# How do you solve 9x^2+3x-2>=0 using a sign chart?

Jan 13, 2017

The answer is x in ] -oo,-2/3 ] uu [1/3, +oo[

#### Explanation:

Let's factorise the LHS

$9 {x}^{2} + 3 x - 2 = \left(3 x - 1\right) \left(3 x + 2\right)$

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

Now we can make 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 a}$$- \infty$$\textcolor{w h i t e}{a a a a}$$- \frac{2}{3}$$\textcolor{w h i t e}{a a a a a a}$$\frac{1}{3}$$\textcolor{w h i t e}{a a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$3 x + 2$$\textcolor{w h i t e}{a a 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 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 a}$$-$$\textcolor{w h i t e}{a a 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}$$f \left(x\right)$$\textcolor{w h i t e}{a a a a 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 a a}$$+$

Therefore,

$f \left(x\right) \ge 0$, when x in ] -oo,-2/3 ] uu [1/3, +oo[