Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

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 $b = 1$ .

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

Step 2

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

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

Step 3

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

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

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Simplify terms.

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Step 5.1

Combine the numerators over the common denominator.

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

Step 5.2

Combine the numerators over the common denominator.

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

Step 5.3

Simplify each term.

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Step 5.3.1

Apply the distributive property.

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

Step 5.3.2

Multiply $x$ by $x$ .

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

Step 5.3.3

Move $- 1$ to the left of $x$ .

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

Step 5.3.4

Rewrite $- 1 x$ as $- x$ .

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

Step 5.3.5

Apply the distributive property.

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

Step 5.3.6

Multiply $- 1$ by $1$ .

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

Step 5.4

Simplify by adding terms.

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Step 5.4.1

Subtract $x$ from $- x$ .

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

Step 5.4.2

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

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Step 5.4.2.1

Add $- 2 x$ and $2 x$ .

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

Step 5.4.2.2

Add $x^{2}$ and $0$ .

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

Step 6

Simplify the numerator.

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Step 6.1

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

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

Step 6.2

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 $b = 1$ .

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

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

Cancel the common factor of $x + 1$ .

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Step 7.1.1

Cancel the common factor.

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

Step 7.1.2

Rewrite the expression.

$\frac{x - 1}{x - 1}$

Step 7.2

Cancel the common factor of $x - 1$ .

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Step 7.2.1

Cancel the common factor.

$\frac{x - 1}{x - 1}$

Step 7.2.2

Rewrite the expression.

$1$