cosx(x)>-√3/2 desigualdades

Se da la desigualdad:
$$x \cos{\left(x \right)} > \frac{\left(-1\right) \sqrt{3}}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$x \cos{\left(x \right)} = \frac{\left(-1\right) \sqrt{3}}{2}$$
Resolvemos:
$$x_{1} = -64.3891991378643$$
$$x_{2} = 45.572098034099$$
$$x_{3} = 20.4626870669311$$
$$x_{4} = -7.74188466920172$$
$$x_{5} = 42.3910699719495$$
$$x_{6} = -95.8095367800758$$
$$x_{7} = 64.4160940432473$$
$$x_{8} = -39.2478408235586$$
$$x_{9} = -54.993619831997$$
$$x_{10} = -45.5340730476022$$
$$x_{11} = 58.1343616049366$$
$$x_{12} = -73.8391561670124$$
$$x_{13} = -92.6863270323791$$
$$x_{14} = 83.2626066401751$$
$$x_{15} = 14.1982002614746$$
$$x_{16} = -4.89041416017729$$
$$x_{17} = 89.5450621701513$$
$$x_{18} = 26.7359350354377$$
$$x_{19} = 76.9802702160215$$
$$x_{20} = -67.5570615725169$$
$$x_{21} = 86.3837724806605$$
$$x_{22} = 73.8156948236903$$
$$x_{23} = 33.0129587650466$$
$$x_{24} = 17.2284712949965$$
$$x_{25} = -98.9689191782982$$
$$x_{26} = -51.8195656815415$$
$$x_{27} = -26.6710612453683$$
$$x_{28} = 23.5251237938033$$
$$x_{29} = -83.2418014003112$$
$$x_{30} = 51.8529811155539$$
$$x_{31} = -61.2751905931337$$
$$x_{32} = 95.827613383739$$
$$x_{33} = -70.6735805215019$$
$$x_{34} = 36.1043264668609$$
$$x_{35} = -76.9577665199981$$
$$x_{36} = 61.2469163716388$$
$$x_{37} = 67.5314176635823$$
$$x_{38} = -17.3287566215911$$
$$x_{39} = -32.960445150861$$
$$x_{40} = 29.8160805418067$$
$$x_{41} = 92.6676376448981$$
$$x_{42} = 10.9161565517939$$
$$x_{43} = 48.6768938881967$$
$$x_{44} = -23.5986512331331$$
$$x_{45} = 183.787882344406$$
$$x_{46} = -20.3778410567194$$
$$x_{47} = -58.1045589361698$$
$$x_{48} = -42.4319120069122$$
$$x_{49} = -48.7124653792196$$
$$x_{50} = -86.4038211404408$$
$$x_{51} = 39.2919507389347$$
$$x_{52} = 4.51958024618035$$
$$x_{53} = -14.075601349021$$
$$x_{54} = 98.951416449997$$
$$x_{55} = 2.01504507250032$$
$$x_{56} = -11.0738587072216$$
$$x_{57} = -89.5257169945252$$
$$x_{58} = -29.8741234195341$$
$$x_{59} = 80.0998006261809$$
$$x_{60} = -80.1214217891022$$
$$x_{61} = 7.96295398117797$$
$$x_{62} = 70.6980846424974$$
$$x_{63} = 54.96211401548$$
$$x_{64} = -36.1522727445826$$
$$x_{1} = -64.3891991378643$$
$$x_{2} = 45.572098034099$$
$$x_{3} = 20.4626870669311$$
$$x_{4} = -7.74188466920172$$
$$x_{5} = 42.3910699719495$$
$$x_{6} = -95.8095367800758$$
$$x_{7} = 64.4160940432473$$
$$x_{8} = -39.2478408235586$$
$$x_{9} = -54.993619831997$$
$$x_{10} = -45.5340730476022$$
$$x_{11} = 58.1343616049366$$
$$x_{12} = -73.8391561670124$$
$$x_{13} = -92.6863270323791$$
$$x_{14} = 83.2626066401751$$
$$x_{15} = 14.1982002614746$$
$$x_{16} = -4.89041416017729$$
$$x_{17} = 89.5450621701513$$
$$x_{18} = 26.7359350354377$$
$$x_{19} = 76.9802702160215$$
$$x_{20} = -67.5570615725169$$
$$x_{21} = 86.3837724806605$$
$$x_{22} = 73.8156948236903$$
$$x_{23} = 33.0129587650466$$
$$x_{24} = 17.2284712949965$$
$$x_{25} = -98.9689191782982$$
$$x_{26} = -51.8195656815415$$
$$x_{27} = -26.6710612453683$$
$$x_{28} = 23.5251237938033$$
$$x_{29} = -83.2418014003112$$
$$x_{30} = 51.8529811155539$$
$$x_{31} = -61.2751905931337$$
$$x_{32} = 95.827613383739$$
$$x_{33} = -70.6735805215019$$
$$x_{34} = 36.1043264668609$$
$$x_{35} = -76.9577665199981$$
