(2x-5)/(x+5)-4>0 inequation

Given the inequality:
$$-4 + \frac{2 x - 5}{x + 5} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$-4 + \frac{2 x - 5}{x + 5} = 0$$
Solve:
Given the equation:
$$-4 + \frac{2 x - 5}{x + 5} = 0$$
Multiply the equation sides by the denominator 5 + x
we get:
$$- 2 x - 25 = 0$$
Move free summands (without x)
from left part to right part, we given:
$$- 2 x = 25$$
Divide both parts of the equation by -2

x = 25 / (-2)

$$x_{1} = - \frac{25}{2}$$
$$x_{1} = - \frac{25}{2}$$
This roots
$$x_{1} = - \frac{25}{2}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{25}{2} + - \frac{1}{10}$$
=
$$- \frac{63}{5}$$
substitute to the expression
$$-4 + \frac{2 x - 5}{x + 5} > 0$$
$$-4 + \frac{\frac{\left(-63\right) 2}{5} - 5}{- \frac{63}{5} + 5} > 0$$

-1/38 > 0

Then
$$x < - \frac{25}{2}$$
no execute
the solution of our inequality is:
$$x > - \frac{25}{2}$$

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