log(2/5)*(2,5-2,5x)>-1 inequation

Given the inequality:
$$\left(\frac{5}{2} - \frac{5 x}{2}\right) \log{\left(\frac{2}{5} \right)} > -1$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\frac{5}{2} - \frac{5 x}{2}\right) \log{\left(\frac{2}{5} \right)} = -1$$
Solve:
Given the equation:

log(2/5)*((5/2)-(5/2)*x) = -1

Expand expressions:

-5*log(5)/2 + 5*log(2)/2 - 5*x*log(2)/2 + 5*x*log(5)/2 = -1

Reducing, you get:

1 - 5*log(5)/2 + 5*log(2)/2 - 5*x*log(2)/2 + 5*x*log(5)/2 = 0

Expand brackets in the left part

1 - 5*log5/2 + 5*log2/2 - 5*x*log2/2 + 5*x*log5/2 = 0

Move free summands (without x)
from left part to right part, we given:
$$- \frac{5 x \log{\left(2 \right)}}{2} + \frac{5 x \log{\left(5 \right)}}{2} - \frac{5 \log{\left(5 \right)}}{2} + \frac{5 \log{\left(2 \right)}}{2} = -1$$
Divide both parts of the equation by (-5*log(5)/2 + 5*log(2)/2 - 5*x*log(2)/2 + 5*x*log(5)/2)/x

x = -1 / ((-5*log(5)/2 + 5*log(2)/2 - 5*x*log(2)/2 + 5*x*log(5)/2)/x)

We get the answer: x = (2 + log(32/3125))/(5*log(2/5))
$$x_{1} = \frac{\log{\left(\frac{32}{3125} \right)} + 2}{5 \log{\left(\frac{2}{5} \right)}}$$
$$x_{1} = \frac{\log{\left(\frac{32}{3125} \right)} + 2}{5 \log{\left(\frac{2}{5} \right)}}$$
This roots
$$x_{1} = \frac{\log{\left(\frac{32}{3125} \right)} + 2}{5 \log{\left(\frac{2}{5} \right)}}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{\log{\left(\frac{32}{3125} \right)} + 2}{5 \log{\left(\frac{2}{5} \right)}}$$
=
$$- \frac{1}{10} + \frac{\log{\left(\frac{32}{3125} \right)} + 2}{5 \log{\left(\frac{2}{5} \right)}}$$
substitute to the expression
$$\left(\frac{5}{2} - \frac{5 x}{2}\right) \log{\left(\frac{2}{5} \right)} > -1$$
$$\left(\frac{5}{2} - \frac{5 \left(- \frac{1}{10} + \frac{\log{\left(\frac{32}{3125} \right)} + 2}{5 \log{\left(\frac{2}{5} \right)}}\right)}{2}\right) \log{\left(\frac{2}{5} \right)} > -1$$

/            / 32 \\              
|     2 + log|----||              
|11          \3125/|          > -1
|-- - -------------|*log(2/5)     
\4      2*log(2/5) /              

Then
$$x < \frac{\log{\left(\frac{32}{3125} \right)} + 2}{5 \log{\left(\frac{2}{5} \right)}}$$
no execute
the solution of our inequality is:
$$x > \frac{\log{\left(\frac{32}{3125} \right)} + 2}{5 \log{\left(\frac{2}{5} \right)}}$$

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