Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

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 $b = 1$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Add $x$ and $2 x$ .

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

Step 6

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

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

Step 7

Write each expression with a common denominator of $(x + 1) (x - 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.1

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

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

Step 7.2

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

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

Simplify the numerator.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Multiply $- 1$ by $1$ .

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

Step 9.3

Subtract $x$ from $3 x$ .

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

Step 9.4

Subtract $1$ from $- 1$ .

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

Step 9.5

Factor $2$ out of $2 x - 2$ .

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Step 9.5.1

Factor $2$ out of $2 x$ .

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

Step 9.5.2

Factor $2$ out of $- 2$ .

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

Step 9.5.3

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

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

Step 10

Cancel the common factor of $x - 1$ .

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Step 10.1

Cancel the common factor.

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

Step 10.2

Rewrite the expression.

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