Se da la desigualdad:
$$\left(x + 2\right) \log{\left(\sqrt{3} \right)} \left(x + 4\right) + \frac{\log{\left(1 \right)}}{3} \left(x + 2\right) < \frac{\log{\left(\sqrt{37} \right)}}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left(x + 2\right) \log{\left(\sqrt{3} \right)} \left(x + 4\right) + \frac{\log{\left(1 \right)}}{3} \left(x + 2\right) = \frac{\log{\left(\sqrt{37} \right)}}{2}$$
Resolvemos:
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$\left(x + 2\right) \log{\left(\sqrt{3} \right)} \left(x + 4\right) + \frac{\log{\left(1 \right)}}{3} \left(x + 2\right) = \frac{\log{\left(\sqrt{37} \right)}}{2}$$
en
$$\left(\left(x + 2\right) \log{\left(\sqrt{3} \right)} \left(x + 4\right) + \frac{\log{\left(1 \right)}}{3} \left(x + 2\right)\right) - \frac{\log{\left(\sqrt{37} \right)}}{2} = 0$$
Abramos la expresión en la ecuación
$$\left(\left(x + 2\right) \log{\left(\sqrt{3} \right)} \left(x + 4\right) + \frac{\log{\left(1 \right)}}{3} \left(x + 2\right)\right) - \frac{\log{\left(\sqrt{37} \right)}}{2} = 0$$
Obtenemos la ecuación cuadrática
$$\frac{x^{2} \log{\left(3 \right)}}{2} + 3 x \log{\left(3 \right)} - \frac{\log{\left(37 \right)}}{4} + 4 \log{\left(3 \right)} = 0$$
Es la ecuación de la forma
a*x^2 + b*x + c = 0
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = \frac{\log{\left(3 \right)}}{2}$$
$$b = 3 \log{\left(3 \right)}$$
$$c = - \frac{\log{\left(37 \right)}}{4} + 4 \log{\left(3 \right)}$$
, entonces
D = b^2 - 4 * a * c =
(3*log(3))^2 - 4 * (log(3)/2) * (4*log(3) - log(37)/4) = 9*log(3)^2 - 2*(4*log(3) - log(37)/4)*log(3)
Como D > 0 la ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{- 3 \log{\left(3 \right)} + \sqrt{- 2 \left(- \frac{\log{\left(37 \right)}}{4} + 4 \log{\left(3 \right)}\right) \log{\left(3 \right)} + 9 \log{\left(3 \right)}^{2}}}{\log{\left(3 \right)}}$$
$$x_{2} = \frac{- 3 \log{\left(3 \right)} - \sqrt{- 2 \left(- \frac{\log{\left(37 \right)}}{4} + 4 \log{\left(3 \right)}\right) \log{\left(3 \right)} + 9 \log{\left(3 \right)}^{2}}}{\log{\left(3 \right)}}$$
$$x_{1} = \frac{- 3 \log{\left(3 \right)} + \sqrt{- 2 \left(- \frac{\log{\left(37 \right)}}{4} + 4 \log{\left(3 \right)}\right) \log{\left(3 \right)} + 9 \log{\left(3 \right)}^{2}}}{\log{\left(3 \right)}}$$
$$x_{2} = \frac{- 3 \log{\left(3 \right)} - \sqrt{- 2 \left(- \frac{\log{\left(37 \right)}}{4} + 4 \log{\left(3 \right)}\right) \log{\left(3 \right)} + 9 \log{\left(3 \right)}^{2}}}{\log{\left(3 \right)}}$$
$$x_{1} = \frac{- 3 \log{\left(3 \right)} + \sqrt{- 2 \left(- \frac{\log{\left(37 \right)}}{4} + 4 \log{\left(3 \right)}\right) \log{\left(3 \right)} + 9 \log{\left(3 \right)}^{2}}}{\log{\left(3 \right)}}$$
$$x_{2} = \frac{- 3 \log{\left(3 \right)} - \sqrt{- 2 \left(- \frac{\log{\left(37 \right)}}{4} + 4 \log{\left(3 \right)}\right) \log{\left(3 \right)} + 9 \log{\left(3 \right)}^{2}}}{\log{\left(3 \right)}}$$
Las raíces dadas
$$x_{2} = \frac{- 3 \log{\left(3 \right)} - \sqrt{- 2 \left(- \frac{\log{\left(37 \right)}}{4} + 4 \log{\left(3 \right)}\right) \log{\left(3 \right)} + 9 \log{\left(3 \right)}^{2}}}{\log{\left(3 \right)}}$$
