(x-3)(8-x)<0 inequation

Given the inequality:
$$\left(8 - x\right) \left(x - 3\right) < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(8 - x\right) \left(x - 3\right) = 0$$
Solve:
Expand the expression in the equation
$$\left(8 - x\right) \left(x - 3\right) = 0$$
We get the quadratic equation
$$- x^{2} + 11 x - 24 = 0$$
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
$$a = -1$$
$$b = 11$$
$$c = -24$$
, then

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

(11)^2 - 4 * (-1) * (-24) = 25

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

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

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

or
$$x_{1} = 3$$
$$x_{2} = 8$$
$$x_{1} = 3$$
$$x_{2} = 8$$
$$x_{1} = 3$$
$$x_{2} = 8$$
This roots
$$x_{1} = 3$$
$$x_{2} = 8$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 3$$
=
$$\frac{29}{10}$$
substitute to the expression
$$\left(8 - x\right) \left(x - 3\right) < 0$$
$$\left(-3 + \frac{29}{10}\right) \left(8 - \frac{29}{10}\right) < 0$$

-51     
---- < 0
100     

one of the solutions of our inequality is:
$$x < 3$$

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

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 3$$
$$x > 8$$