Algebra Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $(x + 1) (x - 3)$ .

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x + 1$ .

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x - 3$ .

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $x + 1$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

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

Step 3.4

Cancel the common factor of $x - 3$ .

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Step 3.4.1

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

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

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

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

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

Step 4.3

Multiply $- 3$ by $- 4$ .

$\frac{- 4 \cdot x + 12 + (x + 1) \cdot 3}{(x - 3) \cdot 4 + (x + 1) \cdot 2}$

Step 4.4

Apply the distributive property.

$\frac{- 4 x + 12 + x \cdot 3 + 1 \cdot 3}{(x - 3) \cdot 4 + (x + 1) \cdot 2}$

Step 4.5

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

$\frac{- 4 x + 12 + 3 \cdot x + 1 \cdot 3}{(x - 3) \cdot 4 + (x + 1) \cdot 2}$

Step 4.6

Multiply $3$ by $1$ .

$\frac{- 4 x + 12 + 3 \cdot x + 3}{(x - 3) \cdot 4 + (x + 1) \cdot 2}$

Step 4.7

Add $- 4 x$ and $3 x$ .

$\frac{- x + 12 + 3}{(x - 3) \cdot 4 + (x + 1) \cdot 2}$

Step 4.8

Add $12$ and $3$ .

$\frac{- x + 15}{(x - 3) \cdot 4 + (x + 1) \cdot 2}$

Step 5

Simplify the denominator.

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Step 5.1

Factor $2$ out of $(x - 3) \cdot 4 + (x + 1) \cdot 2$ .

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Step 5.1.1

Factor $2$ out of $(x - 3) \cdot 4$ .

$\frac{- x + 15}{2 ((x - 3) \cdot 2) + (x + 1) \cdot 2}$

Step 5.1.2

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

$\frac{- x + 15}{2 ((x - 3) \cdot 2) + 2 \cdot (x + 1)}$

Step 5.1.3

Factor $2$ out of $2 ((x - 3) \cdot 2) + 2 \cdot (x + 1)$ .

$\frac{- x + 15}{2 ((x - 3) \cdot 2 + x + 1)}$

Step 5.2

Apply the distributive property.

$\frac{- x + 15}{2 (x \cdot 2 - 3 \cdot 2 + x + 1)}$

Step 5.3

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

$\frac{- x + 15}{2 (2 \cdot x - 3 \cdot 2 + x + 1)}$

Step 5.4

Multiply $- 3$ by $2$ .

$\frac{- x + 15}{2 (2 \cdot x - 6 + x + 1)}$

Step 5.5

Add $2 x$ and $x$ .

$\frac{- x + 15}{2 (3 x - 6 + 1)}$

Step 5.6

Add $- 6$ and $1$ .

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

Step 6

Simplify with factoring out.

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Step 6.1

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

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

Step 6.2

Rewrite $15$ as $- 1 (- 15)$ .

$\frac{- (x) - 1 (- 15)}{2 (3 x - 5)}$

Step 6.3

Factor $- 1$ out of $- (x) - 1 (- 15)$ .

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

Step 6.4

Simplify the expression.

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Step 6.4.1

Rewrite $- (x - 15)$ as $- 1 (x - 15)$ .

$\frac{- 1 (x - 15)}{2 (3 x - 5)}$

Step 6.4.2

Move the negative in front of the fraction.

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