log2/3(x)-2log3(x)<=3 desigualdades

Se da la desigualdad:
$$x \frac{\log{\left(2 \right)}}{3} - 2 \frac{\log{\left(x \right)}}{\log{\left(3 \right)}} \leq 3$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$x \frac{\log{\left(2 \right)}}{3} - 2 \frac{\log{\left(x \right)}}{\log{\left(3 \right)}} = 3$$
Resolvemos:
$$x_{1} = - \frac{6 W\left(- \frac{\sqrt{3} \log{\left(2 \right)} \log{\left(3 \right)}}{54}\right)}{\log{\left(2 \right)} \log{\left(3 \right)}}$$
$$x_{2} = - \frac{6 W_{-1}\left(- \frac{\sqrt{3} \log{\left(2 \right)} \log{\left(3 \right)}}{54}\right)}{\log{\left(2 \right)} \log{\left(3 \right)}}$$
$$x_{1} = - \frac{6 W\left(- \frac{\sqrt{3} \log{\left(2 \right)} \log{\left(3 \right)}}{54}\right)}{\log{\left(2 \right)} \log{\left(3 \right)}}$$
$$x_{2} = - \frac{6 W_{-1}\left(- \frac{\sqrt{3} \log{\left(2 \right)} \log{\left(3 \right)}}{54}\right)}{\log{\left(2 \right)} \log{\left(3 \right)}}$$
Las raíces dadas
$$x_{1} = - \frac{6 W\left(- \frac{\sqrt{3} \log{\left(2 \right)} \log{\left(3 \right)}}{54}\right)}{\log{\left(2 \right)} \log{\left(3 \right)}}$$
$$x_{2} = - \frac{6 W_{-1}\left(- \frac{\sqrt{3} \log{\left(2 \right)} \log{\left(3 \right)}}{54}\right)}{\log{\left(2 \right)} \log{\left(3 \right)}}$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{1}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} - \frac{6 W\left(- \frac{\sqrt{3} \log{\left(2 \right)} \log{\left(3 \right)}}{54}\right)}{\log{\left(2 \right)} \log{\left(3 \right)}}$$
=
$$- \frac{1}{10} - \frac{6 W\left(- \frac{\sqrt{3} \log{\left(2 \right)} \log{\left(3 \right)}}{54}\right)}{\log{\left(2 \right)} \log{\left(3 \right)}}$$
lo sustituimos en la expresión
$$x \frac{\log{\left(2 \right)}}{3} - 2 \frac{\log{\left(x \right)}}{\log{\left(3 \right)}} \leq 3$$
$$- 2 \frac{\log{\left(- \frac{1}{10} - \frac{6 W\left(- \frac{\sqrt{3} \log{\left(2 \right)} \log{\left(3 \right)}}{54}\right)}{\log{\left(2 \right)} \log{\left(3 \right)}} \right)}}{\log{\left(3 \right)}} + \frac{\log{\left(2 \right)}}{3} \left(- \frac{1}{10} - \frac{6 W\left(- \frac{\sqrt{3} \log{\left(2 \right)} \log{\left(3 \right)}}{54}\right)}{\log{\left(2 \right)} \log{\left(3 \right)}}\right) \leq 3$$

       /          /   ___               \\   /          /   ___               \\            
       |          |-\/ 3 *log(2)*log(3) ||   |          |-\/ 3 *log(2)*log(3) ||            
       |       6*W|---------------------||   |       6*W|---------------------||            
       |  1       \          54         /|   |  1       \          54         /|            
  2*log|- -- - --------------------------|   |- -- - --------------------------|*log(2) <= 3
       \  10         log(2)*log(3)       /   \  10         log(2)*log(3)       /            
- ---------------------------------------- + ------------------------------------------     
                   log(3)                                        3                          
     

pero

       /          /   ___               \\   /          /   ___               \\            
       |          |-\/ 3 *log(2)*log(3) ||   |          |-\/ 3 *log(2)*log(3) ||            
       |       6*W|---------------------||   |       6*W|---------------------||            
       |  1       \          54         /|   |  1       \          54         /|            
  2*log|- -- - --------------------------|   |- -- - --------------------------|*log(2) >= 3
       \  10         log(2)*log(3)       /   \  10         log(2)*log(3)       /            
- ---------------------------------------- + ------------------------------------------     
                   log(3)                                        3                          
     

Entonces
$$x \leq - \frac{6 W\left(- \frac{\sqrt{3} \log{\left(2 \right)} \log{\left(3 \right)}}{54}\right)}{\log{\left(2 \right)} \log{\left(3 \right)}}$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x \geq - \frac{6 W\left(- \frac{\sqrt{3} \log{\left(2 \right)} \log{\left(3 \right)}}{54}\right)}{\log{\left(2 \right)} \log{\left(3 \right)}} \wedge x \leq - \frac{6 W_{-1}\left(- \frac{\sqrt{3} \log{\left(2 \right)} \log{\left(3 \right)}}{54}\right)}{\log{\left(2 \right)} \log{\left(3 \right)}}$$

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