log(5)^2x-log√5x-3>=0 desigualdades

Se da la desigualdad:
$$\left(x \log{\left(5 \right)}^{2} - \log{\left(\sqrt{5 x} \right)}\right) - 3 \geq 0$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left(x \log{\left(5 \right)}^{2} - \log{\left(\sqrt{5 x} \right)}\right) - 3 = 0$$
Resolvemos:
$$x_{1} = - \frac{W\left(- \frac{2 \log{\left(5 \right)}^{2}}{5 e^{6}}\right)}{2 \log{\left(5 \right)}^{2}}$$
$$x_{2} = - \frac{W_{-1}\left(- \frac{2 \log{\left(5 \right)}^{2}}{5 e^{6}}\right)}{2 \log{\left(5 \right)}^{2}}$$
$$x_{1} = - \frac{W\left(- \frac{2 \log{\left(5 \right)}^{2}}{5 e^{6}}\right)}{2 \log{\left(5 \right)}^{2}}$$
$$x_{2} = - \frac{W_{-1}\left(- \frac{2 \log{\left(5 \right)}^{2}}{5 e^{6}}\right)}{2 \log{\left(5 \right)}^{2}}$$
Las raíces dadas
$$x_{1} = - \frac{W\left(- \frac{2 \log{\left(5 \right)}^{2}}{5 e^{6}}\right)}{2 \log{\left(5 \right)}^{2}}$$
$$x_{2} = - \frac{W_{-1}\left(- \frac{2 \log{\left(5 \right)}^{2}}{5 e^{6}}\right)}{2 \log{\left(5 \right)}^{2}}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} - \frac{W\left(- \frac{2 \log{\left(5 \right)}^{2}}{5 \left(e^{1}\right)^{6}}\right)}{2 \log{\left(5 \right)}^{2}}$$
=
$$- \frac{1}{10} - \frac{W\left(- \frac{2 \log{\left(5 \right)}^{2}}{5 e^{6}}\right)}{2 \log{\left(5 \right)}^{2}}$$
lo sustituimos en la expresión
$$\left(x \log{\left(5 \right)}^{2} - \log{\left(\sqrt{5 x} \right)}\right) - 3 \geq 0$$
$$\left(- \log{\left(\sqrt{5 \left(- \frac{1}{10} - \frac{W\left(- \frac{2 \log{\left(5 \right)}^{2}}{5 \left(e^{1}\right)^{6}}\right)}{2 \log{\left(5 \right)}^{2}}\right)} \right)} + \left(- \frac{1}{10} - \frac{W\left(- \frac{2 \log{\left(5 \right)}^{2}}{5 \left(e^{1}\right)^{6}}\right)}{2 \log{\left(5 \right)}^{2}}\right) \log{\left(5 \right)}^{2}\right) - 3 \geq 0$$

        /        ___________________________\                                          
        |       /          /      2     -6\ |           /        /      2     -6\\     
        |      /           |-2*log (5)*e  | |           |        |-2*log (5)*e  ||     
        |     /         5*W|--------------| |           |       W|--------------||     
        |    /      1      \      5       / |      2    |  1     \      5       /| >= 0
-3 - log|   /     - - - ------------------- | + log (5)*|- -- - -----------------|     
        |  /        2             2         |           |  10            2       |     
        \\/                  2*log (5)      /           \           2*log (5)    /     
     

Entonces
$$x \leq - \frac{W\left(- \frac{2 \log{\left(5 \right)}^{2}}{5 e^{6}}\right)}{2 \log{\left(5 \right)}^{2}}$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x \geq - \frac{W\left(- \frac{2 \log{\left(5 \right)}^{2}}{5 e^{6}}\right)}{2 \log{\left(5 \right)}^{2}} \wedge x \leq - \frac{W_{-1}\left(- \frac{2 \log{\left(5 \right)}^{2}}{5 e^{6}}\right)}{2 \log{\left(5 \right)}^{2}}$$

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