Precalculus Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

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 $b = 1$ .

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

Step 2

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

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

Step 3

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 3.1

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

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

Step 3.2

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $- 1$ by $1$ .

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

Step 5.3

Subtract $x$ from $x$ .

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

Step 5.4

Subtract $1$ from $0$ .

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

Step 6

Move the negative in front of the fraction.

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