Se da la desigualdad:
$$\left(x \cos^{2}{\left(x \right)} + x 3 \cos{\left(x \right)}\right) - 2 > 0$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left(x \cos^{2}{\left(x \right)} + x 3 \cos{\left(x \right)}\right) - 2 = 0$$
Resolvemos:
$$x_{1} = -26.728693082705$$
$$x_{2} = -33.0070598085472$$
$$x_{3} = -20.4533148165949$$
$$x_{4} = -58.1309768799987$$
$$x_{5} = 11.0547488750862$$
$$x_{6} = 0.682665087517943$$
$$x_{7} = 83.2442179366093$$
$$x_{8} = 95.8116338454143$$
$$x_{9} = 20.3879964679349$$
$$x_{10} = 98.9668898401505$$
$$x_{11} = 58.1080345030785$$
$$x_{12} = 64.3923315009058$$
$$x_{13} = 36.1466479184015$$
$$x_{14} = 73.8364294359671$$
$$x_{15} = -4.55756819579458$$
$$x_{16} = -61.2501324187869$$
$$x_{17} = -64.4130354123689$$
$$x_{18} = 0.930351500067001$$
$$x_{19} = 39.2530186117658$$
$$x_{20} = 26.6787510893005$$
$$x_{21} = 32.9666335529089$$
$$x_{22} = -76.9777057852597$$
$$x_{23} = -80.102266657408$$
$$x_{24} = -95.8255492867835$$
$$x_{25} = 7.77041142824238$$
$$x_{26} = -89.542854490658$$
$$x_{27} = -54.9656929228516$$
$$x_{28} = -51.8491925306413$$
$$x_{29} = 86.4014942219495$$
$$x_{30} = 42.4271332244671$$
$$x_{31} = -42.3956919974488$$
$$x_{32} = 80.1189107746112$$
$$x_{33} = -98.9534161617049$$
$$x_{34} = 29.86728933687$$
$$x_{35} = -23.533339719032$$
$$x_{36} = 14.0905605985215$$
$$x_{37} = 67.554078596124$$
$$x_{38} = -39.2869752362261$$
$$x_{39} = 89.5279624880051$$
$$x_{40} = 48.7083115896913$$
$$x_{41} = -73.8183687073374$$
$$x_{42} = 92.684159065639$$
$$x_{43} = -17.2395670747798$$
$$x_{44} = -1.97009754111268$$
$$x_{45} = -7.94053815319331$$
$$x_{46} = 45.5385240927414$$
$$x_{47} = 4.84459219271314$$
$$x_{48} = -70.6952948192008$$
$$x_{49} = -10.9332655369111$$
$$x_{50} = -86.3860606470558$$
$$x_{51} = -67.534337670891$$
$$x_{52} = 54.9899465589348$$
$$x_{53} = -48.6809279880377$$
$$x_{54} = -83.2602339157403$$
$$x_{55} = 23.5899478660992$$
$$x_{56} = 61.2718982432872$$
$$x_{57} = 76.9603823092665$$
$$x_{58} = -45.5677962742133$$
$$x_{59} = -14.1849435952604$$
$$x_{60} = 51.8234689437341$$
$$x_{61} = 70.6764313816329$$
$$x_{62} = 17.3167852981077$$
$$x_{63} = -29.8226047937162$$
$$x_{64} = -36.1097371682237$$
$$x_{65} = -92.669771880393$$
$$x_{1} = -26.728693082705$$
$$x_{2} = -33.0070598085472$$
$$x_{3} = -20.4533148165949$$
$$x_{4} = -58.1309768799987$$
$$x_{5} = 11.0547488750862$$
$$x_{6} = 0.682665087517943$$
$$x_{7} = 83.2442179366093$$
$$x_{8} = 95.8116338454143$$
$$x_{9} = 20.3879964679349$$
$$x_{10} = 98.9668898401505$$
$$x_{11} = 58.1080345030785$$
$$x_{12} = 64.3923315009058$$
$$x_{13} = 36.1466479184015$$
$$x_{14} = 73.8364294359671$$
$$x_{15} = -4.55756819579458$$
$$x_{16} = -61.2501324187869$$
$$x_{17} = -64.4130354123689$$
$$x_{18} = 0.930351500067001$$
$$x_{19} = 39.2530186117658$$
$$x_{20} = 26.6787510893005$$
$$x_{21} = 32.9666335529089$$
$$x_{22} = -76.9777057852597$$
$$x_{23} = -80.102266657408$$
$$x_{24} = -95.8255492867835$$
$$x_{25} = 7.77041142824238$$
$$x_{26} = -89.542854490658$$
$$x_{27} = -54.9656929228516$$
