Simplify 11/(x+10)-(10x)/(x^2-100)

$\frac{11}{x + 10} - \frac{10 x}{x^{2} - 100}$

Step 1

Simplify the denominator.

Rewrite $100$ as $10^{2}$ .

$\frac{11}{x + 10} - \frac{10 x}{x^{2} - 10^{2}}$

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 $b = 10$ .

$\frac{11}{x + 10} - \frac{10 x}{(x + 10) (x - 10)}$

Step 2

To write $\frac{11}{x + 10}$ as a fraction with a common denominator, multiply by $\frac{x - 10}{x - 10}$ .

$\frac{11}{x + 10} \cdot \frac{x - 10}{x - 10} - \frac{10 x}{(x + 10) (x - 10)}$

Step 3

Simplify terms.

Multiply $\frac{11}{x + 10}$ by $\frac{x - 10}{x - 10}$ .

$\frac{11 (x - 10)}{(x + 10) (x - 10)} - \frac{10 x}{(x + 10) (x - 10)}$

Combine the numerators over the common denominator.

$\frac{11 (x - 10) - 10 x}{(x + 10) (x - 10)}$

Step 4

Simplify the numerator.

Apply the distributive property.

$\frac{11 x + 11 \cdot - 10 - 10 x}{(x + 10) (x - 10)}$

Multiply $11$ by $- 10$ .

$\frac{11 x - 110 - 10 x}{(x + 10) (x - 10)}$

Subtract $10 x$ from $11 x$ .

$\frac{x - 110}{(x + 10) (x - 10)}$