# How do you solve 2w^3-8w<0 using a sign chart?

Oct 10, 2017

The solution is $w \in \left(- \infty , - 2\right) \cup \left(0 , 2\right)$

#### Explanation:

Let's factorise the inequality

$2 {w}^{3} - 8 w < 0$

$2 w \left({w}^{2} - 4\right) < 0$

$2 w \left(w + 2\right) \left(w - 2\right) < 0$

Let $f \left(w\right) = 2 w \left(w + 2\right) \left(w - 2\right)$

Let's build the sign chart

$\textcolor{w h i t e}{a a a a}$$w$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a}$$- 2$$\textcolor{w h i t e}{a a a a}$$0$$\textcolor{w h i t e}{a a a a a a}$$2$$\textcolor{w h i t e}{a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$w + 2$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$w$$\textcolor{w h i t e}{a a a 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 a}$$+$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$w - 2$$\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 a}$$-$$\textcolor{w h i t e}{a a a a}$$+$

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

Therefore,

$f \left(w\right) < 0$, when $w \in \left(- \infty , - 2\right) \cup \left(0 , 2\right)$

graph{2x^3-8x [-14.24, 14.24, -7.12, 7.12]}