# Finding the Solution Set for an Inequality

by Learning_Math
Tags: inequality, solution
 HW Helper P: 1,362 The first step for solving $$|X| > a$$, for any expression X and number a, is to eliminate the absolute values with this: $$X < -a \text{ or } X > a$$ If you need to solve an inequality like (this is entirely made up for illustration) $$\frac x {x+1} > 5$$ your first steps should be \begin{align*} \frac x {x+1} - 5 & > 0 \\ \frac x {x+1} - \frac{5(x+1)}{x+1} & > 0 \\ \frac{x - (5x+5)}{x+1} & > 0 \\ \frac{-4x - 5}{x+1} & > 0 \\ \frac{(-1)(4x+5)}{x+1} & > 0\\ \frac{4x+5}{x+1} & < 0 \end{align*} I passed from the next-to-last to the last line by multiplying by (-1). These steps let you avoid the all-to-common problem of multiplying both sides of an inequality by a variable term when you don't know whether it's positive or negative.