Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

Step 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 $b = 1$ .

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

Step 2

To write $- \frac{2}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

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

Step 3

Simplify terms.

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Step 3.1

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

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $- 2$ by $- 1$ .

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

Step 4.3

Subtract $2 x$ from $x$ .

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

Step 5

Simplify with factoring out.

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Step 5.1

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

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

Step 5.2

Rewrite $2$ as $- 1 (- 2)$ .

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

Step 5.3

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

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

Step 5.4

Simplify the expression.

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Step 5.4.1

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

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

Step 5.4.2

Move the negative in front of the fraction.

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