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- Sistemas de desigualdades paso a paso
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- Resolución de desigualdades exponenciales en línea
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-
¿Cómo usar?
- Desigualdades:
-
x^2>1
-
-x^2+3x-2<0
-
(x-2)/(x-4)>0
-
x+1>0
- Forma canónica:
- =0
-
Expresiones idénticas
- (doce + treinta y dos *sqrt tres ^x)/(tres ^(x/ dos)- tres)+(- dos -3^(x+ dos)/ dos)^ dos >= cero
- (12 más 32 multiplicar por raíz cuadrada de 3 en el grado x) dividir por (3 en el grado (x dividir por 2) menos 3) más ( menos 2 menos 3 en el grado (x más 2) dividir por 2) al cuadrado más o igual a 0
- (doce más treinta y dos multiplicar por raíz cuadrada de tres en el grado x) dividir por (tres en el grado (x dividir por dos) menos tres) más ( menos dos menos 3 en el grado (x más dos) dividir por dos) en el grado dos más o igual a cero
- (12+32*√3^x)/(3^(x/2)-3)+(-2-3^(x+2)/2)^2>=0
- (12+32*sqrt3x)/(3(x/2)-3)+(-2-3(x+2)/2)2>=0
- 12+32*sqrt3x/3x/2-3+-2-3x+2/22>=0
- (12+32*sqrt3^x)/(3^(x/2)-3)+(-2-3^(x+2)/2)²>=0
- (12+32*sqrt3 en el grado x)/(3 en el grado (x/2)-3)+(-2-3 en el grado (x+2)/2) en el grado 2>=0
- (12+32sqrt3^x)/(3^(x/2)-3)+(-2-3^(x+2)/2)^2>=0
- (12+32sqrt3x)/(3(x/2)-3)+(-2-3(x+2)/2)2>=0
- 12+32sqrt3x/3x/2-3+-2-3x+2/22>=0
- 12+32sqrt3^x/3^x/2-3+-2-3^x+2/2^2>=0
- (12+32*sqrt3^x)/(3^(x/2)-3)+(-2-3^(x+2)/2)^2>=O
- (12+32*sqrt3^x) dividir por (3^(x dividir por 2)-3)+(-2-3^(x+2) dividir por 2)^2>=0
-
Expresiones semejantes
- (12+32*sqrt3^x)/(3^(x/2)-3)+(2-3^(x+2)/2)^2>=0
- (12-32*sqrt3^x)/(3^(x/2)-3)+(-2-3^(x+2)/2)^2>=0
- (12+32*sqrt3^x)/(3^(x/2)+3)+(-2-3^(x+2)/2)^2>=0
- (12+32*sqrt3^x)/(3^(x/2)-3)+(-2-3^(x-2)/2)^2>=0
- (12+32*sqrt3^x)/(3^(x/2)-3)-(-2-3^(x+2)/2)^2>=0
- (12+32*sqrt3^x)/(3^(x/2)-3)+(-2+3^(x+2)/2)^2>=0
(12+32*sqrt3^x)/(3^(x/2)-3)+(-2-3^(x+2)/2)^2>=0 desigualdades
Solución
Ha introducido
[src]
x 2
___ / x + 2\
12 + 32*\/ 3 | 3 |
-------------- + |-2 - ------| >= 0
x \ 2 /
-
2
3 - 3
$$\left(- \frac{3^{x + 2}}{2} - 2\right)^{2} + \frac{32 \left(\sqrt{3}\right)^{x} + 12}{3^{\frac{x}{2}} - 3} \geq 0$$
(-3^(x + 2)/2 - 2)^2 + (32*(sqrt(3))^x + 12)/(3^(x/2) - 3) >= 0
Solución detallada
Se da la desigualdad:
$$\left(- \frac{3^{x + 2}}{2} - 2\right)^{2} + \frac{32 \left(\sqrt{3}\right)^{x} + 12}{3^{\frac{x}{2}} - 3} \geq 0$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left(- \frac{3^{x + 2}}{2} - 2\right)^{2} + \frac{32 \left(\sqrt{3}\right)^{x} + 12}{3^{\frac{x}{2}} - 3} = 0$$
Resolvemos:
$$x_{1} = \frac{\log{\left(\frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} \right)}}{\log{\left(3 \right)}}$$
$$x_{2} = \frac{\log{\left(- \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} + \frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} \right)}}{\log{\left(3 \right)}}$$
$$x_{1} = \frac{\log{\left(\frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} \right)}}{\log{\left(3 \right)}}$$
$$x_{2} = \frac{\log{\left(- \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} + \frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} \right)}}{\log{\left(3 \right)}}$$
Las raíces dadas
$$x_{2} = \frac{\log{\left(- \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} + \frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} \right)}}{\log{\left(3 \right)}}$$
$$x_{1} = \frac{\log{\left(\frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} \right)}}{\log{\left(3 \right)}}$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{2}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$\frac{\log{\left(- \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} + \frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} \right)}}{\log{\left(3 \right)}} + - \frac{1}{10}$$
=
$$\frac{\log{\left(- \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} + \frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} \right)}}{\log{\left(3 \right)}} - \frac{1}{10}$$
lo sustituimos en la expresión
$$\left(- \frac{3^{x + 2}}{2} - 2\right)^{2} + \frac{32 \left(\sqrt{3}\right)^{x} + 12}{3^{\frac{x}{2}} - 3} \geq 0$$
$$\frac{12 + 32 \left(\sqrt{3}\right)^{\frac{\log{\left(- \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} + \frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} \right)}}{\log{\left(3 \right)}} - \frac{1}{10}}}{-3 + 3^{\frac{\frac{\log{\left(- \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} + \frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} \right)}}{\log{\left(3 \right)}} - \frac{1}{10}}{2}}} + \left(-2 - \frac{3^{\left(\frac{\log{\left(- \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} + \frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} \right)}}{\log{\left(3 \right)}} - \frac{1}{10}\right) + 2}}{2}\right)^{2} \geq 0$$
pero
Entonces
