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Resolver desigualdades/ absolute(2*x^2-20*x+37)-sqr(ln(cos(5*pi*x)))-5<0

absolute(2*x^2-20*x+37)-sqr(ln(cos(5*pi*x)))-5<0 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src] 
|   2            |      2                     
|2*x  - 20*x + 37| - log (cos(5*pi*x)) - 5 < 0
(−log⁡(cos⁡(5πx))2+∣(2x2−20x)+37∣)−5<0\left(- \log{\left(\cos{\left(5 \pi x \right)} \right)}^{2} + \left|{\left(2 x^{2} - 20 x\right) + 37}\right|\right) - 5 < 0 
-log(cos((5*pi)*x))^2 + |2*x^2 - 20*x + 37| - 5 < 0
Solución detallada
Se da la desigualdad:
(−log⁡(cos⁡(5πx))2+∣(2x2−20x)+37∣)−5<0\left(- \log{\left(\cos{\left(5 \pi x \right)} \right)}^{2} + \left|{\left(2 x^{2} - 20 x\right) + 37}\right|\right) - 5 < 0
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
(−log⁡(cos⁡(5πx))2+∣(2x2−20x)+37∣)−5=0\left(- \log{\left(\cos{\left(5 \pi x \right)} \right)}^{2} + \left|{\left(2 x^{2} - 20 x\right) + 37}\right|\right) - 5 = 0
Resolvemos:
Para cada expresión dentro del módulo en la ecuación
admitimos los casos cuando la expresión correspondiente es ">= 0" o "< 0",
resolvemos las ecuaciones obtenidas.

1.
2x2−20x+37≥02 x^{2} - 20 x + 37 \geq 0
o
(x≤5−262∧−∞<x)∨(262+5≤x∧x<∞)\left(x \leq 5 - \frac{\sqrt{26}}{2} \wedge -\infty < x\right) \vee \left(\frac{\sqrt{26}}{2} + 5 \leq x \wedge x < \infty\right)
obtenemos la ecuación
(2x2−20x+37)−log⁡(cos⁡(5πx))2−5=0\left(2 x^{2} - 20 x + 37\right) - \log{\left(\cos{\left(5 \pi x \right)} \right)}^{2} - 5 = 0
simplificamos, obtenemos
2x2−20x−log⁡(cos⁡(5πx))2+32=02 x^{2} - 20 x - \log{\left(\cos{\left(5 \pi x \right)} \right)}^{2} + 32 = 0
la resolución en este intervalo:

2.
2x2−20x+37<02 x^{2} - 20 x + 37 < 0
o
x<262+5∧5−262<xx < \frac{\sqrt{26}}{2} + 5 \wedge 5 - \frac{\sqrt{26}}{2} < x
obtenemos la ecuación
(−2x2+20x−37)−log⁡(cos⁡(5πx))2−5=0\left(- 2 x^{2} + 20 x - 37\right) - \log{\left(\cos{\left(5 \pi x \right)} \right)}^{2} - 5 = 0
simplificamos, obtenemos
−2x2+20x−log⁡(cos⁡(5πx))2−42=0- 2 x^{2} + 20 x - \log{\left(\cos{\left(5 \pi x \right)} \right)}^{2} - 42 = 0
la resolución en este intervalo:


