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

Se da la desigualdad:
$$\left(- \log{\left(\cos{\left(5 \pi x \right)} \right)}^{2} + \left|{\left(2 x^{2} - 20 x\right) + 37}\right|\right) - 5 < 0$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left(- \log{\left(\cos{\left(5 \pi x \right)} \right)}^{2} + \left|{\left(2 x^{2} - 20 x\right) + 37}\right|\right) - 5 = 0$$
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.
$$2 x^{2} - 20 x + 37 \geq 0$$
o
$$\left(x \leq 5 - \frac{\sqrt{26}}{2} \wedge -\infty < x\right) \vee \left(\frac{\sqrt{26}}{2} + 5 \leq x \wedge x < \infty\right)$$
obtenemos la ecuación
$$\left(2 x^{2} - 20 x + 37\right) - \log{\left(\cos{\left(5 \pi x \right)} \right)}^{2} - 5 = 0$$
simplificamos, obtenemos
$$2 x^{2} - 20 x - \log{\left(\cos{\left(5 \pi x \right)} \right)}^{2} + 32 = 0$$
la resolución en este intervalo:

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


$$x_{1} = 15.6 - 0.964048056576548 i$$
$$x_{2} = 25.6 + 1.88709677246047 i$$
$$x_{3} = 19.6 + 1.3365048240725 i$$
$$x_{4} = -21.2 + 2.39754986841311 i$$
$$x_{5} = -10.8 - 1.44713422707814 i$$
$$x_{6} = 21.2 + 1.48393796925694 i$$
$$x_{7} = 12.8 + 0.696091151451427 i$$
$$x_{8} = 24 + 1.74078871687949 i$$
$$x_{9} = -15.2 + 1.85054497143936 i$$
$$x_{10} = -1.6 + 0.576824003919858 i$$
$$x_{11} = 23.2 + 1.66752247349304 i$$
$$x_{12} = -10.8 + 1.44713422707814 i$$
$$x_{13} = 12.4 - 0.656781587615789 i$$
$$x_{14} = 29.2 + 2.21547199706861 i$$
$$x_{15} = -13.6 + 1.70416597752502 i$$
$$x_{16} = -15.6 + 1.88709677246047 i$$
$$x_{17} = -13.2 + 1.66752247349304 i$$
$$x_{18} = -20 + 2.28832798484984 i$$
$$x_{19} = -7.6 + 1.15117701032558 i$$
$$x_{20} = 20.8 + 1.44713422707814 i$$
$$x_{21} = 7.1258752879164 + 0.0523669109373689 i$$
$$x_{22} = 16 + 1.00166848935189 i$$
$$x_{23} = -2 + 0.617056885947422 i$$
$$x_{24} = -11.2 + 1.48393796925694 i$$
$$x_{25} = -16 + 1.92363339952726 i$$
$$x_{26} = -12.4 + 1.59416728053098 i$$
$$x_{27} = -8.4 + 1.22547661401987 i$$
$$x_{28} = -0.4 - 0.451559258129506 i$$
$$x_{29} = 17.2 + 1.11392542963656 i$$
$$x_{30} = 26.4 + 1.9601557205942 i$$
$$x_{31} = -19.2 + 2.21547199706861 i$$
$$x_{32} = 15.2 + 0.92630544247187 i$$
$$x_{33} = -18.4 + 2.14257886660055 i$$
$$x_{34} = -12 + 1.55745228677979 i$$
$$x_{35} = -14.4 + 1.77739200766274 i$$
$$x_{36} = 9.2 + 0.311948087067143 i$$
$$x_{37} = -20.4 + 2.32474316255281 i$$
$$x_{38} = 18.4 + 1.22547661401987 i$$
$$x_{39} = -9.6 + 1.3365048240725 i$$
$$x_{40} = 13.6 + 0.773732921222498 i$$
$$x_{41} = -21.6 + 2.43394211696928 i$$
$$x_{42} = -9.2 + 1.29954495005496 i$$
$$x_{43} = -13.6 - 1.70416597752502 i$$
$$x_{44} = 6 + 0.201380165400829 i$$
$$x_{45} = 14 + 0.812164029042672 i$$
$$x_{46} = -17.2 + 2.0331605962715 i$$
$$x_{47} = -16.4 + 1.9601557205942 i$$
$$x_{48} = -16.8 + 1.99666453835115 i$$
$$x_{49} = -12.8 + 1.63085676632194 i$$
$$x_{50} = 32.8 + 2.54307832555945 i$$
$$x_{51} = 9.6 + 0.361220847213674 i$$
$$x_{52} = 9.2 - 0.311948087067143 i$$
$$x_{53} = 25.2 + 1.85054497143936 i$$
$$x_{54} = 24.8 + 1.81397705643572 i$$
$$x_{55} = 2$$
$$x_{56} = -8 + 1.18835853325563 i$$
$$x_{57} = 19.2 + 1.29954495005496 i$$
$$x_{58} = -8.8 + 1.26253705120893 i$$
$$x_{59} = 26.8 + 1.99666453835115 i$$
$$x_{60} = 18.8 + 1.26253705120893 i$$
$$x_{61} = -18 + 2.10611714214428 i$$
$$x_{62} = 8$$
$$x_{63} = 12 + 0.617056885947422 i$$
$$x_{64} = -18.8 + 2.17903031237737 i$$
$$x_{65} = 22.4 + 1.59416728053098 i$$
$$x_{66} = -11.6 + 1.52070988246136 i$$
$$x_{67} = -4.4 + 0.850384952750898 i$$
Descartamos las soluciones complejas:
$$x_{1} = 2$$
$$x_{2} = 8$$
Las raíces dadas
$$x_{1} = 2$$
$$x_{2} = 8$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{1}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 2$$
=
$$1.9$$
lo sustituimos en la expresión
$$\left(- \log{\left(\cos{\left(5 \pi x \right)} \right)}^{2} + \left|{\left(2 x^{2} - 20 x\right) + 37}\right|\right) - 5 < 0$$
$$-5 + \left(\left|{\left(- 1.9 \cdot 20 + 2 \cdot 1.9^{2}\right) + 37}\right| - \log{\left(\cos{\left(1.9 \cdot 5 \pi \right)} \right)}^{2}\right) < 0$$

zoo < 0

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

         _____  
        /     \  
-------ο-------ο-------
       x1      x2