(x-2):(3-x)>0 inequation

Given the inequality:
$$\frac{x - 2}{3 - x} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{x - 2}{3 - x} = 0$$
Solve:
Given the equation:
$$\frac{x - 2}{3 - x} = 0$$
Multiply the equation sides by the denominator 3 - x
we get:
$$\frac{\left(2 - x\right) \left(3 - x\right)}{x - 3} = 0$$
Expand brackets in the left part

2+x3+x-3+x = 0

Looking for similar summands in the left part:

(2 - x)*(3 - x)/(-3 + x) = 0

Move free summands (without x)
from left part to right part, we given:
$$\frac{\left(2 - x\right) \left(3 - x\right)}{x - 3} + 3 = 3$$
Divide both parts of the equation by (3 + (2 - x)*(3 - x)/(-3 + x))/x

x = 3 / ((3 + (2 - x)*(3 - x)/(-3 + x))/x)

$$x_{1} = 2$$
$$x_{1} = 2$$
This roots
$$x_{1} = 2$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 2$$
=
$$\frac{19}{10}$$
substitute to the expression
$$\frac{x - 2}{3 - x} > 0$$
$$\frac{-2 + \frac{19}{10}}{3 - \frac{19}{10}} > 0$$

-1/11 > 0

Then
$$x < 2$$
no execute
the solution of our inequality is:
$$x > 2$$

         _____  
        /
-------ο-------
       x1