Algebra Examples

$\frac{- \frac{1}{x + 4} - \frac{3}{x - 3}}{\frac{1}{x + 4} - \frac{4}{x - 3}}$

Step 1

Multiply the numerator and denominator of the fraction by $(x + 4) (x - 3)$ .

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x + 4$ .

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x - 3$ .

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Step 3.2.1

Move the leading negative in $- \frac{3}{x - 3}$ into the numerator.

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

Step 3.2.2

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

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

Step 3.2.3

Cancel the common factor.

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

Step 3.2.4

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $x + 4$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

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

Step 3.4

Cancel the common factor of $x - 3$ .

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Step 3.4.1

Move the leading negative in $- \frac{4}{x - 3}$ into the numerator.

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

Step 3.4.2

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

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

Step 3.4.3

Cancel the common factor.

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

Step 3.4.4

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

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

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

Step 4.3

Multiply $- 3$ by $- 1$ .

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

Step 4.4

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

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

Step 4.5

Apply the distributive property.

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

Step 4.6

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

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

Step 4.7

Multiply $4$ by $- 3$ .

$\frac{- x + 3 - 3 \cdot x - 12}{x - 3 + (x + 4) \cdot - 4}$

Step 4.8

Subtract $3 x$ from $- x$ .

$\frac{- 4 x + 3 - 12}{x - 3 + (x + 4) \cdot - 4}$

Step 4.9

Subtract $12$ from $3$ .

$\frac{- 4 x - 9}{x - 3 + (x + 4) \cdot - 4}$

Step 5

Simplify the denominator.

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Step 5.1

Apply the distributive property.

$\frac{- 4 x - 9}{x - 3 + x \cdot - 4 + 4 \cdot - 4}$

Step 5.2

Move $- 4$ to the left of $x$ .

$\frac{- 4 x - 9}{x - 3 - 4 \cdot x + 4 \cdot - 4}$

Step 5.3

Multiply $4$ by $- 4$ .

$\frac{- 4 x - 9}{x - 3 - 4 \cdot x - 16}$

Step 5.4

Subtract $4 x$ from $x$ .

$\frac{- 4 x - 9}{- 3 x - 3 - 16}$

Step 5.5

Subtract $16$ from $- 3$ .

$\frac{- 4 x - 9}{- 3 x - 19}$

Step 6

Simplify terms.

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Step 6.1

Factor $- 1$ out of $- 4 x$ .

$\frac{- (4 x) - 9}{- 3 x - 19}$

Step 6.2

Rewrite $- 9$ as $- 1 (9)$ .

$\frac{- (4 x) - 1 (9)}{- 3 x - 19}$

Step 6.3

Factor $- 1$ out of $- (4 x) - 1 (9)$ .

$\frac{- (4 x + 9)}{- 3 x - 19}$

Step 6.4

Rewrite $- (4 x + 9)$ as $- 1 (4 x + 9)$ .

$\frac{- 1 (4 x + 9)}{- 3 x - 19}$

Step 6.5

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

$\frac{- 1 (4 x + 9)}{- (3 x) - 19}$

Step 6.6

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

$\frac{- 1 (4 x + 9)}{- (3 x) - 1 (19)}$

Step 6.7

Factor $- 1$ out of $- (3 x) - 1 (19)$ .

$\frac{- 1 (4 x + 9)}{- (3 x + 19)}$

Step 6.8

Rewrite $- (3 x + 19)$ as $- 1 (3 x + 19)$ .

$\frac{- 1 (4 x + 9)}{- 1 (3 x + 19)}$

Step 6.9

Cancel the common factor.

$\frac{- 1 (4 x + 9)}{- 1 (3 x + 19)}$

Step 6.10

Rewrite the expression.

$\frac{4 x + 9}{3 x + 19}$