$$x_{28} = -51.8491925306413$$
$$x_{29} = 86.4014942219495$$
$$x_{30} = 42.4271332244671$$
$$x_{31} = -42.3956919974488$$
$$x_{32} = 80.1189107746112$$
$$x_{33} = -98.9534161617049$$
$$x_{34} = 29.86728933687$$
$$x_{35} = -23.533339719032$$
$$x_{36} = 14.0905605985215$$
$$x_{37} = 67.554078596124$$
$$x_{38} = -39.2869752362261$$
$$x_{39} = 89.5279624880051$$
$$x_{40} = 48.7083115896913$$
$$x_{41} = -73.8183687073374$$
$$x_{42} = 92.684159065639$$
$$x_{43} = -17.2395670747798$$
$$x_{44} = -1.97009754111268$$
$$x_{45} = -7.94053815319331$$
$$x_{46} = 45.5385240927414$$
$$x_{47} = 4.84459219271314$$
$$x_{48} = -70.6952948192008$$
$$x_{49} = -10.9332655369111$$
$$x_{50} = -86.3860606470558$$
$$x_{51} = -67.534337670891$$
$$x_{52} = 54.9899465589348$$
$$x_{53} = -48.6809279880377$$
$$x_{54} = -83.2602339157403$$
$$x_{55} = 23.5899478660992$$
$$x_{56} = 61.2718982432872$$
$$x_{57} = 76.9603823092665$$
$$x_{58} = -45.5677962742133$$
$$x_{59} = -14.1849435952604$$
$$x_{60} = 51.8234689437341$$
$$x_{61} = 70.6764313816329$$
$$x_{62} = 17.3167852981077$$
$$x_{63} = -29.8226047937162$$
$$x_{64} = -36.1097371682237$$
$$x_{65} = -92.669771880393$$
Las raíces dadas
$$x_{33} = -98.9534161617049$$
$$x_{24} = -95.8255492867835$$
$$x_{65} = -92.669771880393$$
$$x_{26} = -89.542854490658$$
$$x_{50} = -86.3860606470558$$
$$x_{54} = -83.2602339157403$$
$$x_{23} = -80.102266657408$$
$$x_{22} = -76.9777057852597$$
$$x_{41} = -73.8183687073374$$
$$x_{48} = -70.6952948192008$$
$$x_{51} = -67.534337670891$$
$$x_{17} = -64.4130354123689$$
$$x_{16} = -61.2501324187869$$
$$x_{4} = -58.1309768799987$$
$$x_{27} = -54.9656929228516$$
$$x_{28} = -51.8491925306413$$
$$x_{53} = -48.6809279880377$$
$$x_{58} = -45.5677962742133$$
$$x_{31} = -42.3956919974488$$
$$x_{38} = -39.2869752362261$$
$$x_{64} = -36.1097371682237$$
$$x_{2} = -33.0070598085472$$
$$x_{63} = -29.8226047937162$$
$$x_{1} = -26.728693082705$$
$$x_{35} = -23.533339719032$$
$$x_{3} = -20.4533148165949$$
$$x_{43} = -17.2395670747798$$
$$x_{59} = -14.1849435952604$$
$$x_{49} = -10.9332655369111$$
$$x_{45} = -7.94053815319331$$
$$x_{15} = -4.55756819579458$$
$$x_{44} = -1.97009754111268$$
$$x_{6} = 0.682665087517943$$
$$x_{18} = 0.930351500067001$$
$$x_{47} = 4.84459219271314$$
$$x_{25} = 7.77041142824238$$
$$x_{5} = 11.0547488750862$$
$$x_{36} = 14.0905605985215$$
$$x_{62} = 17.3167852981077$$
$$x_{9} = 20.3879964679349$$
$$x_{55} = 23.5899478660992$$
$$x_{20} = 26.6787510893005$$
$$x_{34} = 29.86728933687$$
$$x_{21} = 32.9666335529089$$
$$x_{13} = 36.1466479184015$$
$$x_{19} = 39.2530186117658$$
$$x_{30} = 42.4271332244671$$
$$x_{46} = 45.5385240927414$$
$$x_{40} = 48.7083115896913$$
$$x_{60} = 51.8234689437341$$
$$x_{52} = 54.9899465589348$$
$$x_{11} = 58.1080345030785$$
$$x_{56} = 61.2718982432872$$
$$x_{12} = 64.3923315009058$$
$$x_{37} = 67.554078596124$$
$$x_{61} = 70.6764313816329$$
$$x_{14} = 73.8364294359671$$
$$x_{57} = 76.9603823092665$$
$$x_{32} = 80.1189107746112$$
$$x_{7} = 83.2442179366093$$
