4x^2-9≤0 inequation

Given the inequality:
$$4 x^{2} - 9 \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$4 x^{2} - 9 = 0$$
Solve:
This equation is of the form

a*x^2 + b*x + c = 0

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 4$$
$$b = 0$$
$$c = -9$$
, then

D = b^2 - 4 * a * c = 

(0)^2 - 4 * (4) * (-9) = 144

Because D > 0, then the equation has two roots.

x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

or
$$x_{1} = \frac{3}{2}$$
Simplify
$$x_{2} = - \frac{3}{2}$$
Simplify
$$x_{1} = \frac{3}{2}$$
$$x_{2} = - \frac{3}{2}$$
$$x_{1} = \frac{3}{2}$$
$$x_{2} = - \frac{3}{2}$$
This roots
$$x_{2} = - \frac{3}{2}$$
$$x_{1} = \frac{3}{2}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{3}{2} - \frac{1}{10}$$
=
$$- \frac{8}{5}$$
substitute to the expression
$$4 x^{2} - 9 \leq 0$$
$$\left(-1\right) 9 + 4 \left(- \frac{8}{5}\right)^{2} \leq 0$$

31     
-- <= 0
25     

but

31     
-- >= 0
25     

Then
$$x \leq - \frac{3}{2}$$
no execute
one of the solutions of our inequality is:
$$x \geq - \frac{3}{2} \wedge x \leq \frac{3}{2}$$

         _____  
        /     \  
-------•-------•-------
       x_2      x_1