2x^2-9x-4<0 inequation

Given the inequality:
$$\left(2 x^{2} - 9 x\right) - 4 < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(2 x^{2} - 9 x\right) - 4 = 0$$
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
$$a = 2$$
$$b = -9$$
$$c = -4$$
, then

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

(-9)^2 - 4 * (2) * (-4) = 113

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

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

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

or
$$x_{1} = \frac{9}{4} + \frac{\sqrt{113}}{4}$$
$$x_{2} = \frac{9}{4} - \frac{\sqrt{113}}{4}$$
$$x_{1} = \frac{9}{4} + \frac{\sqrt{113}}{4}$$
$$x_{2} = \frac{9}{4} - \frac{\sqrt{113}}{4}$$
$$x_{1} = \frac{9}{4} + \frac{\sqrt{113}}{4}$$
$$x_{2} = \frac{9}{4} - \frac{\sqrt{113}}{4}$$
This roots
$$x_{2} = \frac{9}{4} - \frac{\sqrt{113}}{4}$$
$$x_{1} = \frac{9}{4} + \frac{\sqrt{113}}{4}$$
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
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$\left(\frac{9}{4} - \frac{\sqrt{113}}{4}\right) + - \frac{1}{10}$$
=
$$\frac{43}{20} - \frac{\sqrt{113}}{4}$$
substitute to the expression
$$\left(2 x^{2} - 9 x\right) - 4 < 0$$
$$-4 + \left(2 \left(\frac{43}{20} - \frac{\sqrt{113}}{4}\right)^{2} - 9 \left(\frac{43}{20} - \frac{\sqrt{113}}{4}\right)\right) < 0$$

                        2                
          /       _____\        _____    
  467     |43   \/ 113 |    9*\/ 113  < 0
- --- + 2*|-- - -------|  + ---------    
   20     \20      4   /        4        

but

                        2                
          /       _____\        _____    
  467     |43   \/ 113 |    9*\/ 113  > 0
- --- + 2*|-- - -------|  + ---------    
   20     \20      4   /        4        

Then
$$x < \frac{9}{4} - \frac{\sqrt{113}}{4}$$
no execute
one of the solutions of our inequality is:
$$x > \frac{9}{4} - \frac{\sqrt{113}}{4} \wedge x < \frac{9}{4} + \frac{\sqrt{113}}{4}$$

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