x1=15.6−0.964048056576548ix_{1} = 15.6 - 0.964048056576548 i
x2=25.6+1.88709677246047ix_{2} = 25.6 + 1.88709677246047 i
x3=19.6+1.3365048240725ix_{3} = 19.6 + 1.3365048240725 i
x4=−21.2+2.39754986841311ix_{4} = -21.2 + 2.39754986841311 i
x5=−10.8−1.44713422707814ix_{5} = -10.8 - 1.44713422707814 i
x6=21.2+1.48393796925694ix_{6} = 21.2 + 1.48393796925694 i
x7=12.8+0.696091151451427ix_{7} = 12.8 + 0.696091151451427 i
x8=24+1.74078871687949ix_{8} = 24 + 1.74078871687949 i
x9=−15.2+1.85054497143936ix_{9} = -15.2 + 1.85054497143936 i
x10=−1.6+0.576824003919858ix_{10} = -1.6 + 0.576824003919858 i
x11=23.2+1.66752247349304ix_{11} = 23.2 + 1.66752247349304 i
x12=−10.8+1.44713422707814ix_{12} = -10.8 + 1.44713422707814 i
x13=12.4−0.656781587615789ix_{13} = 12.4 - 0.656781587615789 i
x14=29.2+2.21547199706861ix_{14} = 29.2 + 2.21547199706861 i
x15=−13.6+1.70416597752502ix_{15} = -13.6 + 1.70416597752502 i
x16=−15.6+1.88709677246047ix_{16} = -15.6 + 1.88709677246047 i
x17=−13.2+1.66752247349304ix_{17} = -13.2 + 1.66752247349304 i
x18=−20+2.28832798484984ix_{18} = -20 + 2.28832798484984 i
x19=−7.6+1.15117701032558ix_{19} = -7.6 + 1.15117701032558 i
x20=20.8+1.44713422707814ix_{20} = 20.8 + 1.44713422707814 i
x21=7.1258752879164+0.0523669109373689ix_{21} = 7.1258752879164 + 0.0523669109373689 i
x22=16+1.00166848935189ix_{22} = 16 + 1.00166848935189 i
x23=−2+0.617056885947422ix_{23} = -2 + 0.617056885947422 i
x24=−11.2+1.48393796925694ix_{24} = -11.2 + 1.48393796925694 i
x25=−16+1.92363339952726ix_{25} = -16 + 1.92363339952726 i
x26=−12.4+1.59416728053098ix_{26} = -12.4 + 1.59416728053098 i
x27=−8.4+1.22547661401987ix_{27} = -8.4 + 1.22547661401987 i
x28=−0.4−0.451559258129506ix_{28} = -0.4 - 0.451559258129506 i
x29=17.2+1.11392542963656ix_{29} = 17.2 + 1.11392542963656 i
x30=26.4+1.9601557205942ix_{30} = 26.4 + 1.9601557205942 i
x31=−19.2+2.21547199706861ix_{31} = -19.2 + 2.21547199706861 i
x32=15.2+0.92630544247187ix_{32} = 15.2 + 0.92630544247187 i
x33=−18.4+2.14257886660055ix_{33} = -18.4 + 2.14257886660055 i
x34=−12+1.55745228677979ix_{34} = -12 + 1.55745228677979 i
x35=−14.4+1.77739200766274ix_{35} = -14.4 + 1.77739200766274 i
x36=9.2+0.311948087067143ix_{36} = 9.2 + 0.311948087067143 i
x37=−20.4+2.32474316255281ix_{37} = -20.4 + 2.32474316255281 i
x38=18.4+1.22547661401987ix_{38} = 18.4 + 1.22547661401987 i
x39=−9.6+1.3365048240725ix_{39} = -9.6 + 1.3365048240725 i
x40=13.6+0.773732921222498ix_{40} = 13.6 + 0.773732921222498 i
x41=−21.6+2.43394211696928ix_{41} = -21.6 + 2.43394211696928 i
x42=−9.2+1.29954495005496ix_{42} = -9.2 + 1.29954495005496 i
x43=−13.6−1.70416597752502ix_{43} = -13.6 - 1.70416597752502 i
x44=6+0.201380165400829ix_{44} = 6 + 0.201380165400829 i
x45=14+0.812164029042672ix_{45} = 14 + 0.812164029042672 i
x46=−17.2+2.0331605962715ix_{46} = -17.2 + 2.0331605962715 i
x47=−16.4+1.9601557205942ix_{47} = -16.4 + 1.9601557205942 i
x48=−16.8+1.99666453835115ix_{48} = -16.8 + 1.99666453835115 i
x49=−12.8+1.63085676632194ix_{49} = -12.8 + 1.63085676632194 i
x50=32.8+2.54307832555945ix_{50} = 32.8 + 2.54307832555945 i
x51=9.6+0.361220847213674ix_{51} = 9.6 + 0.361220847213674 i
x52=9.2−0.311948087067143ix_{52} = 9.2 - 0.311948087067143 i
x53=25.2+1.85054497143936ix_{53} = 25.2 + 1.85054497143936 i
x54=24.8+1.81397705643572ix_{54} = 24.8 + 1.81397705643572 i
x55=2x_{55} = 2
x56=−8+1.18835853325563ix_{56} = -8 + 1.18835853325563 i
x57=19.2+1.29954495005496ix_{57} = 19.2 + 1.29954495005496 i
x58=−8.8+1.26253705120893ix_{58} = -8.8 + 1.26253705120893 i
x59=26.8+1.99666453835115ix_{59} = 26.8 + 1.99666453835115 i
x60=18.8+1.26253705120893ix_{60} = 18.8 + 1.26253705120893 i
x61=−18+2.10611714214428ix_{61} = -18 + 2.10611714214428 i
x62=8x_{62} = 8
x63=12+0.617056885947422ix_{63} = 12 + 0.617056885947422 i
x64=−18.8+2.17903031237737ix_{64} = -18.8 + 2.17903031237737 i
x65=22.4+1.59416728053098ix_{65} = 22.4 + 1.59416728053098 i
x66=−11.6+1.52070988246136ix_{66} = -11.6 + 1.52070988246136 i
x67=−4.4+0.850384952750898ix_{67} = -4.4 + 0.850384952750898 i
Descartamos las soluciones complejas:
x1=2x_{1} = 2
x2=8x_{2} = 8
Las raíces dadas
x1=2x_{1} = 2
x2=8x_{2} = 8
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x0<x1x_{0} < x_{1}
Consideremos, por ejemplo, el punto
x0=x1−110x_{0} = x_{1} - \frac{1}{10}
=
−110+2- \frac{1}{10} + 2
=
1.91.9
lo sustituimos en la expresión
(−log⁡(cos⁡(5πx))2+∣(2x2−20x)+37∣)−5<0\left(- \log{\left(\cos{\left(5 \pi x \right)} \right)}^{2} + \left|{\left(2 x^{2} - 20 x\right) + 37}\right|\right) - 5 < 0
−5+(∣(−1.9⋅20+2⋅1.92)+37∣−log⁡(cos⁡(1.9⋅5π))2)<0-5 + \left(\left|{\left(- 1.9 \cdot 20 + 2 \cdot 1.9^{2}\right) + 37}\right| - \log{\left(\cos{\left(1.9 \cdot 5 \pi \right)} \right)}^{2}\right) < 0
zoo < 0

Entonces
x<2x < 2
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
x>2∧x<8x > 2 \wedge x < 8
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