$$x \leq \frac{\log{\left(- \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} + \frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} \right)}}{\log{\left(3 \right)}}$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x \geq \frac{\log{\left(- \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} + \frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} \right)}}{\log{\left(3 \right)}} \wedge x \leq \frac{\log{\left(\frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} \right)}}{\log{\left(3 \right)}}$$
$$\left(- \frac{3^{x + 2}}{2} - 2\right)^{2} + \frac{32 \left(\sqrt{3}\right)^{x} + 12}{3^{\frac{x}{2}} - 3} \geq 0$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left(- \frac{3^{x + 2}}{2} - 2\right)^{2} + \frac{32 \left(\sqrt{3}\right)^{x} + 12}{3^{\frac{x}{2}} - 3} = 0$$
Resolvemos:
$$x_{1} = \frac{\log{\left(\frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} \right)}}{\log{\left(3 \right)}}$$
$$x_{2} = \frac{\log{\left(- \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} + \frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} \right)}}{\log{\left(3 \right)}}$$
$$x_{1} = \frac{\log{\left(\frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} \right)}}{\log{\left(3 \right)}}$$
$$x_{2} = \frac{\log{\left(- \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} + \frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} \right)}}{\log{\left(3 \right)}}$$
Las raíces dadas
$$x_{2} = \frac{\log{\left(- \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} + \frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} \right)}}{\log{\left(3 \right)}}$$
$$x_{1} = \frac{\log{\left(\frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} \right)}}{\log{\left(3 \right)}}$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{2}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$\frac{\log{\left(- \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} + \frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} \right)}}{\log{\left(3 \right)}} + - \frac{1}{10}$$
=
$$\frac{\log{\left(- \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} + \frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} \right)}}{\log{\left(3 \right)}} - \frac{1}{10}$$
lo sustituimos en la expresión
$$\left(- \frac{3^{x + 2}}{2} - 2\right)^{2} + \frac{32 \left(\sqrt{3}\right)^{x} + 12}{3^{\frac{x}{2}} - 3} \geq 0$$
$$\frac{12 + 32 \left(\sqrt{3}\right)^{\frac{\log{\left(- \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} + \frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} \right)}}{\log{\left(3 \right)}} - \frac{1}{10}}}{-3 + 3^{\frac{\frac{\log{\left(- \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} + \frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} \right)}}{\log{\left(3 \right)}} - \frac{1}{10}}{2}}} + \left(-2 - \frac{3^{\left(\frac{\log{\left(- \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} + \frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} \right)}}{\log{\left(3 \right)}} - \frac{1}{10}\right) + 2}}{2}\right)^{2} \geq 0$$
2
/ / _____________________________________________________________________________________________________________________________________________________________________________________________\\ / _____________________________________________________________________________________________________________________________________________________________________________________________\
| | / ____________________________ || | / ____________________________ |
| | / / _________ || | / / _________ |
| | / 20227 / 244173680 16*\/ 5135937 289184 745 || | / 20227 / 244173680 16*\/ 5135937 289184 745 |
| | / ----- - 2*3 / --------- + -------------- - --------------------------------------- + --------------------------------------------------------------------------------------------------- || | / ----- - 2*3 / --------- + -------------- - --------------------------------------- + --------------------------------------------------------------------------------------------------- |
| | _______________________________________________________________________________________ / 486 \/ 14348907 2187 ____________________________ _______________________________________________________________________________________ || | _______________________________________________________________________________________ / 486 \/ 14348907 2187 ____________________________ _______________________________________________________________________________________ |
| | / ____________________________ / / _________ / ____________________________ || | / ____________________________ / / _________ / ____________________________ |
| | / / _________ / / 244173680 16*\/ 5135937 / / _________ || | / / _________ / / 244173680 16*\/ 5135937 / / _________ |
