Algebra Examples

$\frac{- \frac{2}{x + 2} - \frac{4}{x + 1}}{- \frac{4}{x + 2} - \frac{2}{x + 1}}$

Step 1

Multiply the numerator and denominator of the fraction by $(x + 2) (x + 1)$ .

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Step 1.1

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

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

Step 1.2

Combine.

$\frac{(x + 2) (x + 1) (- \frac{2}{x + 2} - \frac{4}{x + 1})}{(x + 2) (x + 1) (- \frac{4}{x + 2} - \frac{2}{x + 1})}$

Step 2

Apply the distributive property.

$\frac{(x + 2) (x + 1) (- \frac{2}{x + 2}) + (x + 2) (x + 1) (- \frac{4}{x + 1})}{(x + 2) (x + 1) (- \frac{4}{x + 2}) + (x + 2) (x + 1) (- \frac{2}{x + 1})}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x + 2$ .

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Step 3.1.1

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

$\frac{(x + 2) (x + 1) \frac{- 2}{x + 2} + (x + 2) (x + 1) (- \frac{4}{x + 1})}{(x + 2) (x + 1) (- \frac{4}{x + 2}) + (x + 2) (x + 1) (- \frac{2}{x + 1})}$

Step 3.1.2

Cancel the common factor.

$\frac{(x + 2) (x + 1) \frac{- 2}{x + 2} + (x + 2) (x + 1) (- \frac{4}{x + 1})}{(x + 2) (x + 1) (- \frac{4}{x + 2}) + (x + 2) (x + 1) (- \frac{2}{x + 1})}$

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x + 1$ .

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Cancel the common factor.

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

Step 3.2.4

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $x + 2$ .

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

Cancel the common factor of $x + 1$ .

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Step 3.4.1

Move the leading negative in $- \frac{2}{x + 1}$ into the numerator.

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

Step 3.4.2

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

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

Step 3.4.3

Cancel the common factor.

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

Step 3.4.4

Rewrite the expression.

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

Step 4

Cancel the common factor of $(x + 1) \cdot - 2 + (x + 2) \cdot - 4$ and $(x + 1) \cdot - 4 + (x + 2) \cdot - 2$ .

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Step 4.1

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

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

Step 4.2

Factor $2$ out of $(x + 2) \cdot - 4$ .

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

Step 4.3

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

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

Step 4.4

Cancel the common factors.

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Step 4.4.1

Factor $2$ out of $(x + 1) \cdot - 4$ .

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

Step 4.4.2

Factor $2$ out of $(x + 2) \cdot - 2$ .

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

Step 4.4.3

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

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

Step 4.4.4

Cancel the common factor.

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

Step 4.4.5

Rewrite the expression.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Move $- 1$ to the left of $x$ .

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

Step 5.3

Multiply $- 1$ by $1$ .

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

Step 5.4

Rewrite $- 1 x$ as $- x$ .

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

Step 5.5

Apply the distributive property.

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

Step 5.6

Move $- 2$ to the left of $x$ .

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

Step 5.7

Multiply $2$ by $- 2$ .

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

Step 5.8

Subtract $2 x$ from $- x$ .

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

Step 5.9

Subtract $4$ from $- 1$ .

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

Step 6

Simplify the denominator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Move $- 2$ to the left of $x$ .

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

Step 6.3

Multiply $- 2$ by $1$ .

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

Step 6.4

Apply the distributive property.

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

Step 6.5

Move $- 1$ to the left of $x$ .

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

Step 6.6

Multiply $2$ by $- 1$ .

$\frac{- 3 x - 5}{- 2 x - 2 - 1 \cdot x - 2}$

Step 6.7

Rewrite $- 1 x$ as $- x$ .

$\frac{- 3 x - 5}{- 2 x - 2 - x - 2}$

Step 6.8

Subtract $x$ from $- 2 x$ .

$\frac{- 3 x - 5}{- 3 x - 2 - 2}$

Step 6.9

Subtract $2$ from $- 2$ .

$\frac{- 3 x - 5}{- 3 x - 4}$

Step 7

Simplify terms.

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Step 7.1

Factor $- 1$ out of $- 3 x$ .

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

Step 7.2

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{- (3 x) - 1 (5)}{- 3 x - 4}$

Step 7.3

Factor $- 1$ out of $- (3 x) - 1 (5)$ .

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

Step 7.4

Rewrite $- (3 x + 5)$ as $- 1 (3 x + 5)$ .

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

Step 7.5

Factor $- 1$ out of $- 3 x$ .

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

Step 7.6

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 7.7

Factor $- 1$ out of $- (3 x) - 1 (4)$ .

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

Step 7.8

Rewrite $- (3 x + 4)$ as $- 1 (3 x + 4)$ .

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

Step 7.9

Cancel the common factor.

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

Step 7.10

Rewrite the expression.

$\frac{3 x + 5}{3 x + 4}$