### Home > PC3 > Chapter 5 > Lesson 5.1.2 > Problem5-24

5-24.

Solve for $x$: $\frac { 1 } { x + 2 } - \frac { 1 } { x } > 2$

This equation has rational expressions, so the function can "change sign" at $x$-intercepts or vertical asymptotes.
Set this inequality equal to $0$.
Create a common denominator and then a single fraction.
Determine the boundary points by setting each factor equal to $0$. Mark these boundary points on a number line.
Test a point in each interval to determine the solution interval(s).

$\frac{1}{x+2}-\frac{1}{x}-2>0$

$\frac{x}{x(x+2)}-\frac{x+2}{x(x+2)}-\frac{2x(x+2)}{x(x+2)}>0$

$\frac{-2-2x^2-4x}{x(x+2)}>0$

$-\frac{2x^2+4x+2}{x(x+2)}>0$

$-\frac{2(x+1)^2}{x(x+2)}>0$

The boundary points are $x=−1$, $x=0$, and $x=−2$.
Mark these points on a number line and test a point in each interval.
You only need to think about each factor as being positive or negative since the entire expression is being compared to $0$.