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Resolver desigualdades/ absolute(x^2-5x+4/(x^2-4))<=1

absolute(x^2-5x+4/(x^2-4))<=1 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src] 
| 2           4   |     
|x  - 5*x + ------| <= 1
|            2    |     
|           x  - 4|
∣(x2−5x)+4x2−4∣≤1\left|{\left(x^{2} - 5 x\right) + \frac{4}{x^{2} - 4}}\right| \leq 1 
Abs(x^2 - 5*x + 4/(x^2 - 4)) <= 1
Solución detallada
Se da la desigualdad:
∣(x2−5x)+4x2−4∣≤1\left|{\left(x^{2} - 5 x\right) + \frac{4}{x^{2} - 4}}\right| \leq 1
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
∣(x2−5x)+4x2−4∣=1\left|{\left(x^{2} - 5 x\right) + \frac{4}{x^{2} - 4}}\right| = 1
Resolvemos:
x1=0x_{1} = 0
x2=4.74361443417914x_{2} = 4.74361443417914
x3=5.15954362649092x_{3} = 5.15954362649092
x4=−1.91667869499989x_{4} = -1.91667869499989
x5=−0.378749156555439x_{5} = -0.378749156555439
x1=0x_{1} = 0
x2=4.74361443417914x_{2} = 4.74361443417914
x3=5.15954362649092x_{3} = 5.15954362649092
x4=−1.91667869499989x_{4} = -1.91667869499989
x5=−0.378749156555439x_{5} = -0.378749156555439
Las raíces dadas
x4=−1.91667869499989x_{4} = -1.91667869499989
x5=−0.378749156555439x_{5} = -0.378749156555439
x1=0x_{1} = 0
x2=4.74361443417914x_{2} = 4.74361443417914
x3=5.15954362649092x_{3} = 5.15954362649092
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x0≤x4x_{0} \leq x_{4}
Consideremos, por ejemplo, el punto
x0=x4−110x_{0} = x_{4} - \frac{1}{10}
=
−1.91667869499989+−110-1.91667869499989 + - \frac{1}{10}
=
−2.01667869499989-2.01667869499989
lo sustituimos en la expresión
∣(x2−5x)+4x2−4∣≤1\left|{\left(x^{2} - 5 x\right) + \frac{4}{x^{2} - 4}}\right| \leq 1
∣((−2.01667869499989)2−−2.01667869499989⋅5)+4−4+(−2.01667869499989)2∣≤1\left|{\left(\left(-2.01667869499989\right)^{2} - - 2.01667869499989 \cdot 5\right) + \frac{4}{-4 + \left(-2.01667869499989\right)^{2}}}\right| \leq 1
73.8581537527153 <= 1

pero
73.8581537527153 >= 1

Entonces
x≤−1.91667869499989x \leq -1.91667869499989
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
x≥−1.91667869499989∧x≤−0.378749156555439x \geq -1.91667869499989 \wedge x \leq -0.378749156555439
         _____           _____           _____  
        /     \         /     \         /
-------•-------•-------•-------•-------•-------
       x4      x5      x1      x2      x3