$$x_{36} = 61.2469163716388$$
$$x_{37} = 67.5314176635823$$
$$x_{38} = -17.3287566215911$$
$$x_{39} = -32.960445150861$$
$$x_{40} = 29.8160805418067$$
$$x_{41} = 92.6676376448981$$
$$x_{42} = 10.9161565517939$$
$$x_{43} = 48.6768938881967$$
$$x_{44} = -23.5986512331331$$
$$x_{45} = 183.787882344406$$
$$x_{46} = -20.3778410567194$$
$$x_{47} = -58.1045589361698$$
$$x_{48} = -42.4319120069122$$
$$x_{49} = -48.7124653792196$$
$$x_{50} = -86.4038211404408$$
$$x_{51} = 39.2919507389347$$
$$x_{52} = 4.51958024618035$$
$$x_{53} = -14.075601349021$$
$$x_{54} = 98.951416449997$$
$$x_{55} = 2.01504507250032$$
$$x_{56} = -11.0738587072216$$
$$x_{57} = -89.5257169945252$$
$$x_{58} = -29.8741234195341$$
$$x_{59} = 80.0998006261809$$
$$x_{60} = -80.1214217891022$$
$$x_{61} = 7.96295398117797$$
$$x_{62} = 70.6980846424974$$
$$x_{63} = 54.96211401548$$
$$x_{64} = -36.1522727445826$$
Las raíces dadas
$$x_{25} = -98.9689191782982$$
$$x_{6} = -95.8095367800758$$
$$x_{13} = -92.6863270323791$$
$$x_{57} = -89.5257169945252$$
$$x_{50} = -86.4038211404408$$
$$x_{29} = -83.2418014003112$$
$$x_{60} = -80.1214217891022$$
$$x_{35} = -76.9577665199981$$
$$x_{12} = -73.8391561670124$$
$$x_{33} = -70.6735805215019$$
$$x_{20} = -67.5570615725169$$
$$x_{1} = -64.3891991378643$$
$$x_{31} = -61.2751905931337$$
$$x_{47} = -58.1045589361698$$
$$x_{9} = -54.993619831997$$
$$x_{26} = -51.8195656815415$$
$$x_{49} = -48.7124653792196$$
$$x_{10} = -45.5340730476022$$
$$x_{48} = -42.4319120069122$$
$$x_{8} = -39.2478408235586$$
$$x_{64} = -36.1522727445826$$
$$x_{39} = -32.960445150861$$
$$x_{58} = -29.8741234195341$$
$$x_{27} = -26.6710612453683$$
$$x_{44} = -23.5986512331331$$
$$x_{46} = -20.3778410567194$$
$$x_{38} = -17.3287566215911$$
$$x_{53} = -14.075601349021$$
$$x_{56} = -11.0738587072216$$
$$x_{4} = -7.74188466920172$$
$$x_{16} = -4.89041416017729$$
$$x_{55} = 2.01504507250032$$
$$x_{52} = 4.51958024618035$$
$$x_{61} = 7.96295398117797$$
$$x_{42} = 10.9161565517939$$
$$x_{15} = 14.1982002614746$$
$$x_{24} = 17.2284712949965$$
$$x_{3} = 20.4626870669311$$
$$x_{28} = 23.5251237938033$$
$$x_{18} = 26.7359350354377$$
$$x_{40} = 29.8160805418067$$
$$x_{23} = 33.0129587650466$$
$$x_{34} = 36.1043264668609$$
$$x_{51} = 39.2919507389347$$
$$x_{5} = 42.3910699719495$$
$$x_{2} = 45.572098034099$$
$$x_{43} = 48.6768938881967$$
$$x_{30} = 51.8529811155539$$
$$x_{63} = 54.96211401548$$
$$x_{11} = 58.1343616049366$$
$$x_{36} = 61.2469163716388$$
$$x_{7} = 64.4160940432473$$
$$x_{37} = 67.5314176635823$$
$$x_{62} = 70.6980846424974$$
$$x_{22} = 73.8156948236903$$
$$x_{19} = 76.9802702160215$$
$$x_{59} = 80.0998006261809$$
$$x_{14} = 83.2626066401751$$
$$x_{21} = 86.3837724806605$$
$$x_{17} = 89.5450621701513$$
$$x_{41} = 92.6676376448981$$
$$x_{32} = 95.827613383739$$
$$x_{54} = 98.951416449997$$
$$x_{45} = 183.787882344406$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{25}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{25} - \frac{1}{10}$$
=
$$-98.9689191782982 + - \frac{1}{10}$$
=
$$-99.0689191782982$$
lo sustituimos en la expresión
$$x \cos{\left(x \right)} > \frac{\left(-1\right) \sqrt{3}}{2}$$
$$\left(-99.0689191782982\right) \cos{\left(-99.0689191782982 \right)} > \frac{\left(-1\right) \sqrt{3}}{2}$$

