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Resolver desigualdades/ abs(1/(lg0,5/lg((3-x)^2)+2))*(x^2-16)<=0

abs(1/(lg0,5/lg((3-x)^2)+2))*(x^2-16)<=0 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src] 
|        1        | / 2     \     
|-----------------|*\x  - 16/ <= 0
|   log(0.5)      |               
|------------- + 2|               
|   /       2\    |               
|log\(3 - x) /    |
(x2−16)∣12+log⁡(0.5)log⁡((3−x)2)∣≤0\left(x^{2} - 16\right) \left|{\frac{1}{2 + \frac{\log{\left(0.5 \right)}}{\log{\left(\left(3 - x\right)^{2} \right)}}}}\right| \leq 0 
(x^2 - 16)*Abs(1/(2 + log(0.5)/log((3 - x)^2))) <= 0
Solución detallada
Se da la desigualdad:
(x2−16)∣12+log⁡(0.5)log⁡((3−x)2)∣≤0\left(x^{2} - 16\right) \left|{\frac{1}{2 + \frac{\log{\left(0.5 \right)}}{\log{\left(\left(3 - x\right)^{2} \right)}}}}\right| \leq 0
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
(x2−16)∣12+log⁡(0.5)log⁡((3−x)2)∣=0\left(x^{2} - 16\right) \left|{\frac{1}{2 + \frac{\log{\left(0.5 \right)}}{\log{\left(\left(3 - x\right)^{2} \right)}}}}\right| = 0
Resolvemos:
x1=4.00000020009241x_{1} = 4.00000020009241
x2=3.99999987346384x_{2} = 3.99999987346384
x3=4.00000001051126x_{3} = 4.00000001051126
x4=3.99999983948161x_{4} = 3.99999983948161
x5=3.99999988388641x_{5} = 3.99999988388641
x6=−4x_{6} = -4
x7=4.00000012343569x_{7} = 4.00000012343569
x1=4.00000020009241x_{1} = 4.00000020009241
x2=3.99999987346384x_{2} = 3.99999987346384
x3=4.00000001051126x_{3} = 4.00000001051126
x4=3.99999983948161x_{4} = 3.99999983948161
x5=3.99999988388641x_{5} = 3.99999988388641
x6=−4x_{6} = -4
x7=4.00000012343569x_{7} = 4.00000012343569
Las raíces dadas
x6=−4x_{6} = -4
x4=3.99999983948161x_{4} = 3.99999983948161
x2=3.99999987346384x_{2} = 3.99999987346384
x5=3.99999988388641x_{5} = 3.99999988388641
x3=4.00000001051126x_{3} = 4.00000001051126
x7=4.00000012343569x_{7} = 4.00000012343569
x1=4.00000020009241x_{1} = 4.00000020009241
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x0≤x6x_{0} \leq x_{6}
Consideremos, por ejemplo, el punto
x0=x6−110x_{0} = x_{6} - \frac{1}{10}
=
−4+−110-4 + - \frac{1}{10}
=
−4.1-4.1
lo sustituimos en la expresión
(x2−16)∣12+log⁡(0.5)log⁡((3−x)2)∣≤0\left(x^{2} - 16\right) \left|{\frac{1}{2 + \frac{\log{\left(0.5 \right)}}{\log{\left(\left(3 - x\right)^{2} \right)}}}}\right| \leq 0
(−16+(−4.1)2)∣1log⁡(0.5)log⁡((3−−4.1)2)+2∣≤0\left(-16 + \left(-4.1\right)^{2}\right) \left|{\frac{1}{\frac{\log{\left(0.5 \right)}}{\log{\left(\left(3 - -4.1\right)^{2} \right)}} + 2}}\right| \leq 0
0.444277388766664 <= 0

pero
0.444277388766664 >= 0

Entonces
x≤−4x \leq -4
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
x≥−4∧x≤3.99999983948161x \geq -4 \wedge x \leq 3.99999983948161
         _____           _____           _____           _____  
        /     \         /     \         /     \         /
-------•-------•-------•-------•-------•-------•-------•-------
       x6      x4      x2      x5      x3      x7      x1

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x≥−4∧x≤3.99999983948161x \geq -4 \wedge x \leq 3.99999983948161
x≥3.99999987346384∧x≤3.99999988388641x \geq 3.99999987346384 \wedge x \leq 3.99999988388641
x≥4.00000001051126∧x≤4.00000012343569x \geq 4.00000001051126 \wedge x \leq 4.00000012343569
x≥4.00000020009241x \geq 4.00000020009241
Solución de la desigualdad en el gráfico
0123456-5-4-3-2-1-5050
Respuesta rápida 2 [src] 
[-4, 1.81079288499728) U (1.81079288499728, 2) U (2, 4)
x in [−4,1.81079288499728)∪(1.81079288499728,2)∪(2,4)x\ in\ \left[-4, 1.81079288499728\right) \cup \left(1.81079288499728, 2\right) \cup \left(2, 4\right) 
x in Union(Interval.Ropen(-4, 1.81079288499728), Interval.open(1.81079288499728, 2), Interval.open(2, 4))
Respuesta rápida [src] 
Or(And(-4 <= x, x < 1.81079288499728), And(1.81079288499728 < x, x < 2), And(2 < x, x < 4))
(−4≤x∧x<1.81079288499728)∨(1.81079288499728<x∧x<2)∨(2<x∧x<4)\left(-4 \leq x \wedge x < 1.81079288499728\right) \vee \left(1.81079288499728 < x \wedge x < 2\right) \vee \left(2 < x \wedge x < 4\right) 
((-4 <= x)∧(x < 1.81079288499728))∨((1.81079288499728 < x)∧(x < 2))∨((2 < x)∧(x < 4))
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