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log(x^2+1)(2*9^x-19*3^x+40)*1/(9^x+11*3^x+24)>=0
Resolver desigualdades/ log(x^2+1)(2*9^x-19*3^x+40)*1/(9^x+11*3^x+24)>=0

log(x^2+1)(2*9^x-19*3^x+40)*1/(9^x+11*3^x+24)>=0 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src] 
   / 2    \ /   x       x     \     
log\x  + 1/*\2*9  - 19*3  + 40/     
------------------------------- >= 0
         x       x                  
        9  + 11*3  + 24
((−19⋅3x+2⋅9x)+40)log⁡(x2+1)(11⋅3x+9x)+24≥0\frac{\left(\left(- 19 \cdot 3^{x} + 2 \cdot 9^{x}\right) + 40\right) \log{\left(x^{2} + 1 \right)}}{\left(11 \cdot 3^{x} + 9^{x}\right) + 24} \geq 0 
((-19*3^x + 2*9^x + 40)*log(x^2 + 1))/(11*3^x + 9^x + 24) >= 0
Solución detallada
Se da la desigualdad:
((−19⋅3x+2⋅9x)+40)log⁡(x2+1)(11⋅3x+9x)+24≥0\frac{\left(\left(- 19 \cdot 3^{x} + 2 \cdot 9^{x}\right) + 40\right) \log{\left(x^{2} + 1 \right)}}{\left(11 \cdot 3^{x} + 9^{x}\right) + 24} \geq 0
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
((−19⋅3x+2⋅9x)+40)log⁡(x2+1)(11⋅3x+9x)+24=0\frac{\left(\left(- 19 \cdot 3^{x} + 2 \cdot 9^{x}\right) + 40\right) \log{\left(x^{2} + 1 \right)}}{\left(11 \cdot 3^{x} + 9^{x}\right) + 24} = 0
Resolvemos:
x1=log⁡(194−414)log⁡(3)x_{1} = \frac{\log{\left(\frac{19}{4} - \frac{\sqrt{41}}{4} \right)}}{\log{\left(3 \right)}}
x2=log⁡(414+194)log⁡(3)x_{2} = \frac{\log{\left(\frac{\sqrt{41}}{4} + \frac{19}{4} \right)}}{\log{\left(3 \right)}}
x1=log⁡(194−414)log⁡(3)x_{1} = \frac{\log{\left(\frac{19}{4} - \frac{\sqrt{41}}{4} \right)}}{\log{\left(3 \right)}}
x2=log⁡(414+194)log⁡(3)x_{2} = \frac{\log{\left(\frac{\sqrt{41}}{4} + \frac{19}{4} \right)}}{\log{\left(3 \right)}}
Las raíces dadas
x1=log⁡(194−414)log⁡(3)x_{1} = \frac{\log{\left(\frac{19}{4} - \frac{\sqrt{41}}{4} \right)}}{\log{\left(3 \right)}}
x2=log⁡(414+194)log⁡(3)x_{2} = \frac{\log{\left(\frac{\sqrt{41}}{4} + \frac{19}{4} \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:
x0≤x1x_{0} \leq x_{1}
Consideremos, por ejemplo, el punto
x0=x1−110x_{0} = x_{1} - \frac{1}{10}
=
−110+log⁡(194−414)log⁡(3)- \frac{1}{10} + \frac{\log{\left(\frac{19}{4} - \frac{\sqrt{41}}{4} \right)}}{\log{\left(3 \right)}}
=
−110+log⁡(194−414)log⁡(3)- \frac{1}{10} + \frac{\log{\left(\frac{19}{4} - \frac{\sqrt{41}}{4} \right)}}{\log{\left(3 \right)}}
lo sustituimos en la expresión
((−19⋅3x+2⋅9x)+40)log⁡(x2+1)(11⋅3x+9x)+24≥0\frac{\left(\left(- 19 \cdot 3^{x} + 2 \cdot 9^{x}\right) + 40\right) \log{\left(x^{2} + 1 \right)}}{\left(11 \cdot 3^{x} + 9^{x}\right) + 24} \geq 0
((−19⋅3−110+log⁡(194−414)log⁡(3)+2⋅9−110+log⁡(194−414)log⁡(3))+40)log⁡((−110+log⁡(194−414)log⁡(3))2+1)24+(9−110+log⁡(194−414)log⁡(3)+11⋅3−110+log⁡(194−414)log⁡(3))≥0\frac{\left(\left(- 19 \cdot 3^{- \frac{1}{10} + \frac{\log{\left(\frac{19}{4} - \frac{\sqrt{41}}{4} \right)}}{\log{\left(3 \right)}}} + 2 \cdot 9^{- \frac{1}{10} + \frac{\log{\left(\frac{19}{4} - \frac{\sqrt{41}}{4} \right)}}{\log{\left(3 \right)}}}\right) + 40\right) \log{\left(\left(- \frac{1}{10} + \frac{\log{\left(\frac{19}{4} - \frac{\sqrt{41}}{4} \right)}}{\log{\left(3 \right)}}\right)^{2} + 1 \right)}}{24 + \left(9^{- \frac{1}{10} + \frac{\log{\left(\frac{19}{4} - \frac{\sqrt{41}}{4} \right)}}{\log{\left(3 \right)}}} + 11 \cdot 3^{- \frac{1}{10} + \frac{\log{\left(\frac{19}{4} - \frac{\sqrt{41}}{4} \right)}}{\log{\left(3 \right)}}}\right)} \geq 0
/                   /       ____\                /       ____\\                                         
|                   |19   \/ 41 |                |19   \/ 41 ||    /                             2\     
|                log|-- - ------|             log|-- - ------||    |    /          /       ____\\ |     
|           1       \4      4   /        1       \4      4   /|    |    |          |19   \/ 41 || |     
|         - -- + ----------------      - -- + ----------------|    |    |       log|-- - ------|| |     
|           10        log(3)             10        log(3)     |    |    |  1       \4      4   /| |     
\40 - 19*3                        + 2*9                       /*log|1 + |- -- + ----------------| |     
                                                                   \    \  10        log(3)     / /     
--------------------------------------------------------------------------------------------------- >= 0
                                    /       ____\                 /       ____\                         
                                    |19   \/ 41 |                 |19   \/ 41 |                         
                                 log|-- - ------|              log|-- - ------|                         
                            1       \4      4   /         1       \4      4   /                         
                          - -- + ----------------       - -- + ----------------                         
                            10        log(3)              10        log(3)                              
                    24 + 9                        + 11*3

