9x^2-24x+16>0 inequation

Given the inequality:
$$\left(9 x^{2} - 24 x\right) + 16 > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(9 x^{2} - 24 x\right) + 16 = 0$$
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
$$a = 9$$
$$b = -24$$
$$c = 16$$
, then

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

(-24)^2 - 4 * (9) * (16) = 0

Because D = 0, then the equation has one root.

x = -b/2a = --24/2/(9)

$$x_{1} = \frac{4}{3}$$
$$x_{1} = \frac{4}{3}$$
$$x_{1} = \frac{4}{3}$$
This roots
$$x_{1} = \frac{4}{3}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{4}{3}$$
=
$$\frac{37}{30}$$
substitute to the expression
$$\left(9 x^{2} - 24 x\right) + 16 > 0$$
$$\left(- \frac{24 \cdot 37}{30} + 9 \left(\frac{37}{30}\right)^{2}\right) + 16 > 0$$

9/100 > 0

the solution of our inequality is:
$$x < \frac{4}{3}$$

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