log((3/5)*x-1)/(3^x-4)*((|x|)-2)<0 desigualdades

Se da la desigualdad:
$$\frac{\log{\left(\frac{3 x}{5} - 1 \right)}}{3^{x} - 4} \left(\left|{x}\right| - 2\right) < 0$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\frac{\log{\left(\frac{3 x}{5} - 1 \right)}}{3^{x} - 4} \left(\left|{x}\right| - 2\right) = 0$$
Resolvemos:
$$x_{1} = 2$$
$$x_{2} = 99.5489461011519$$
$$x_{3} = 93.569076935984$$
$$x_{4} = -1.58824836707847 + 1.21551105485412 i$$
$$x_{5} = 69.6955656021939$$
$$x_{6} = 89.5843484961848$$
$$x_{7} = 47.986138217215$$
$$x_{8} = 53.8721104571523$$
$$x_{9} = 34.5865198984739$$
$$x_{10} = 38.3274738944155$$
$$x_{11} = 101.542871331041$$
$$x_{12} = 0.239810685711167 - 1.98557065727179 i$$
$$x_{13} = 117.502985828178$$
$$x_{14} = 44.0911138123982$$
$$x_{15} = 77.6422682789124$$
$$x_{16} = 46.0348974386323$$
$$x_{17} = 111.516357806476$$
$$x_{18} = 75.6542283996978$$
$$x_{19} = 107.526259258136$$
$$x_{20} = -1.99183624728629 + 0.180522475045236 i$$
$$x_{21} = 91.5765077934754$$
$$x_{22} = 63.7475479215625$$
$$x_{23} = 113.511715370856$$
$$x_{24} = 61.7679847128898$$
$$x_{25} = 81.6205699126418$$
$$x_{26} = 103.537077665171$$
$$x_{27} = 79.6310724235353$$
$$x_{28} = 57.8149061867567$$
$$x_{29} = -1.90471137738747 + 0.609979154439506 i$$
$$x_{30} = 40.2342227391959$$
$$x_{31} = -1.66293534725546 + 1.11114626888109 i$$
$$x_{32} = 73.6670337421039$$
$$x_{33} = 31.0371486419629$$
$$x_{34} = 115.507261797432$$
$$x_{35} = 29.4315516048316$$
$$x_{36} = -1.91639660213643 + 0.572209807081247 i$$
$$x_{37} = -1.86464911627047 - 0.723245237241096 i$$
$$x_{38} = 95.5620246791833$$
$$x_{39} = 97.5553228758457$$
$$x_{40} = 71.6807771448587$$
$$x_{41} = 67.7115230823567$$
$$x_{42} = 59.7903423536149$$
$$x_{43} = 85.6014026112032$$
$$x_{44} = 105.531546087423$$
$$x_{45} = 32.7757404339411$$
$$x_{46} = 87.5926338246734$$
$$x_{47} = 42.1566825563676$$
$$x_{48} = 49.9434275339244$$
$$x_{49} = 51.9056947463735$$
$$x_{50} = -2$$
$$x_{51} = -1.75828888460461 + 0.953110800629115 i$$
$$x_{52} = 83.6106983486007$$
$$x_{53} = 119.498877081524$$
$$x_{54} = 36.4419983188166$$
$$x_{55} = 65.7287940517843$$
$$x_{56} = 55.8420215236371$$
$$x_{57} = 109.521201333264$$
Descartamos las soluciones complejas:
$$x_{1} = 2$$
$$x_{2} = 99.5489461011519$$
$$x_{3} = 93.569076935984$$
$$x_{4} = 69.6955656021939$$
$$x_{5} = 89.5843484961848$$
$$x_{6} = 47.986138217215$$
$$x_{7} = 53.8721104571523$$
$$x_{8} = 34.5865198984739$$
$$x_{9} = 38.3274738944155$$
$$x_{10} = 101.542871331041$$
$$x_{11} = 117.502985828178$$
$$x_{12} = 44.0911138123982$$
$$x_{13} = 77.6422682789124$$
$$x_{14} = 46.0348974386323$$
$$x_{15} = 111.516357806476$$
$$x_{16} = 75.6542283996978$$
$$x_{17} = 107.526259258136$$
$$x_{18} = 91.5765077934754$$
$$x_{19} = 63.7475479215625$$
$$x_{20} = 113.511715370856$$
$$x_{21} = 61.7679847128898$$
$$x_{22} = 81.6205699126418$$
$$x_{23} = 103.537077665171$$
$$x_{24} = 79.6310724235353$$
$$x_{25} = 57.8149061867567$$
$$x_{26} = 40.2342227391959$$
$$x_{27} = 73.6670337421039$$
$$x_{28} = 31.0371486419629$$
$$x_{29} = 115.507261797432$$
$$x_{30} = 29.4315516048316$$
$$x_{31} = 95.5620246791833$$
$$x_{32} = 97.5553228758457$$
$$x_{33} = 71.6807771448587$$
$$x_{34} = 67.7115230823567$$
$$x_{35} = 59.7903423536149$$
$$x_{36} = 85.6014026112032$$
$$x_{37} = 105.531546087423$$
$$x_{38} = 32.7757404339411$$
$$x_{39} = 87.5926338246734$$
$$x_{40} = 42.1566825563676$$
$$x_{41} = 49.9434275339244$$
$$x_{42} = 51.9056947463735$$
$$x_{43} = -2$$
$$x_{44} = 83.6106983486007$$
$$x_{45} = 119.498877081524$$
$$x_{46} = 36.4419983188166$$
$$x_{47} = 65.7287940517843$$
$$x_{48} = 55.8420215236371$$
$$x_{49} = 109.521201333264$$
Las raíces dadas
$$x_{43} = -2$$
$$x_{1} = 2$$
$$x_{30} = 29.4315516048316$$
$$x_{28} = 31.0371486419629$$
$$x_{38} = 32.7757404339411$$
$$x_{8} = 34.5865198984739$$
$$x_{46} = 36.4419983188166$$
$$x_{9} = 38.3274738944155$$
$$x_{26} = 40.2342227391959$$
$$x_{40} = 42.1566825563676$$
$$x_{12} = 44.0911138123982$$
$$x_{14} = 46.0348974386323$$
$$x_{6} = 47.986138217215$$
$$x_{41} = 49.9434275339244$$
$$x_{42} = 51.9056947463735$$
$$x_{7} = 53.8721104571523$$
$$x_{48} = 55.8420215236371$$
$$x_{25} = 57.8149061867567$$
$$x_{35} = 59.7903423536149$$
$$x_{21} = 61.7679847128898$$
$$x_{19} = 63.7475479215625$$
$$x_{47} = 65.7287940517843$$
$$x_{34} = 67.7115230823567$$
$$x_{4} = 69.6955656021939$$
$$x_{33} = 71.6807771448587$$
$$x_{27} = 73.6670337421039$$
$$x_{16} = 75.6542283996978$$
$$x_{13} = 77.6422682789124$$
$$x_{24} = 79.6310724235353$$
$$x_{22} = 81.6205699126418$$
$$x_{44} = 83.6106983486007$$
$$x_{36} = 85.6014026112032$$
$$x_{39} = 87.5926338246734$$
$$x_{5} = 89.5843484961848$$
$$x_{18} = 91.5765077934754$$
$$x_{3} = 93.569076935984$$
$$x_{31} = 95.5620246791833$$
$$x_{32} = 97.5553228758457$$
$$x_{2} = 99.5489461011519$$
$$x_{10} = 101.542871331041$$
$$x_{23} = 103.537077665171$$
$$x_{37} = 105.531546087423$$
$$x_{17} = 107.526259258136$$
$$x_{49} = 109.521201333264$$
$$x_{15} = 111.516357806476$$
$$x_{20} = 113.511715370856$$
$$x_{29} = 115.507261797432$$
$$x_{11} = 117.502985828178$$
$$x_{45} = 119.498877081524$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{43}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{43} - \frac{1}{10}$$
=
$$-2 + - \frac{1}{10}$$
=
$$-2.1$$
lo sustituimos en la expresión
$$\frac{\log{\left(\frac{3 x}{5} - 1 \right)}}{3^{x} - 4} \left(\left|{x}\right| - 2\right) < 0$$
$$\frac{\log{\left(\frac{\left(-2.1\right) 3}{5} - 1 \right)}}{-4 + 3^{-2.1}} \left(-2 + \left|{-2.1}\right|\right) < 0$$

