x³-3x²-x+3>0 inequation

Given the inequality:
$$x^{3} - 3 x^{2} - x + 3 > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$x^{3} - 3 x^{2} - x + 3 = 0$$
Solve:
$$x_{1} = 1$$
$$x_{2} = 3$$
$$x_{3} = -1$$
$$x_{1} = 1$$
$$x_{2} = 3$$
$$x_{3} = -1$$
This roots
$$x_{3} = -1$$
$$x_{1} = 1$$
$$x_{2} = 3$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{3}$$
For example, let's take the point
$$x_{0} = x_{3} - \frac{1}{10}$$
=
$$-1 - \frac{1}{10}$$
=
$$- \frac{11}{10}$$
substitute to the expression
$$x^{3} - 3 x^{2} - x + 3 > 0$$
$$- 3 \left(- \frac{11}{10}\right)^{2} + \left(- \frac{11}{10}\right)^{3} - - \frac{11}{10} + 3 > 0$$

-861     
----- > 0
 1000    

Then
$$x < -1$$
no execute
one of the solutions of our inequality is:
$$x > -1 \wedge x < 1$$

         _____           _____  
        /     \         /
-------ο-------ο-------ο-------
       x_3      x_1      x_2

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x > -1 \wedge x < 1$$
$$x > 3$$