Question

Simplify

In the following exercises, use either method.

$\frac{x-{\displaystyle \frac{3x}{x+2}}}{{\displaystyle \frac{3}{x+2}}+{\displaystyle \frac{3}{x-2}}}$.

Short answer

The simplified expression for$\frac{x-{\displaystyle \frac{3x}{x+2}}}{{\displaystyle \frac{3}{x+2}+\frac{3}{x-2}}}$is,$\frac{\left(x-1\right)\left(x-2\right)}{6}$.

Step-by-step solution

Step 1. Given information

$\frac{x-{\displaystyle \frac{3x}{x+2}}}{{\displaystyle \frac{3}{x+2}+\frac{3}{x-2}}}$.

We need to simplify the given expression.

Step 2.Find the LCD and add the fractions in the numerator.

Find the LCD and add the fractions in the denominator.

$\frac{x-{\displaystyle \frac{3x}{x+2}}}{{\displaystyle \frac{3}{x+2}+\frac{3}{x-2}}}=\frac{{\displaystyle \frac{x\left(x+2\right)}{x+2}}-{\displaystyle \frac{3x}{x+2}}}{{\displaystyle \frac{3\left(x-2\right)}{\left(x-2\right)(x+2)}+\frac{3\left(x+2\right)}{\left(x+2\right)(x-2)}}}$.

Now simplify both the numerators and the denominators.

$$\begin{gathered} \frac{{\displaystyle \frac{x\left(x+2\right)}{x+2}-\frac{3x}{x+2}}}{{\displaystyle \frac{3\left(x-2\right)}{\left(x-2\right)(x+2)}+\frac{3\left(x+2\right)}{\left(x+2\right)(x-2)}}}=\frac{{\displaystyle \frac{\left(x^2+2x\right)}{x+2}-\frac{3x}{x+2}}}{{\displaystyle \frac{\left(3x-6\right)}{\left(x-2\right)(x+2)}+\frac{\left(3x+6\right)}{\left(x+2\right)(x-2)}}} \\ =\frac{{\displaystyle \frac{x^2+2x-3x}{x+2}}}{{\displaystyle \frac{3x-6+3x+6}{\left(x-2\right)\left(x+2\right)}}} \end{gathered}$$

Step 3. Now simplify the obtained expression

$$\begin{gathered} \frac{{\displaystyle \frac{x^2+2x-3x}{x+2}}}{{\displaystyle \frac{3x-6+3x+6}{\left(x-2\right)\left(x+2\right)}}}=\frac{{\displaystyle \frac{x^2-x}{x+2}}}{{\displaystyle \frac{6x}{\left(x-2\right)\left(x+2\right)}}} \\ =\frac{x^2-x}{{\displaystyle \frac{6x}{\left(x-2\right)}}} \\ =\frac{\left(x^2-x\right)\left(x-2\right)}{6x} \\ =\frac{x\left(x-1\right)\left(x-2\right)}{6x} \\ =\frac{\left(x-1\right)\left(x-2\right)}{6} \end{gathered}$$