How do you solve 5x<=2-3x^2 using a sign chart?

Apr 23, 2017

The solution is $x \in \left[- 2 , \frac{1}{3}\right]$

Explanation:

Let's rearrange and factorise the inequality

$5 x \le 2 - 3 {x}^{2}$

$3 {x}^{2} + 5 x - 2 \le 0$

$\left(3 x - 1\right) \left(x + 2\right) \le 0$

Let $f \left(x\right) = \left(3 x - 1\right) \left(x + 2\right)$

We 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}$$- 2$$\textcolor{w h i t e}{a a a a a}$$\frac{1}{3}$$\textcolor{w h i t e}{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}$$-$$\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}$$3 x - 1$$\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}$$+$

$\textcolor{w h i t e}{a a a a}$f(x9$\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}$$+$

Therefore,

$f \left(x\right) \le 0$ when $x \in \left[- 2 , \frac{1}{3}\right]$