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

Given the inequality:
$$3 + \frac{x \left(x - 2\right)}{x} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$3 + \frac{x \left(x - 2\right)}{x} = 0$$
Solve:
Given the equation:
$$3 + \frac{x \left(x - 2\right)}{x} = 0$$
transform:
$$x + 1 = 0$$
Move free summands (without x)
from left part to right part, we given:
$$x = -1$$
$$x_{1} = -1$$
$$x_{1} = -1$$
This roots
$$x_{1} = -1$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-1 + - \frac{1}{10}$$
=
$$- \frac{11}{10}$$
substitute to the expression
$$3 + \frac{x \left(x - 2\right)}{x} > 0$$
$$\frac{\left(-1\right) \frac{11}{10} \left(-2 - \frac{11}{10}\right)}{- \frac{11}{10}} + 3 > 0$$

-1/10 > 0

Then
$$x < -1$$
no execute
the solution of our inequality is:
$$x > -1$$

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