(x+19,2)/(x-5)<0 inequation

Given the inequality:
$$\frac{x + \frac{96}{5}}{x - 5} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{x + \frac{96}{5}}{x - 5} = 0$$
Solve:
Given the equation:
$$\frac{x + \frac{96}{5}}{x - 5} = 0$$
Multiply the equation sides by the denominator -5 + x
we get:
$$x + \frac{96}{5} = 0$$
Move free summands (without x)
from left part to right part, we given:
$$x = - \frac{96}{5}$$
$$x_{1} = - \frac{96}{5}$$
$$x_{1} = - \frac{96}{5}$$
This roots
$$x_{1} = - \frac{96}{5}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{96}{5} + - \frac{1}{10}$$
=
$$- \frac{193}{10}$$
substitute to the expression
$$\frac{x + \frac{96}{5}}{x - 5} < 0$$
$$\frac{- \frac{193}{10} + \frac{96}{5}}{- \frac{193}{10} - 5} < 0$$

1/243 < 0

but

1/243 > 0

Then
$$x < - \frac{96}{5}$$
no execute
the solution of our inequality is:
$$x > - \frac{96}{5}$$

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