Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

Rewrite $1$ as $1^{2}$ .

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

Step 1.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

Step 1.2

Move the negative in front of the fraction.

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

Step 2

Find the common denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify each term.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $- 1$ by $1$ .

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

Step 5

Simplify by adding terms.

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Step 5.1

Combine the opposite terms in $- 2 + x - 1 - x - 1$ .

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Step 5.1.1

Subtract $x$ from $x$ .

$\frac{- 2 + 0 - 1 - 1}{(x + 1) (x - 1)}$

Step 5.1.2

Add $- 2$ and $0$ .

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

Step 5.2

Simplify the expression.

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Step 5.2.1

Subtract $1$ from $- 2$ .

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

Step 5.2.2

Subtract $1$ from $- 3$ .

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

Step 5.2.3

Move the negative in front of the fraction.

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