Question

In the following exercise, perform the indicated operation:

$\frac{5}{x^2+8x-9}-\frac{4}{x^2+10x+9}$

Short answer

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

Step-by-step solution

Step 1. Given information.

The following expression is given:

$\frac{5}{x^2+8x-9}-\frac{4}{x^2+10x+9}$

Step 2. Simplification of the given expression.

Factor the denominators:

$\frac{5}{\left(x+9\right)\left(x-1\right)}-\frac{4}{\left(x+9\right)\left(x+1\right)}$

Find the LCD of the denominators:

$\left(x+9\right)\left(x-1\right)\left(x+1\right)$

Rewrite the expressions with the LCD at the denominator:

$\frac{5\left(x+1\right)}{\left(x+9\right)\left(x-1\right)\left(x+1\right)}-\frac{4\left(x-1\right)}{\left(x+9\right)\left(x+1\right)\left(x-1\right)}$

Simplify:

$$\begin{gathered} \frac{5\left(x+1\right)-4\left(x-1\right)}{\left(x+9\right)\left(x-1\right)\left(x+1\right)} \\ \frac{5x+5-4x+4}{\left(x+9\right)\left(x-1\right)\left(x+1\right)} \end{gathered}$$

Simplify and cancel the common factors from the numerator and the denominator:

$$\begin{gathered} \frac{\left(x+9\right)}{\left(x+9\right)\left(x-1\right)\left(x+1\right)} \\ \frac{1}{\left(x-1\right)\left(x+1\right)} \end{gathered}$$