x + 7)}$ into the numerator.

$\frac{- (- 7 (x + 6) - 2 (x + 7))}{(x + 6) (x + 7)} \cdot \frac{x + 7}{6}$

Step 5.3.2

Factor $x + 7$ out of $(x + 6) (x + 7)$ .

$\frac{- (- 7 (x + 6) - 2 (x + 7))}{(x + 7) (x + 6)} \cdot \frac{x + 7}{6}$

Step 5.3.3

Cancel the common factor.

$\frac{- (- 7 (x + 6) - 2 (x + 7))}{(x + 7) (x + 6)} \cdot \frac{x + 7}{6}$

Step 5.3.4

Rewrite the expression.

$\frac{- (- 7 (x + 6) - 2 (x + 7))}{x + 6} \cdot \frac{1}{6}$

Step 5.4

Multiply $\frac{- (- 7 (x + 6) - 2 (x + 7))}{x + 6}$ by $\frac{1}{6}$ .

$\frac{- (- 7 (x + 6) - 2 (x + 7))}{(x + 6) \cdot 6}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $- 1$ out of $- 7 (x + 6) - 2 (x + 7)$ .

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Step 6.1.1

Factor $- 1$ out of $- 7 (x + 6)$ .

$\frac{- (- (7 (x + 6)) - 2 (x + 7))}{(x + 6) \cdot 6}$

Step 6.1.2

Factor $- 1$ out of $- 2 (x + 7)$ .

$\frac{- (- (7 (x + 6)) - (2 (x + 7)))}{(x + 6) \cdot 6}$

Step 6.1.3

Factor $- 1$ out of $- (7 (x + 6)) - (2 (x + 7))$ .

$\frac{- - (7 (x + 6) + 2 (x + 7))}{(x + 6) \cdot 6}$

Step 6.2

Apply the distributive property.

$\frac{- - (7 x + 7 \cdot 6 + 2 (x + 7))}{(x + 6) \cdot 6}$

Step 6.3

Multiply $7$ by $6$ .

$\frac{- - (7 x + 42 + 2 (x + 7))}{(x + 6) \cdot 6}$

Step 6.4

Apply the distributive property.

$\frac{- - (7 x + 42 + 2 x + 2 \cdot 7)}{(x + 6) \cdot 6}$

Step 6.5

Multiply $2$ by $7$ .

$\frac{- - (7 x + 42 + 2 x + 14)}{(x + 6) \cdot 6}$

Step 6.6

Add $7 x$ and $2 x$ .

$\frac{- - (9 x + 42 + 14)}{(x + 6) \cdot 6}$

Step 6.7

Add $42$ and $14$ .

$\frac{- - 1 (9 x + 56)}{(x + 6) \cdot 6}$

Step 6.8

Combine exponents.

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Step 6.8.1

Factor out negative.

$\frac{- - (9 x + 56)}{(x + 6) \cdot 6}$

Step 6.8.2

Multiply $- 1$ by $- 1$ .

$\frac{1 (9 x + 56)}{(x + 6) \cdot 6}$

Step 6.8.3

Multiply $9 x + 56$ by $1$ .

$\frac{9 x + 56}{(x + 6) \cdot 6}$

Step 7

Move $6$ to the left of $x + 6$ .

$\frac{9 x + 56}{6 (x + 6)}$