# How do you solve 6/x<2 using a sign chart?

Jun 11, 2017

The solution is $x \in \left(- \infty , 0\right) \cup \left(3 , + \infty\right)$

#### Explanation:

Let's rewrite and simplify the inequality

We cannot do crossing over

$\frac{6}{x} < 2$

$\frac{6}{x} - 2 < 0$

$\frac{6 - 2 x}{x} < 0$

$\frac{2 \left(3 - x\right)}{x} < 0$

Let $f \left(x\right) = \frac{2 \left(3 - x\right)}{x}$

Let's build the sign chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a a a}$$0$$\textcolor{w h i t e}{a a a a a a a}$$3$$\textcolor{w h i t e}{a a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$\left(x\right)$$\textcolor{w h i t e}{a a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$| |$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a a a}$$+$

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

$\textcolor{w h i t e}{a a a a}$$f \left(x\right)$$\textcolor{w h i t e}{a a a a a a}$$-$$\textcolor{w h i t e}{a a a}$$| |$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a a a}$$-$

Therefore,

$f \left(x\right) < 0$, when $x \in \left(- \infty , 0\right) \cup \left(3 , + \infty\right)$ graph{(6/x)-2 [-22.81, 22.8, -11.4, 11.42]}