# How do you solve (3-x)/(x+5)<=0 using a sign chart?

Oct 2, 2016

$\frac{3 - x}{x + 5} \le 0$ when either $x < - 5$ or $x \ge 3$, which is the solution for the inequality. Note that at $x = - 5$, it is not defined.

#### Explanation:

The sign of $\frac{3 - x}{x + 5}$ depends on the signs of binomials $\left(3 - x\right)$ and $x + 5$, which should change around the values $- 5$ and $3$ respectively. In sign chart we divide the real number line using these values, i.e. below $- 5$, between $- 5$ and $3$ and above $3$ and see how the sign of $\frac{3 - x}{x + 5}$ changes.

Sign Chart

$\textcolor{w h i t e}{X X X X X X X X X X X} - 5 \textcolor{w h i t e}{X X X X X} 3$

$\left(3 - x\right) \textcolor{w h i t e}{X X X} + i v e \textcolor{w h i t e}{X X X X} + i v e \textcolor{w h i t e}{X X X X} - i v e$

$\left(x + 5\right) \textcolor{w h i t e}{X X X} - i v e \textcolor{w h i t e}{X X X X} + i v e \textcolor{w h i t e}{X X X X} + i v e$

$\frac{3 - x}{x + 5} \textcolor{w h i t e}{X X X} - i v e \textcolor{w h i t e}{X X X X} + i v e \textcolor{w h i t e}{X X X X} - i v e$

It is observed that $\frac{3 - x}{x + 5} \le 0$ when either $x < - 5$ or $x \ge 3$, which is the solution for the inequality. Note that at $x = - 5$, it is not defined.
graph{(3-x)/(x+5) [-20, 20, -10, 10]}