2x^3-x^4>0 inequation

Given the inequality:
$$- x^{4} + 2 x^{3} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$- x^{4} + 2 x^{3} = 0$$
Solve:
Given the equation
$$- x^{2} + 2 x = 0$$
Obviously:

x0 = 0

next,
transform
$$\frac{1}{x} = \frac{1}{2}$$
Because equation degree is equal to = -1 - does not contain even numbers in the numerator, then
the equation has single real root.
Get the root -1-th degree of the equation sides:
We get:

 1     1
--- = 2 
/1\     
|-|     
\x/     

or
$$x = 2$$
We get the answer: x = 2

$$x_{1} = 0$$
$$x_{2} = 0$$
$$x_{3} = 2$$
$$x_{4} = 2$$
$$x_{1} = 0$$
$$x_{3} = 2$$
This roots
$$x_{1} = 0$$
$$x_{3} = 2$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10}$$
=
$$- \frac{1}{10}$$
substitute to the expression
$$- x^{4} + 2 x^{3} > 0$$
$$2 \left(- \frac{1}{10}\right)^{3} - \left(- \frac{1}{10}\right)^{4} > 0$$

 -21     
----- > 0
10000    

Then
$$x < 0$$
no execute
one of the solutions of our inequality is:
$$x > 0 \wedge x < 2$$

         _____  
        /     \  
-------ο-------ο-------
       x1      x3

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x > 0 \wedge x < 2$$