significa que una de las soluciones de nuestra ecuación será con:
x≤log⁡(194−414)log⁡(3)x \leq \frac{\log{\left(\frac{19}{4} - \frac{\sqrt{41}}{4} \right)}}{\log{\left(3 \right)}}
 _____           _____          
      \         /
-------•-------•-------
       x1      x2

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x≤log⁡(194−414)log⁡(3)x \leq \frac{\log{\left(\frac{19}{4} - \frac{\sqrt{41}}{4} \right)}}{\log{\left(3 \right)}}
x≥log⁡(414+194)log⁡(3)x \geq \frac{\log{\left(\frac{\sqrt{41}}{4} + \frac{19}{4} \right)}}{\log{\left(3 \right)}}
Solución de la desigualdad en el gráfico
02468-8-6-4-21020-10
Respuesta rápida 2 [src] 
         /       ____\        /       ____\     
         |19   \/ 41 |        |19   \/ 41 |     
      log|-- - ------|     log|-- + ------|     
         \4      4   /        \4      4   /     
(-oo, ----------------] U [----------------, oo)
           log(3)               log(3)
x in (−∞,log⁡(194−414)log⁡(3)]∪[log⁡(414+194)log⁡(3),∞)x\ in\ \left(-\infty, \frac{\log{\left(\frac{19}{4} - \frac{\sqrt{41}}{4} \right)}}{\log{\left(3 \right)}}\right] \cup \left[\frac{\log{\left(\frac{\sqrt{41}}{4} + \frac{19}{4} \right)}}{\log{\left(3 \right)}}, \infty\right) 
x in Union(Interval(-oo, log(19/4 - sqrt(41)/4)/log(3)), Interval(log(sqrt(41)/4 + 19/4)/log(3), oo))
Respuesta rápida [src] 
  /   /   /       ____\             \          /       ____\\
  |   |   |19   \/ 41 |             |          |19   \/ 41 ||
  |   |log|-- + ------|             |       log|-- - ------||
  |   |   \4      4   /             |          \4      4   /|
Or|And|---------------- <= x, x < oo|, x <= ----------------|
  \   \     log(3)                  /            log(3)     /
(log⁡(414+194)log⁡(3)≤x∧x<∞)∨x≤log⁡(194−414)log⁡(3)\left(\frac{\log{\left(\frac{\sqrt{41}}{4} + \frac{19}{4} \right)}}{\log{\left(3 \right)}} \leq x \wedge x < \infty\right) \vee x \leq \frac{\log{\left(\frac{19}{4} - \frac{\sqrt{41}}{4} \right)}}{\log{\left(3 \right)}} 
(x <= log(19/4 - sqrt(41)/4)/log(3))∨((x < oo)∧(log(19/4 + sqrt(41)/4)/log(3) <= x))
Gráfico
log(x^2+1)(2*9^x-19*3^x+40)*1/(9^x+11*3^x+24)>=0 desigualdades
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