x-5x+5<0 inequation

Given the inequality:
$$\left(- 5 x + x\right) + 5 < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(- 5 x + x\right) + 5 = 0$$
Solve:
Given the linear equation:

x-5*x+5 = 0

Looking for similar summands in the left part:

5 - 4*x = 0

Move free summands (without x)
from left part to right part, we given:
$$- 4 x = -5$$
Divide both parts of the equation by -4

x = -5 / (-4)

$$x_{1} = \frac{5}{4}$$
$$x_{1} = \frac{5}{4}$$
This roots
$$x_{1} = \frac{5}{4}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{5}{4}$$
=
$$\frac{23}{20}$$
substitute to the expression
$$\left(- 5 x + x\right) + 5 < 0$$
$$\left(\frac{23}{20} - \frac{5 \cdot 23}{20}\right) + 5 < 0$$

2/5 < 0

but

2/5 > 0

Then
$$x < \frac{5}{4}$$
no execute
the solution of our inequality is:
$$x > \frac{5}{4}$$

         _____  
        /
-------ο-------
       x1