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logsqrt(3)(x+2)(x+4)+log1/3(x+2)<0.5*logsqrt37

logsqrt(3)(x+2)(x+4)+log1/3(x+2)<0.5*logsqrt37 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src]
                                                 /  ____\
   /  ___\                   log(1)           log\\/ 37 /
log\\/ 3 /*(x + 2)*(x + 4) + ------*(x + 2) < -----------
                               3                   2     
$$\left(x + 2\right) \log{\left(\sqrt{3} \right)} \left(x + 4\right) + \frac{\log{\left(1 \right)}}{3} \left(x + 2\right) < \frac{\log{\left(\sqrt{37} \right)}}{2}$$
((x + 2)*log(sqrt(3)))*(x + 4) + (log(1)/3)*(x + 2) < log(sqrt(37))/2
Solución detallada
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
Solución de la desigualdad en el gráfico
Respuesta rápida [src]
   /           ___   ____________________         ___   ____________________    \
   |         \/ 2 *\/ 2*log(3) + log(37)        \/ 2 *\/ 2*log(3) + log(37)     |
And|x < -3 + ----------------------------, -3 - ---------------------------- < x|
   |                     ________                           ________            |
   \                 2*\/ log(3)                        2*\/ log(3)             /
$$x < -3 + \frac{\sqrt{2} \sqrt{2 \log{\left(3 \right)} + \log{\left(37 \right)}}}{2 \sqrt{\log{\left(3 \right)}}} \wedge -3 - \frac{\sqrt{2} \sqrt{2 \log{\left(3 \right)} + \log{\left(37 \right)}}}{2 \sqrt{\log{\left(3 \right)}}} < x$$
(x < -3 + sqrt(2)*sqrt(2*log(3) + log(37))/(2*sqrt(log(3))))∧(-3 - sqrt(2)*sqrt(2*log(3) + log(37))/(2*sqrt(log(3))) < x)
Respuesta rápida 2 [src]
        ___   ____________________         ___   ____________________ 
      \/ 2 *\/ 2*log(3) + log(37)        \/ 2 *\/ 2*log(3) + log(37)  
(-3 - ----------------------------, -3 + ----------------------------)
                  ________                           ________         
              2*\/ log(3)                        2*\/ log(3)          
$$x\ in\ \left(-3 - \frac{\sqrt{2} \sqrt{2 \log{\left(3 \right)} + \log{\left(37 \right)}}}{2 \sqrt{\log{\left(3 \right)}}}, -3 + \frac{\sqrt{2} \sqrt{2 \log{\left(3 \right)} + \log{\left(37 \right)}}}{2 \sqrt{\log{\left(3 \right)}}}\right)$$
x in Interval.open(-3 - sqrt(2)*sqrt(2*log(3) + log(37))/(2*sqrt(log(3))), -3 + sqrt(2)*sqrt(2*log(3) + log(37))/(2*sqrt(log(3))))
Gráfico
logsqrt(3)(x+2)(x+4)+log1/3(x+2)<0.5*logsqrt37 desigualdades