$$x_{29} = 86.4014942219495$$
$$x_{39} = 89.5279624880051$$
$$x_{42} = 92.684159065639$$
$$x_{8} = 95.8116338454143$$
$$x_{10} = 98.9668898401505$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{33}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{33} - \frac{1}{10}$$
=
$$-98.9534161617049 + - \frac{1}{10}$$
=
$$-99.0534161617049$$
lo sustituimos en la expresión
$$\left(x \cos^{2}{\left(x \right)} + x 3 \cos{\left(x \right)}\right) - 2 > 0$$
$$\left(\left(-99.0534161617049\right) 3 \cos{\left(-99.0534161617049 \right)} + \left(-99.0534161617049\right) \cos^{2}{\left(-99.0534161617049 \right)}\right) - 2 > 0$$
-30.5281203839216 > 0
Entonces
$$x < -98.9534161617049$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x > -98.9534161617049 \wedge x < -95.8255492867835$$
_____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
-------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-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------
x33 x24 x65 x26 x50 x54 x23 x22 x41 x48 x51 x17 x16 x4 x27 x28 x53 x58 x31 x38 x64 x2 x63 x1 x35 x3 x43 x59 x49 x45 x15 x44 x6 x18 x47 x25 x5 x36 x62 x9 x55 x20 x34 x21 x13 x19 x30 x46 x40 x60 x52 x11 x56 x12 x37 x61 x14 x57 x32 x7 x29 x39 x42 x8 x10
Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x > -98.9534161617049 \wedge x < -95.8255492867835$$
$$x > -92.669771880393 \wedge x < -89.542854490658$$
$$x > -86.3860606470558 \wedge x < -83.2602339157403$$
$$x > -80.102266657408 \wedge x < -76.9777057852597$$
$$x > -73.8183687073374 \wedge x < -70.6952948192008$$
$$x > -67.534337670891 \wedge x < -64.4130354123689$$
$$x > -61.2501324187869 \wedge x < -58.1309768799987$$
$$x > -54.9656929228516 \wedge x < -51.8491925306413$$
$$x > -48.6809279880377 \wedge x < -45.5677962742133$$
$$x > -42.3956919974488 \wedge x < -39.2869752362261$$
$$x > -36.1097371682237 \wedge x < -33.0070598085472$$
$$x > -29.8226047937162 \wedge x < -26.728693082705$$
$$x > -23.533339719032 \wedge x < -20.4533148165949$$
$$x > -17.2395670747798 \wedge x < -14.1849435952604$$
$$x > -10.9332655369111 \wedge x < -7.94053815319331$$
$$x > -4.55756819579458 \wedge x < -1.97009754111268$$
$$x > 0.682665087517943 \wedge x < 0.930351500067001$$
$$x > 4.84459219271314 \wedge x < 7.77041142824238$$
$$x > 11.0547488750862 \wedge x < 14.0905605985215$$
$$x > 17.3167852981077 \wedge x < 20.3879964679349$$
$$x > 23.5899478660992 \wedge x < 26.6787510893005$$
$$x > 29.86728933687 \wedge x < 32.9666335529089$$
$$x > 36.1466479184015 \wedge x < 39.2530186117658$$
$$x > 42.4271332244671 \wedge x < 45.5385240927414$$
$$x > 48.7083115896913 \wedge x < 51.8234689437341$$
$$x > 54.9899465589348 \wedge x < 58.1080345030785$$
$$x > 61.2718982432872 \wedge x < 64.3923315009058$$
$$x > 67.554078596124 \wedge x < 70.6764313816329$$
$$x > 73.8364294359671 \wedge x < 76.9603823092665$$
$$x > 80.1189107746112 \wedge x < 83.2442179366093$$
$$x > 86.4014942219495 \wedge x < 89.5279624880051$$
$$x > 92.684159065639 \wedge x < 95.8116338454143$$
$$x > 98.9668898401505$$