log(1/2)log(sqrt(5(x-4)))>-1 inequation

Given the inequality:
$$\log{\left(\frac{1}{2} \right)} \log{\left(\sqrt{5 \left(x - 4\right)} \right)} > -1$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(\frac{1}{2} \right)} \log{\left(\sqrt{5 \left(x - 4\right)} \right)} = -1$$
Solve:
$$x_{1} = \frac{e^{\frac{2}{\log{\left(2 \right)}}}}{5} + 4$$
$$x_{1} = \frac{e^{\frac{2}{\log{\left(2 \right)}}}}{5} + 4$$
This roots
$$x_{1} = \frac{e^{\frac{2}{\log{\left(2 \right)}}}}{5} + 4$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \left(\frac{e^{\frac{2}{\log{\left(2 \right)}}}}{5} + 4\right)$$
=
$$\frac{e^{\frac{2}{\log{\left(2 \right)}}}}{5} + \frac{39}{10}$$
substitute to the expression
$$\log{\left(\frac{1}{2} \right)} \log{\left(\sqrt{5 \left(x - 4\right)} \right)} > -1$$
$$\log{\left(\frac{1}{2} \right)} \log{\left(\sqrt{5 \left(-4 + \left(\frac{e^{\frac{2}{\log{\left(2 \right)}}}}{5} + \frac{39}{10}\right)\right)} \right)} > -1$$

           /      _______________\     
           |     /          2    |     
           |    /         ------ |     
           |   /     1    log(2) | > -1
-log(2)*log|  /    - - + e       |     
           \\/       2           /     
     

the solution of our inequality is:
$$x < \frac{e^{\frac{2}{\log{\left(2 \right)}}}}{5} + 4$$

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