Simplify (7x)/(x+1)+7/(x^2-1)

$\frac{7 x}{x + 1} + \frac{7}{x^{2} - 1}$

Step 1

Simplify the denominator.

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

$\frac{7 x}{x + 1} + \frac{7}{x^{2} - 1^{2}}$

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 $b = 1$ .

$\frac{7 x}{x + 1} + \frac{7}{(x + 1) (x - 1)}$

Step 2

To write $\frac{7 x}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{7 x}{x + 1} \cdot \frac{x - 1}{x - 1} + \frac{7}{(x + 1) (x - 1)}$

Step 3

Simplify terms.

Multiply $\frac{7 x}{x + 1}$ by $\frac{x - 1}{x - 1}$ .

$\frac{7 x (x - 1)}{(x + 1) (x - 1)} + \frac{7}{(x + 1) (x - 1)}$

Combine the numerators over the common denominator.

$\frac{7 x (x - 1) + 7}{(x + 1) (x - 1)}$

Step 4

Simplify the numerator.

Factor $7$ out of $7 x (x - 1) + 7$ .

Factor $7$ out of $7 x (x - 1)$ .

$\frac{7 (x (x - 1)) + 7}{(x + 1) (x - 1)}$

Factor $7$ out of $7$ .

$\frac{7 (x (x - 1)) + 7 (1)}{(x + 1) (x - 1)}$

Factor $7$ out of $7 (x (x - 1)) + 7 (1)$ .

$\frac{7 (x (x - 1) + 1)}{(x + 1) (x - 1)}$

Apply the distributive property.

$\frac{7 (x \cdot x + x \cdot - 1 + 1)}{(x + 1) (x - 1)}$

Multiply $x$ by $x$ .

$\frac{7 (x^{2} + x \cdot - 1 + 1)}{(x + 1) (x - 1)}$

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

$\frac{7 (x^{2} - 1 \cdot x + 1)}{(x + 1) (x - 1)}$

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

$\frac{7 (x^{2} - x + 1)}{(x + 1) (x - 1)}$