Algebra Examples

$\frac{- \frac{5}{x + 1} - \frac{4}{x - 7}}{- \frac{6}{x - 7}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$(- \frac{5}{x + 1} - \frac{4}{x - 7}) (- \frac{x - 7}{6})$

Step 2

To write $- \frac{5}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x - 7}{x - 7}$ .

$(- \frac{5}{x + 1} \cdot \frac{x - 7}{x - 7} - \frac{4}{x - 7}) (- \frac{x - 7}{6})$

Step 3

To write $- \frac{4}{x - 7}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$(- \frac{5}{x + 1} \cdot \frac{x - 7}{x - 7} - \frac{4}{x - 7} \cdot \frac{x + 1}{x + 1}) (- \frac{x - 7}{6})$

Step 4

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

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Step 4.1

Multiply $\frac{5}{x + 1}$ by $\frac{x - 7}{x - 7}$ .

$(- \frac{5 (x - 7)}{(x + 1) (x - 7)} - \frac{4}{x - 7} \cdot \frac{x + 1}{x + 1}) (- \frac{x - 7}{6})$

Step 4.2

Multiply $\frac{4}{x - 7}$ by $\frac{x + 1}{x + 1}$ .

$(- \frac{5 (x - 7)}{(x + 1) (x - 7)} - \frac{4 (x + 1)}{(x - 7) (x + 1)}) (- \frac{x - 7}{6})$

Step 4.3

Reorder the factors of $(x - 7) (x + 1)$ .

$(- \frac{5 (x - 7)}{(x + 1) (x - 7)} - \frac{4 (x + 1)}{(x + 1) (x - 7)}) (- \frac{x - 7}{6})$

Step 5

Simplify terms.

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Step 5.1

Combine the numerators over the common denominator.

$\frac{- 5 (x - 7) - 4 (x + 1)}{(x + 1) (x - 7)} (- \frac{x - 7}{6})$

Step 5.2

Rewrite using the commutative property of multiplication.

$- \frac{- 5 (x - 7) - 4 (x + 1)}{(x + 1) (x - 7)} \cdot \frac{x - 7}{6}$

Step 5.3

Cancel the common factor of $x - 7$ .

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Step 5.3.1

Move the leading negative in $- \frac{- 5 (x - 7) - 4 (x + 1)}{(x + 1) (x - 7)}$ into the numerator.

$\frac{- (- 5 (x - 7) - 4 (x + 1))}{(x + 1) (x - 7)} \cdot \frac{x - 7}{6}$

Step 5.3.2

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

$\frac{- (- 5 (x - 7) - 4 (x + 1))}{(x - 7) (x + 1)} \cdot \frac{x - 7}{6}$

Step 5.3.3

Cancel the common factor.

$\frac{- (- 5 (x - 7) - 4 (x + 1))}{(x - 7) (x + 1)} \cdot \frac{x - 7}{6}$

Step 5.3.4

Rewrite the expression.

$\frac{- (- 5 (x - 7) - 4 (x + 1))}{x + 1} \cdot \frac{1}{6}$

Step 5.4

Multiply $\frac{- (- 5 (x - 7) - 4 (x + 1))}{x + 1}$ by $\frac{1}{6}$ .

$\frac{- (- 5 (x - 7) - 4 (x + 1))}{(x + 1) \cdot 6}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $- 1$ out of $- 5 (x - 7) - 4 (x + 1)$ .

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Step 6.1.1

Factor $- 1$ out of $- 5 (x - 7)$ .

$\frac{- (- (5 (x - 7)) - 4 (x + 1))}{(x + 1) \cdot 6}$

Step 6.1.2

Factor $- 1$ out of $- 4 (x + 1)$ .

$\frac{- (- (5 (x - 7)) - (4 (x + 1)))}{(x + 1) \cdot 6}$

Step 6.1.3

Factor $- 1$ out of $- (5 (x - 7)) - (4 (x + 1))$ .

$\frac{- - (5 (x - 7) + 4 (x + 1))}{(x + 1) \cdot 6}$

Step 6.2

Apply the distributive property.

$\frac{- - (5 x + 5 \cdot - 7 + 4 (x + 1))}{(x + 1) \cdot 6}$

Step 6.3

Multiply $5$ by $- 7$ .

$\frac{- - (5 x - 35 + 4 (x + 1))}{(x + 1) \cdot 6}$

Step 6.4

Apply the distributive property.

$\frac{- - (5 x - 35 + 4 x + 4 \cdot 1)}{(x + 1) \cdot 6}$

Step 6.5

Multiply $4$ by $1$ .

$\frac{- - (5 x - 35 + 4 x + 4)}{(x + 1) \cdot 6}$

Step 6.6

Add $5 x$ and $4 x$ .

$\frac{- - (9 x - 35 + 4)}{(x + 1) \cdot 6}$

Step 6.7

Add $- 35$ and $4$ .

$\frac{- - 1 (9 x - 31)}{(x + 1) \cdot 6}$

Step 6.8

Combine exponents.

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Step 6.8.1

Factor out negative.

$\frac{- - (9 x - 31)}{(x + 1) \cdot 6}$

Step 6.8.2

Multiply $- 1$ by $- 1$ .

$\frac{1 (9 x - 31)}{(x + 1) \cdot 6}$

Step 6.8.3

Multiply $9 x - 31$ by $1$ .

$\frac{9 x - 31}{(x + 1) \cdot 6}$

Step 7

Move $6$ to the left of $x + 1$ .

$\frac{9 x - 31}{6 (x + 1)}$