log(1/5)x<=log(1/5)1/8 inequation

Given the inequality:
$$x \log{\left(\frac{1}{5} \right)} \leq \frac{\log{\left(\frac{1}{5} \right)}}{8}$$
To solve this inequality, we must first solve the corresponding equation:
$$x \log{\left(\frac{1}{5} \right)} = \frac{\log{\left(\frac{1}{5} \right)}}{8}$$
Solve:
Given the linear equation:

log(1/5)*x = log(1/5)*1/8

Expand brackets in the left part

log1/5x = log(1/5)*1/8

Expand brackets in the right part

log1/5x = log1/5*1/8

Divide both parts of the equation by -log(5)

x = -log(5)/8 / (-log(5))

$$x_{1} = \frac{1}{8}$$
$$x_{1} = \frac{1}{8}$$
This roots
$$x_{1} = \frac{1}{8}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{1}{8}$$
=
$$\frac{1}{40}$$
substitute to the expression
$$x \log{\left(\frac{1}{5} \right)} \leq \frac{\log{\left(\frac{1}{5} \right)}}{8}$$
$$\frac{\log{\left(\frac{1}{5} \right)}}{40} \leq \frac{\log{\left(\frac{1}{5} \right)}}{8}$$

-log(5)     -log(5) 
-------- <= --------
   40          8    

but

-log(5)     -log(5) 
-------- >= --------
   40          8    

Then
$$x \leq \frac{1}{8}$$
no execute
the solution of our inequality is:
$$x \geq \frac{1}{8}$$

         _____  
        /
-------•-------
       x1