# How do you solve x^3<=4x^2 using a sign chart?

Jan 9, 2018

The solution is $x \in \left(- \infty , 4\right]$

#### Explanation:

The inequality is

${x}^{3} \le 4 {x}^{2}$

Rearranging

${x}^{3} - 4 {x}^{2} \le 0$

${x}^{2} \left(x - 4\right) \le 0$

Let $f \left(x\right) = {x}^{2} \left(x - 4\right)$

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 a}$$- \infty$$\textcolor{w h i t e}{a a a a a}$$0$$\textcolor{w h i t e}{a a a a a a a a a}$$4$$\textcolor{w h i t e}{a a a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$${x}^{2}$$\textcolor{w h i t e}{a a a a a a a}$$+$$\textcolor{w h i t e}{a a a}$$0$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$x - 4$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a a}$color(white)(aaaa)-$\textcolor{w h i t e}{a a a}$$0$$\textcolor{w h i t e}{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}$$0$$\textcolor{w h i t e}{a a a}$$-$$\textcolor{w h i t e}{a a a}$$0$$\textcolor{w h i t e}{a a a}$$+$

Therefore,

$f \left(x\right) \le 0$ when $x \in \left(- \infty , 4\right]$

graph{x^3-4x^2 [-14.29, 14.18, -10.41, 3.83]}