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Resolver desigualdades/ log2^2(3x+1)+log(3x+1)^22-2log2(3x+1)^2-2log(3x+1)4+6<=0

log2^2(3x+1)+log(3x+1)^22-2log2(3x+1)^2-2log(3x+1)4+6<=0 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src] 
                                                     2                            
   2                   22              /log(3*x + 1)\                             
log (2)*(3*x + 1) + log  (3*x + 1) - 2*|------------|  - 2*log(3*x + 1)*4 + 6 <= 0
                                       \   log(2)   /
((−2(log⁡(3x+1)log⁡(2))2+((3x+1)log⁡(2)2+log⁡(3x+1)22))−4⋅2log⁡(3x+1))+6≤0\left(\left(- 2 \left(\frac{\log{\left(3 x + 1 \right)}}{\log{\left(2 \right)}}\right)^{2} + \left(\left(3 x + 1\right) \log{\left(2 \right)}^{2} + \log{\left(3 x + 1 \right)}^{22}\right)\right) - 4 \cdot 2 \log{\left(3 x + 1 \right)}\right) + 6 \leq 0 
-2*log(3*x + 1)^2/log(2)^2 + (3*x + 1)*log(2)^2 + log(3*x + 1)^22 - 4*2*log(3*x + 1) + 6 <= 0
Solución detallada
Se da la desigualdad:
((−2(log⁡(3x+1)log⁡(2))2+((3x+1)log⁡(2)2+log⁡(3x+1)22))−4⋅2log⁡(3x+1))+6≤0\left(\left(- 2 \left(\frac{\log{\left(3 x + 1 \right)}}{\log{\left(2 \right)}}\right)^{2} + \left(\left(3 x + 1\right) \log{\left(2 \right)}^{2} + \log{\left(3 x + 1 \right)}^{22}\right)\right) - 4 \cdot 2 \log{\left(3 x + 1 \right)}\right) + 6 \leq 0
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
((−2(log⁡(3x+1)log⁡(2))2+((3x+1)log⁡(2)2+log⁡(3x+1)22))−4⋅2log⁡(3x+1))+6=0\left(\left(- 2 \left(\frac{\log{\left(3 x + 1 \right)}}{\log{\left(2 \right)}}\right)^{2} + \left(\left(3 x + 1\right) \log{\left(2 \right)}^{2} + \log{\left(3 x + 1 \right)}^{22}\right)\right) - 4 \cdot 2 \log{\left(3 x + 1 \right)}\right) + 6 = 0
Resolvemos:
x1=0.654431990671493x_{1} = 0.654431990671493
x2=0.303220633176932x_{2} = 0.303220633176932
x1=0.654431990671493x_{1} = 0.654431990671493
x2=0.303220633176932x_{2} = 0.303220633176932
Las raíces dadas
x2=0.303220633176932x_{2} = 0.303220633176932
x1=0.654431990671493x_{1} = 0.654431990671493
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x0≤x2x_{0} \leq x_{2}
Consideremos, por ejemplo, el punto
x0=x2−110x_{0} = x_{2} - \frac{1}{10}
=
−110+0.303220633176932- \frac{1}{10} + 0.303220633176932
=
0.2032206331769320.203220633176932
lo sustituimos en la expresión
((−2(log⁡(3x+1)log⁡(2))2+((3x+1)log⁡(2)2+log⁡(3x+1)22))−4⋅2log⁡(3x+1))+6≤0\left(\left(- 2 \left(\frac{\log{\left(3 x + 1 \right)}}{\log{\left(2 \right)}}\right)^{2} + \left(\left(3 x + 1\right) \log{\left(2 \right)}^{2} + \log{\left(3 x + 1 \right)}^{22}\right)\right) - 4 \cdot 2 \log{\left(3 x + 1 \right)}\right) + 6 \leq 0
(−4⋅2log⁡(0.203220633176932⋅3+1)+(−2(log⁡(0.203220633176932⋅3+1)log⁡(2))2+(log⁡(0.203220633176932⋅3+1)22+(0.203220633176932⋅3+1)log⁡(2)2)))+6≤0\left(- 4 \cdot 2 \log{\left(0.203220633176932 \cdot 3 + 1 \right)} + \left(- 2 \left(\frac{\log{\left(0.203220633176932 \cdot 3 + 1 \right)}}{\log{\left(2 \right)}}\right)^{2} + \left(\log{\left(0.203220633176932 \cdot 3 + 1 \right)}^{22} + \left(0.203220633176932 \cdot 3 + 1\right) \log{\left(2 \right)}^{2}\right)\right)\right) + 6 \leq 0
                                      2      0.4531979954321     
2.19180682766505 + 1.6096618995308*log (2) - ---------------     
                                                    2        <= 0
                                                 log (2)

pero
                                      2      0.4531979954321     
2.19180682766505 + 1.6096618995308*log (2) - ---------------     
                                                    2        >= 0
                                                 log (2)

Entonces
x≤0.303220633176932x \leq 0.303220633176932
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
x≥0.303220633176932∧x≤0.654431990671493x \geq 0.303220633176932 \wedge x \leq 0.654431990671493
         _____  
        /     \  
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       x2      x1
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