9x^2+8x-16<0 inequation

Given the inequality:
$$\left(9 x^{2} + 8 x\right) - 16 < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(9 x^{2} + 8 x\right) - 16 = 0$$
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
$$a = 9$$
$$b = 8$$
$$c = -16$$
, then

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

(8)^2 - 4 * (9) * (-16) = 640

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

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

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

or
$$x_{1} = - \frac{4}{9} + \frac{4 \sqrt{10}}{9}$$
$$x_{2} = - \frac{4 \sqrt{10}}{9} - \frac{4}{9}$$
$$x_{1} = - \frac{4}{9} + \frac{4 \sqrt{10}}{9}$$
$$x_{2} = - \frac{4 \sqrt{10}}{9} - \frac{4}{9}$$
$$x_{1} = - \frac{4}{9} + \frac{4 \sqrt{10}}{9}$$
$$x_{2} = - \frac{4 \sqrt{10}}{9} - \frac{4}{9}$$
This roots
$$x_{2} = - \frac{4 \sqrt{10}}{9} - \frac{4}{9}$$
$$x_{1} = - \frac{4}{9} + \frac{4 \sqrt{10}}{9}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$\left(- \frac{4 \sqrt{10}}{9} - \frac{4}{9}\right) + - \frac{1}{10}$$
=
$$- \frac{4 \sqrt{10}}{9} - \frac{49}{90}$$
substitute to the expression
$$\left(9 x^{2} + 8 x\right) - 16 < 0$$
$$-16 + \left(8 \left(- \frac{4 \sqrt{10}}{9} - \frac{49}{90}\right) + 9 \left(- \frac{4 \sqrt{10}}{9} - \frac{49}{90}\right)^{2}\right) < 0$$

                           2                
          /           ____\         ____    
  916     |  49   4*\/ 10 |    32*\/ 10  < 0
- --- + 9*|- -- - --------|  - ---------    
   45     \  90      9    /        9        

but

                           2                
          /           ____\         ____    
  916     |  49   4*\/ 10 |    32*\/ 10  > 0
- --- + 9*|- -- - --------|  - ---------    
   45     \  90      9    /        9        

Then
$$x < - \frac{4 \sqrt{10}}{9} - \frac{4}{9}$$
no execute
one of the solutions of our inequality is:
$$x > - \frac{4 \sqrt{10}}{9} - \frac{4}{9} \wedge x < - \frac{4}{9} + \frac{4 \sqrt{10}}{9}$$

         _____  
        /     \  
-------ο-------ο-------
       x2      x1