Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

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 $b = 1$ .

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

Step 2

To write $\frac{3}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

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

Step 3

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

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

Step 4

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 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Simplify terms.

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Step 5.1

Combine the numerators over the common denominator.

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

Step 5.2

Combine the numerators over the common denominator.

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

Step 6

Simplify each term.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $3$ by $- 1$ .

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

Step 6.3

Apply the distributive property.

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

Step 6.4

Multiply $- 1$ by $1$ .

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

Step 7

Simplify by adding terms.

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Step 7.1

Subtract $x$ from $3 x$ .

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

Step 7.2

Combine the opposite terms in $2 x - 3 - 1 - 2 x$ .

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Step 7.2.1

Subtract $2 x$ from $2 x$ .

$\frac{0 - 3 - 1}{(x + 1) (x - 1)}$

Step 7.2.2

Subtract $3$ from $0$ .

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

Step 7.3

Simplify the expression.

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Step 7.3.1

Subtract $1$ from $- 3$ .

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

Step 7.3.2

Move the negative in front of the fraction.

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