$$x_{1} = \frac{- 3 \log{\left(3 \right)} + \sqrt{- 2 \left(- \frac{\log{\left(37 \right)}}{4} + 4 \log{\left(3 \right)}\right) \log{\left(3 \right)} + 9 \log{\left(3 \right)}^{2}}}{\log{\left(3 \right)}}$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{2}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$\frac{- 3 \log{\left(3 \right)} - \sqrt{- 2 \left(- \frac{\log{\left(37 \right)}}{4} + 4 \log{\left(3 \right)}\right) \log{\left(3 \right)} + 9 \log{\left(3 \right)}^{2}}}{\log{\left(3 \right)}} + - \frac{1}{10}$$
=
$$\frac{- 3 \log{\left(3 \right)} - \sqrt{- 2 \left(- \frac{\log{\left(37 \right)}}{4} + 4 \log{\left(3 \right)}\right) \log{\left(3 \right)} + 9 \log{\left(3 \right)}^{2}}}{\log{\left(3 \right)}} - \frac{1}{10}$$
lo sustituimos en la expresión
$$\left(x + 2\right) \log{\left(\sqrt{3} \right)} \left(x + 4\right) + \frac{\log{\left(1 \right)}}{3} \left(x + 2\right) < \frac{\log{\left(\sqrt{37} \right)}}{2}$$
$$\frac{\log{\left(1 \right)}}{3} \left(\left(\frac{- 3 \log{\left(3 \right)} - \sqrt{- 2 \left(- \frac{\log{\left(37 \right)}}{4} + 4 \log{\left(3 \right)}\right) \log{\left(3 \right)} + 9 \log{\left(3 \right)}^{2}}}{\log{\left(3 \right)}} - \frac{1}{10}\right) + 2\right) + \left(\left(\frac{- 3 \log{\left(3 \right)} - \sqrt{- 2 \left(- \frac{\log{\left(37 \right)}}{4} + 4 \log{\left(3 \right)}\right) \log{\left(3 \right)} + 9 \log{\left(3 \right)}^{2}}}{\log{\left(3 \right)}} - \frac{1}{10}\right) + 2\right) \log{\left(\sqrt{3} \right)} \left(\left(\frac{- 3 \log{\left(3 \right)} - \sqrt{- 2 \left(- \frac{\log{\left(37 \right)}}{4} + 4 \log{\left(3 \right)}\right) \log{\left(3 \right)} + 9 \log{\left(3 \right)}^{2}}}{\log{\left(3 \right)}} - \frac{1}{10}\right) + 4\right) < \frac{\log{\left(\sqrt{37} \right)}}{2}$$
/ ___________________________________________ \ / ___________________________________________ \
| / 2 / log(37)\ | | / 2 / log(37)\ | / ____\
| - / 9*log (3) - 2*|4*log(3) - -------|*log(3) - 3*log(3)| | - / 9*log (3) - 2*|4*log(3) - -------|*log(3) - 3*log(3)| log\\/ 37 /
|19 \/ \ 4 / | |39 \/ \ 4 / | / ___\ < -----------
|-- + ------------------------------------------------------------|*|-- + ------------------------------------------------------------|*log\\/ 3 / 2
\10 log(3) / \10 log(3) /
pero
/ ___________________________________________ \ / ___________________________________________ \
| / 2 / log(37)\ | | / 2 / log(37)\ | / ____\
| - / 9*log (3) - 2*|4*log(3) - -------|*log(3) - 3*log(3)| | - / 9*log (3) - 2*|4*log(3) - -------|*log(3) - 3*log(3)| log\\/ 37 /
|19 \/ \ 4 / | |39 \/ \ 4 / | / ___\ > -----------
|-- + ------------------------------------------------------------|*|-- + ------------------------------------------------------------|*log\\/ 3 / 2
\10 log(3) / \10 log(3) /
Entonces
$$x < \frac{- 3 \log{\left(3 \right)} - \sqrt{- 2 \left(- \frac{\log{\left(37 \right)}}{4} + 4 \log{\left(3 \right)}\right) \log{\left(3 \right)} + 9 \log{\left(3 \right)}^{2}}}{\log{\left(3 \right)}}$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x > \frac{- 3 \log{\left(3 \right)} - \sqrt{- 2 \left(- \frac{\log{\left(37 \right)}}{4} + 4 \log{\left(3 \right)}\right) \log{\left(3 \right)} + 9 \log{\left(3 \right)}^{2}}}{\log{\left(3 \right)}} \wedge x < \frac{- 3 \log{\left(3 \right)} + \sqrt{- 2 \left(- \frac{\log{\left(37 \right)}}{4} + 4 \log{\left(3 \right)}\right) \log{\left(3 \right)} + 9 \log{\left(3 \right)}^{2}}}{\log{\left(3 \right)}}$$
_____
/ \
-------ο-------ο-------
x2 x1