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cos(x)^2*x+3*cos(x)*x-2>0 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src]
   2                          
cos (x)*x + 3*cos(x)*x - 2 > 0
$$\left(x \cos^{2}{\left(x \right)} + x 3 \cos{\left(x \right)}\right) - 2 > 0$$
x*cos(x)^2 + x*(3*cos(x)) - 2 > 0
Solución detallada
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:
$$x > -98.9534161617049 \wedge x < -95.8255492867835$$
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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-------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:
$$x > -98.9534161617049 \wedge x < -95.8255492867835$$
$$x > -92.669771880393 \wedge x < -89.542854490658$$
$$x > -86.3860606470558 \wedge x < -83.2602339157403$$
$$x > -80.102266657408 \wedge x < -76.9777057852597$$
$$x > -73.8183687073374 \wedge x < -70.6952948192008$$
$$x > -67.534337670891 \wedge x < -64.4130354123689$$
$$x > -61.2501324187869 \wedge x < -58.1309768799987$$
$$x > -54.9656929228516 \wedge x < -51.8491925306413$$
$$x > -48.6809279880377 \wedge x < -45.5677962742133$$
$$x > -42.3956919974488 \wedge x < -39.2869752362261$$
$$x > -36.1097371682237 \wedge x < -33.0070598085472$$
$$x > -29.8226047937162 \wedge x < -26.728693082705$$
$$x > -23.533339719032 \wedge x < -20.4533148165949$$
$$x > -17.2395670747798 \wedge x < -14.1849435952604$$
$$x > -10.9332655369111 \wedge x < -7.94053815319331$$
$$x > -4.55756819579458 \wedge x < -1.97009754111268$$
$$x > 0.682665087517943 \wedge x < 0.930351500067001$$
$$x > 4.84459219271314 \wedge x < 7.77041142824238$$
$$x > 11.0547488750862 \wedge x < 14.0905605985215$$
$$x > 17.3167852981077 \wedge x < 20.3879964679349$$
$$x > 23.5899478660992 \wedge x < 26.6787510893005$$
$$x > 29.86728933687 \wedge x < 32.9666335529089$$
$$x > 36.1466479184015 \wedge x < 39.2530186117658$$
$$x > 42.4271332244671 \wedge x < 45.5385240927414$$
$$x > 48.7083115896913 \wedge x < 51.8234689437341$$
$$x > 54.9899465589348 \wedge x < 58.1080345030785$$
$$x > 61.2718982432872 \wedge x < 64.3923315009058$$
$$x > 67.554078596124 \wedge x < 70.6764313816329$$
$$x > 73.8364294359671 \wedge x < 76.9603823092665$$
$$x > 80.1189107746112 \wedge x < 83.2442179366093$$
$$x > 86.4014942219495 \wedge x < 89.5279624880051$$
$$x > 92.684159065639 \wedge x < 95.8116338454143$$
$$x > 98.9668898401505$$
Solución de la desigualdad en el gráfico