| | / 20227 / 244173680 16*\/ 5135937 289184 / 59049*3 / --------- + -------------- / 20227 / 244173680 16*\/ 5135937 289184 || | / 20227 / 244173680 16*\/ 5135937 289184 / 59049*3 / --------- + -------------- / 20227 / 244173680 16*\/ 5135937 289184 |
| | / ----- + 2*3 / --------- + -------------- + --------------------------------------- / \/ 14348907 2187 4* / ----- + 2*3 / --------- + -------------- + --------------------------------------- || | / ----- + 2*3 / --------- + -------------- + --------------------------------------- / \/ 14348907 2187 4* / ----- + 2*3 / --------- + -------------- + --------------------------------------- |
| | / 972 \/ 14348907 2187 ____________________________ / / 972 \/ 14348907 2187 ____________________________ || | / 972 \/ 14348907 2187 ____________________________ / / 972 \/ 14348907 2187 ____________________________ |
| | / / _________ / / / _________ || | / / _________ / / / _________ |
| | / / 244173680 16*\/ 5135937 / / / 244173680 16*\/ 5135937 || | / / 244173680 16*\/ 5135937 / / / 244173680 16*\/ 5135937 |
| | / 59049*3 / --------- + -------------- / / 59049*3 / --------- + -------------- || | / 59049*3 / --------- + -------------- / / 59049*3 / --------- + -------------- |
| |65 \/ \/ 14348907 2187 \/ \/ \/ 14348907 2187 || |65 \/ \/ 14348907 2187 \/ \/ \/ 14348907 2187 |
| log|-- + ------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|| log|-- + ------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
| 19 \36 2 2 /| 1 \36 2 2 /
| -- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| - -- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
| 10 log(3) | 20 2*log(3)
| 3 | 12 + 32*3
|-2 - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ >= 0
\ 2 / / _____________________________________________________________________________________________________________________________________________________________________________________________\
| / ____________________________ |
| / / _________ |
| / 20227 / 244173680 16*\/ 5135937 289184 745 |
| / ----- - 2*3 / --------- + -------------- - --------------------------------------- + --------------------------------------------------------------------------------------------------- |
| _______________________________________________________________________________________ / 486 \/ 14348907 2187 ____________________________ _______________________________________________________________________________________ |
| / ____________________________ / / _________ / ____________________________ |
| / / _________ / / 244173680 16*\/ 5135937 / / _________ |
| / 20227 / 244173680 16*\/ 5135937 289184 / 59049*3 / --------- + -------------- / 20227 / 244173680 16*\/ 5135937 289184 |
| / ----- + 2*3 / --------- + -------------- + --------------------------------------- / \/ 14348907 2187 4* / ----- + 2*3 / --------- + -------------- + --------------------------------------- |
| / 972 \/ 14348907 2187 ____________________________ / / 972 \/ 14348907 2187 ____________________________ |
| / / _________ / / / _________ |
| / / 244173680 16*\/ 5135937 / / / 244173680 16*\/ 5135937 |
| / 59049*3 / --------- + -------------- / / 59049*3 / --------- + -------------- |
|65 \/ \/ 14348907 2187 \/ \/ \/ 14348907 2187 |
log|-- + ------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
1 \36 2 2 /
- -- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
20 2*log(3)
-3 + 3
pero
2
/ / _____________________________________________________________________________________________________________________________________________________________________________________________\\ / _____________________________________________________________________________________________________________________________________________________________________________________________\
| | / ____________________________ || | / ____________________________ |
| | / / _________ || | / / _________ |
| | / 20227 / 244173680 16*\/ 5135937 289184 745 || | / 20227 / 244173680 16*\/ 5135937 289184 745 |
| | / ----- - 2*3 / --------- + -------------- - --------------------------------------- + --------------------------------------------------------------------------------------------------- || | / ----- - 2*3 / --------- + -------------- - --------------------------------------- + --------------------------------------------------------------------------------------------------- |
| | _______________________________________________________________________________________ / 486 \/ 14348907 2187 ____________________________ _______________________________________________________________________________________ || | _______________________________________________________________________________________ / 486 \/ 14348907 2187 ____________________________ _______________________________________________________________________________________ |
