Simplify x/(x^2-1)-x/(x-1)

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

Simplify the denominator.

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

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

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 $b = 1$ .

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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

Raise $x$ to the power of $1$ .

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

Factor $x$ out of $x^{1}$ .

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

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

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

Factor $x$ out of $x \cdot 1 + x (- (x + 1))$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $1$ .

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

Subtract $1$ from $1$ .

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

Add $- x$ and $0$ .

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

Combine exponents.

Factor out negative.

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

Raise $x$ to the power of $1$ .

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

Raise $x$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

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

Move the negative in front of the fraction.

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

Simplify x/(x^2-1)-x/(x-1)

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