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x≥−1.91667869499989∧x≤−0.378749156555439x \geq -1.91667869499989 \wedge x \leq -0.378749156555439
x≥0∧x≤4.74361443417914x \geq 0 \wedge x \leq 4.74361443417914
x≥5.15954362649092x \geq 5.15954362649092
Solución de la desigualdad en el gráfico
0123456789-5-4-3-2-10200
Respuesta rápida [src] 
  /   /               / 4      3      2              \     \     /            / 3      2              \         / 4      3      2              \     \     /            / 4      3      2              \         / 3      2              \     \     /            / 4      3      2              \         / 3      2              \     \\
Or\And\x <= 0, CRootOf\x  - 5*x  - 5*x  + 20*x + 8, 1/ <= x/, And\x <= CRootOf\x  - 5*x  - 3*x + 20, 1/, CRootOf\x  - 5*x  - 5*x  + 20*x + 8, 2/ <= x/, And\x <= CRootOf\x  - 5*x  - 5*x  + 20*x + 8, 0/, CRootOf\x  - 5*x  - 3*x + 20, 0/ <= x/, And\x <= CRootOf\x  - 5*x  - 5*x  + 20*x + 8, 3/, CRootOf\x  - 5*x  - 3*x + 20, 2/ <= x//
(x≤0∧CRootOf⁡(x4−5x3−5x2+20x+8,1)≤x)∨(x≤CRootOf⁡(x3−5x2−3x+20,1)∧CRootOf⁡(x4−5x3−5x2+20x+8,2)≤x)∨(x≤CRootOf⁡(x4−5x3−5x2+20x+8,0)∧CRootOf⁡(x3−5x2−3x+20,0)≤x)∨(x≤CRootOf⁡(x4−5x3−5x2+20x+8,3)∧CRootOf⁡(x3−5x2−3x+20,2)≤x)\left(x \leq 0 \wedge \operatorname{CRootOf} {\left(x^{4} - 5 x^{3} - 5 x^{2} + 20 x + 8, 1\right)} \leq x\right) \vee \left(x \leq \operatorname{CRootOf} {\left(x^{3} - 5 x^{2} - 3 x + 20, 1\right)} \wedge \operatorname{CRootOf} {\left(x^{4} - 5 x^{3} - 5 x^{2} + 20 x + 8, 2\right)} \leq x\right) \vee \left(x \leq \operatorname{CRootOf} {\left(x^{4} - 5 x^{3} - 5 x^{2} + 20 x + 8, 0\right)} \wedge \operatorname{CRootOf} {\left(x^{3} - 5 x^{2} - 3 x + 20, 0\right)} \leq x\right) \vee \left(x \leq \operatorname{CRootOf} {\left(x^{4} - 5 x^{3} - 5 x^{2} + 20 x + 8, 3\right)} \wedge \operatorname{CRootOf} {\left(x^{3} - 5 x^{2} - 3 x + 20, 2\right)} \leq x\right) 
((x <= 0)∧(CRootOf(x^4 - 5*x^3 - 5*x^2 + 20*x + 8, 1) <= x))∨((x <= CRootOf(x^3 - 5*x^2 - 3*x + 20, 1))∧(CRootOf(x^4 - 5*x^3 - 5*x^2 + 20*x + 8, 2) <= x))∨((CRootOf(x^3 - 5*x^2 - 3*x + 20, 0) <= x)∧(x <= CRootOf(x^4 - 5*x^3 - 5*x^2 + 20*x + 8, 0)))∨((CRootOf(x^3 - 5*x^2 - 3*x + 20, 2) <= x)∧(x <= CRootOf(x^4 - 5*x^3 - 5*x^2 + 20*x + 8, 3)))
Respuesta rápida 2 [src] 
        / 3      2              \         / 4      3      2              \            / 4      3      2              \               / 4      3      2              \         / 3      2              \            / 3      2              \         / 4      3      2              \ 
[CRootOf\x  - 5*x  - 3*x + 20, 0/, CRootOf\x  - 5*x  - 5*x  + 20*x + 8, 0/] U [CRootOf\x  - 5*x  - 5*x  + 20*x + 8, 1/, 0] U [CRootOf\x  - 5*x  - 5*x  + 20*x + 8, 2/, CRootOf\x  - 5*x  - 3*x + 20, 1/] U [CRootOf\x  - 5*x  - 3*x + 20, 2/, CRootOf\x  - 5*x  - 5*x  + 20*x + 8, 3/]
x in [CRootOf⁡(x3−5x2−3x+20,0),CRootOf⁡(x4−5x3−5x2+20x+8,0)]∪[CRootOf⁡(x4−5x3−5x2+20x+8,1),0]∪[CRootOf⁡(x4−5x3−5x2+20x+8,2),CRootOf⁡(x3−5x2−3x+20,1)]∪[CRootOf⁡(x3−5x2−3x+20,2),CRootOf⁡(x4−5x3−5x2+20x+8,3)]x\ in\ \left[\operatorname{CRootOf} {\left(x^{3} - 5 x^{2} - 3 x + 20, 0\right)}, \operatorname{CRootOf} {\left(x^{4} - 5 x^{3} - 5 x^{2} + 20 x + 8, 0\right)}\right] \cup \left[\operatorname{CRootOf} {\left(x^{4} - 5 x^{3} - 5 x^{2} + 20 x + 8, 1\right)}, 0\right] \cup \left[\operatorname{CRootOf} {\left(x^{4} - 5 x^{3} - 5 x^{2} + 20 x + 8, 2\right)}, \operatorname{CRootOf} {\left(x^{3} - 5 x^{2} - 3 x + 20, 1\right)}\right] \cup \left[\operatorname{CRootOf} {\left(x^{3} - 5 x^{2} - 3 x + 20, 2\right)}, \operatorname{CRootOf} {\left(x^{4} - 5 x^{3} - 5 x^{2} + 20 x + 8, 3\right)}\right] 
x in Union(Interval(CRootOf(x^3 - 5*x^2 - 3*x + 20, 0), CRootOf(x^4 - 5*x^3 - 5*x^2 + 20*x + 8, 0)), Interval(CRootOf(x^3 - 5*x^2 - 3*x + 20, 2), CRootOf(x^4 - 5*x^3 - 5*x^2 + 20*x + 8, 3)), Interval(CRootOf(x^4 - 5*x^3 - 5*x^2 + 20*x + 8, 1), 0), Interval(CRootOf(x^4 - 5*x^3 - 5*x^2 + 20*x + 8, 2), CRootOf(x^3 - 5*x^2 - 3*x + 20, 1)))
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