Simplify (8x)/(x^2-1)-2/(x-1)

$\frac{8 x}{x^{2} - 1} - \frac{2}{x - 1}$

Step 1

Simplify the denominator.

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

$\frac{8 x}{x^{2} - 1^{2}} - \frac{2}{x - 1}$

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 $b = 1$ .

$\frac{8 x}{(x + 1) (x - 1)} - \frac{2}{x - 1}$

Step 2

To write $- \frac{2}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$\frac{8 x}{(x + 1) (x - 1)} - \frac{2}{x - 1} \cdot \frac{x + 1}{x + 1}$

Step 3

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

Multiply $\frac{2}{x - 1}$ by $\frac{x + 1}{x + 1}$ .

$\frac{8 x}{(x + 1) (x - 1)} - \frac{2 (x + 1)}{(x - 1) (x + 1)}$

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

$\frac{8 x}{(x + 1) (x - 1)} - \frac{2 (x + 1)}{(x + 1) (x - 1)}$

Step 4

Combine the numerators over the common denominator.

$\frac{8 x - 2 (x + 1)}{(x + 1) (x - 1)}$

Step 5

Simplify the numerator.

Factor $2$ out of $8 x - 2 (x + 1)$ .

Factor $2$ out of $8 x$ .

$\frac{2 (4 x) - 2 (x + 1)}{(x + 1) (x - 1)}$

Factor $2$ out of $- 2 (x + 1)$ .

$\frac{2 (4 x) + 2 (- (x + 1))}{(x + 1) (x - 1)}$

Factor $2$ out of $2 (4 x) + 2 (- (x + 1))$ .

$\frac{2 (4 x - (x + 1))}{(x + 1) (x - 1)}$

Apply the distributive property.

$\frac{2 (4 x - x - 1 \cdot 1)}{(x + 1) (x - 1)}$

Multiply $- 1$ by $1$ .

$\frac{2 (4 x - x - 1)}{(x + 1) (x - 1)}$

Subtract $x$ from $4 x$ .

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