                       ___ 
                    -\/ 3  
-10.7525795802343 > -------
                       2   
                    

Entonces
$$x < -98.9689191782982$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x > -98.9689191782982 \wedge x < -95.8095367800758$$

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       x25      x6      x13      x57      x50      x29      x60      x35      x12      x33      x20      x1      x31      x47      x9      x26      x49      x10      x48      x8      x64      x39      x58      x27      x44      x46      x38      x53      x56      x4      x16      x55      x52      x61      x42      x15      x24      x3      x28      x18      x40      x23      x34      x51      x5      x2      x43      x30      x63      x11      x36      x7      x37      x62      x22      x19      x59      x14      x21      x17      x41      x32      x54      x45

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x > -98.9689191782982 \wedge x < -95.8095367800758$$
$$x > -92.6863270323791 \wedge x < -89.5257169945252$$
$$x > -86.4038211404408 \wedge x < -83.2418014003112$$
$$x > -80.1214217891022 \wedge x < -76.9577665199981$$
$$x > -73.8391561670124 \wedge x < -70.6735805215019$$
$$x > -67.5570615725169 \wedge x < -64.3891991378643$$
$$x > -61.2751905931337 \wedge x < -58.1045589361698$$
$$x > -54.993619831997 \wedge x < -51.8195656815415$$
$$x > -48.7124653792196 \wedge x < -45.5340730476022$$
$$x > -42.4319120069122 \wedge x < -39.2478408235586$$
$$x > -36.1522727445826 \wedge x < -32.960445150861$$
$$x > -29.8741234195341 \wedge x < -26.6710612453683$$
$$x > -23.5986512331331 \wedge x < -20.3778410567194$$
$$x > -17.3287566215911 \wedge x < -14.075601349021$$
$$x > -11.0738587072216 \wedge x < -7.74188466920172$$
$$x > -4.89041416017729 \wedge x < 2.01504507250032$$
$$x > 4.51958024618035 \wedge x < 7.96295398117797$$
$$x > 10.9161565517939 \wedge x < 14.1982002614746$$
$$x > 17.2284712949965 \wedge x < 20.4626870669311$$
$$x > 23.5251237938033 \wedge x < 26.7359350354377$$
$$x > 29.8160805418067 \wedge x < 33.0129587650466$$
$$x > 36.1043264668609 \wedge x < 39.2919507389347$$
$$x > 42.3910699719495 \wedge x < 45.572098034099$$
$$x > 48.6768938881967 \wedge x < 51.8529811155539$$
$$x > 54.96211401548 \wedge x < 58.1343616049366$$
$$x > 61.2469163716388 \wedge x < 64.4160940432473$$
$$x > 67.5314176635823 \wedge x < 70.6980846424974$$
$$x > 73.8156948236903 \wedge x < 76.9802702160215$$
$$x > 80.0998006261809 \wedge x < 83.2626066401751$$
$$x > 86.3837724806605 \wedge x < 89.5450621701513$$
$$x > 92.6676376448981 \wedge x < 95.827613383739$$
$$x > 98.951416449997 \wedge x < 183.787882344406$$