log2x-3(10-3x)>=0 inequation

Given the inequality:
$$- 3 \left(10 - 3 x\right) + \log{\left(2 x \right)} \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$- 3 \left(10 - 3 x\right) + \log{\left(2 x \right)} = 0$$
Solve:
$$x_{1} = \frac{W\left(\frac{9 e^{30}}{2}\right)}{9}$$
$$x_{1} = \frac{W\left(\frac{9 e^{30}}{2}\right)}{9}$$
This roots
$$x_{1} = \frac{W\left(\frac{9 e^{30}}{2}\right)}{9}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{W\left(\frac{9 e^{30}}{2}\right)}{9}$$
=
$$- \frac{1}{10} + \frac{W\left(\frac{9 e^{30}}{2}\right)}{9}$$
substitute to the expression
$$- 3 \left(10 - 3 x\right) + \log{\left(2 x \right)} \geq 0$$
$$\log{\left(2 \left(- \frac{1}{10} + \frac{W\left(\frac{9 e^{30}}{2}\right)}{9}\right) \right)} - 3 \left(10 - 3 \left(- \frac{1}{10} + \frac{W\left(\frac{9 e^{30}}{2}\right)}{9}\right)\right) \geq 0$$

                      /         /   30\\     
                      |         |9*e  ||     
         /   30\      |      2*W|-----||     
  309    |9*e  |      |  1      \  2  /| >= 0
- --- + W|-----| + log|- - + ----------|     
   10    \  2  /      \  5       9     /     
     

but

                      /         /   30\\    
                      |         |9*e  ||    
         /   30\      |      2*W|-----||    
  309    |9*e  |      |  1      \  2  /| < 0
- --- + W|-----| + log|- - + ----------|    
   10    \  2  /      \  5       9     /    
    

Then
$$x \leq \frac{W\left(\frac{9 e^{30}}{2}\right)}{9}$$
no execute
the solution of our inequality is:
$$x \geq \frac{W\left(\frac{9 e^{30}}{2}\right)}{9}$$

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       x1