# How do you solve (x-1)/(x+2)<0 using a sign chart?

Sep 23, 2016

The solution for the inequality is $- 2 < x < 1$

#### Explanation:

As $\frac{x - 1}{x + 2} < 0$, $x$ cannot take value of $- 2$.

From this we know that the product $\frac{x - 1}{x + 2}$ is negative. It is apparent that sign of binomials $\left(x + 2\right)$ and $x - 1$ will change around the values $- 2$ and $1$ respectively. In sign chart we divide the real number line using these values, i.e. below $- 2$, between $- 2$ and $1$ and above $1$ and see how the sign of $\frac{x - 1}{x + 2}$ changes.

Sign Chart

$\textcolor{w h i t e}{X X X X X X X X X X X} - 2 \textcolor{w h i t e}{X X X X X} 1$

$\left(x + 2\right) \textcolor{w h i t e}{X X X} - i v e \textcolor{w h i t e}{X X X X} + i v e \textcolor{w h i t e}{X X X X} + i v e$

$\left(x - 1\right) \textcolor{w h i t e}{X X X} - i v e \textcolor{w h i t e}{X X X X} - i v e \textcolor{w h i t e}{X X X X} + i v e$

$\frac{x - 1}{x + 2} \textcolor{w h i t e}{X X X} + i v e \textcolor{w h i t e}{X X X} - i v e \textcolor{w h i t e}{X X X X} + i v e$

It is observed that $\frac{x - 1}{x + 2} < 0$ between $- 2$ and $1$ i.e. $- 2 < x < 1$, which is the solution for the inequality.