log((x-2)⁸/2)+log((x+4)/2)>=3 inequation

Given the inequality:
$$\log{\left(\frac{\left(x - 2\right)^{8}}{2} \right)} + \log{\left(\frac{x + 4}{2} \right)} \geq 3$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(\frac{\left(x - 2\right)^{8}}{2} \right)} + \log{\left(\frac{x + 4}{2} \right)} = 3$$
Solve:
$$x_{1} = 1.02800045621935 - 1.02199144603853 i$$
$$x_{2} = 0.568997767166696$$
$$x_{3} = 3.34847614652908$$
Exclude the complex solutions:
$$x_{1} = 0.568997767166696$$
$$x_{2} = 3.34847614652908$$
This roots
$$x_{1} = 0.568997767166696$$
$$x_{2} = 3.34847614652908$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 0.568997767166696$$
=
$$0.468997767166696$$
substitute to the expression
$$\log{\left(\frac{\left(x - 2\right)^{8}}{2} \right)} + \log{\left(\frac{x + 4}{2} \right)} \geq 3$$
$$\log{\left(\frac{0.468997767166696 + 4}{2} \right)} + \log{\left(\frac{\left(-2 + 0.468997767166696\right)^{8}}{2} \right)} \geq 3$$

3.51825040991979 >= 3

one of the solutions of our inequality is:
$$x \leq 0.568997767166696$$

 _____           _____          
      \         /
-------•-------•-------
       x1      x2

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq 0.568997767166696$$
$$x \geq 3.34847614652908$$