log(0,8)(3-5x)=>0 inequation

Given the inequality:
$$\left(3 - 5 x\right) \log{\left(\frac{4}{5} \right)} \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(3 - 5 x\right) \log{\left(\frac{4}{5} \right)} = 0$$
Solve:
Given the equation:

log((4/5))*(3-5*x) = 0

Expand expressions:

-3*log(5) + 6*log(2) - 10*x*log(2) + 5*x*log(5) = 0

Reducing, you get:

-3*log(5) + 6*log(2) - 10*x*log(2) + 5*x*log(5) = 0

Expand brackets in the left part

-3*log5 + 6*log2 - 10*x*log2 + 5*x*log5 = 0

Divide both parts of the equation by (-3*log(5) + 6*log(2) - 10*x*log(2) + 5*x*log(5))/x

x = 0 / ((-3*log(5) + 6*log(2) - 10*x*log(2) + 5*x*log(5))/x)

We get the answer: x = 3/5
$$x_{1} = \frac{3}{5}$$
$$x_{1} = \frac{3}{5}$$
This roots
$$x_{1} = \frac{3}{5}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{3}{5}$$
=
$$\frac{1}{2}$$
substitute to the expression
$$\left(3 - 5 x\right) \log{\left(\frac{4}{5} \right)} \geq 0$$
$$\left(3 - \frac{5}{2}\right) \log{\left(\frac{4}{5} \right)} \geq 0$$

log(4/5)     
-------- >= 0
   2         

but

log(4/5)    
-------- < 0
   2        

Then
$$x \leq \frac{3}{5}$$
no execute
the solution of our inequality is:
$$x \geq \frac{3}{5}$$

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