Simplify 1/(x+1)-1/(x-1)

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

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

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

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

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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 $x$ from $x$ .

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

Subtract $1$ from $0$ .

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

Subtract $1$ from $- 1$ .

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

Step 6

Move the negative in front of the fraction.

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