-0.0209043830784063 - 0.0256380735810831*pi*I < 0

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

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-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------
       x43      x1      x30      x28      x38      x8      x46      x9      x26      x40      x12      x14      x6      x41      x42      x7      x48      x25      x35      x21      x19      x47      x34      x4      x33      x27      x16      x13      x24      x22      x44      x36      x39      x5      x18      x3      x31      x32      x2      x10      x23      x37      x17      x49      x15      x20      x29      x11      x45

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x > -2 \wedge x < 2$$
$$x > 29.4315516048316 \wedge x < 31.0371486419629$$
$$x > 32.7757404339411 \wedge x < 34.5865198984739$$
$$x > 36.4419983188166 \wedge x < 38.3274738944155$$
$$x > 40.2342227391959 \wedge x < 42.1566825563676$$
$$x > 44.0911138123982 \wedge x < 46.0348974386323$$
$$x > 47.986138217215 \wedge x < 49.9434275339244$$
$$x > 51.9056947463735 \wedge x < 53.8721104571523$$
$$x > 55.8420215236371 \wedge x < 57.8149061867567$$
$$x > 59.7903423536149 \wedge x < 61.7679847128898$$
$$x > 63.7475479215625 \wedge x < 65.7287940517843$$
$$x > 67.7115230823567 \wedge x < 69.6955656021939$$
$$x > 71.6807771448587 \wedge x < 73.6670337421039$$
$$x > 75.6542283996978 \wedge x < 77.6422682789124$$
$$x > 79.6310724235353 \wedge x < 81.6205699126418$$
$$x > 83.6106983486007 \wedge x < 85.6014026112032$$
$$x > 87.5926338246734 \wedge x < 89.5843484961848$$
$$x > 91.5765077934754 \wedge x < 93.569076935984$$
$$x > 95.5620246791833 \wedge x < 97.5553228758457$$
$$x > 99.5489461011519 \wedge x < 101.542871331041$$
$$x > 103.537077665171 \wedge x < 105.531546087423$$
$$x > 107.526259258136 \wedge x < 109.521201333264$$
$$x > 111.516357806476 \wedge x < 113.511715370856$$
$$x > 115.507261797432 \wedge x < 117.502985828178$$
$$x > 119.498877081524$$