| | / ____________________________ / / _________ / ____________________________ || | / ____________________________ / / _________ / ____________________________ |
| | / / _________ / / 244173680 16*\/ 5135937 / / _________ || | / / _________ / / 244173680 16*\/ 5135937 / / _________ |
| | / 20227 / 244173680 16*\/ 5135937 289184 / 59049*3 / --------- + -------------- / 20227 / 244173680 16*\/ 5135937 289184 || | / 20227 / 244173680 16*\/ 5135937 289184 / 59049*3 / --------- + -------------- / 20227 / 244173680 16*\/ 5135937 289184 |
| | / ----- + 2*3 / --------- + -------------- + --------------------------------------- / \/ 14348907 2187 4* / ----- + 2*3 / --------- + -------------- + --------------------------------------- || | / ----- + 2*3 / --------- + -------------- + --------------------------------------- / \/ 14348907 2187 4* / ----- + 2*3 / --------- + -------------- + --------------------------------------- |
| | / 972 \/ 14348907 2187 ____________________________ / / 972 \/ 14348907 2187 ____________________________ || | / 972 \/ 14348907 2187 ____________________________ / / 972 \/ 14348907 2187 ____________________________ |
| | / / _________ / / / _________ || | / / _________ / / / _________ |
| | / / 244173680 16*\/ 5135937 / / / 244173680 16*\/ 5135937 || | / / 244173680 16*\/ 5135937 / / / 244173680 16*\/ 5135937 |
| | / 59049*3 / --------- + -------------- / / 59049*3 / --------- + -------------- || | / 59049*3 / --------- + -------------- / / 59049*3 / --------- + -------------- |
| |65 \/ \/ 14348907 2187 \/ \/ \/ 14348907 2187 || |65 \/ \/ 14348907 2187 \/ \/ \/ 14348907 2187 |
| log|-- + ------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|| log|-- + ------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
| 19 \36 2 2 /| 1 \36 2 2 /
| -- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| - -- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
| 10 log(3) | 20 2*log(3)
| 3 | 12 + 32*3
|-2 - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ < 0
\ 2 / / _____________________________________________________________________________________________________________________________________________________________________________________________\
| / ____________________________ |
| / / _________ |
| / 20227 / 244173680 16*\/ 5135937 289184 745 |
| / ----- - 2*3 / --------- + -------------- - --------------------------------------- + --------------------------------------------------------------------------------------------------- |
| _______________________________________________________________________________________ / 486 \/ 14348907 2187 ____________________________ _______________________________________________________________________________________ |
| / ____________________________ / / _________ / ____________________________ |
| / / _________ / / 244173680 16*\/ 5135937 / / _________ |
| / 20227 / 244173680 16*\/ 5135937 289184 / 59049*3 / --------- + -------------- / 20227 / 244173680 16*\/ 5135937 289184 |
| / ----- + 2*3 / --------- + -------------- + --------------------------------------- / \/ 14348907 2187 4* / ----- + 2*3 / --------- + -------------- + --------------------------------------- |
| / 972 \/ 14348907 2187 ____________________________ / / 972 \/ 14348907 2187 ____________________________ |
| / / _________ / / / _________ |
| / / 244173680 16*\/ 5135937 / / / 244173680 16*\/ 5135937 |
| / 59049*3 / --------- + -------------- / / 59049*3 / --------- + -------------- |
|65 \/ \/ 14348907 2187 \/ \/ \/ 14348907 2187 |
log|-- + ------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
1 \36 2 2 /
- -- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
20 2*log(3)
-3 + 3
Entonces
$$x \leq \frac{\log{\left(- \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} + \frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} \right)}}{\log{\left(3 \right)}}$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x \geq \frac{\log{\left(- \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} + \frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} \right)}}{\log{\left(3 \right)}} \wedge x \leq \frac{\log{\left(\frac{65}{36} + \frac{\sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} - \frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + \frac{745}{4 \sqrt{\frac{289184}{59049 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}}} + 2 \sqrt[3]{\frac{16 \sqrt{5135937}}{2187} + \frac{244173680}{14348907}} + \frac{20227}{972}}} + \frac{20227}{486}}}{2} \right)}}{\log{\left(3 \right)}}$$
_____
/ \
-------•-------•-------
x2 x1
Solución de la desigualdad en el gráfico