Se da la desigualdad:
$$\frac{x \cot{\left(x \right)}}{3} \leq - \sqrt{3}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\frac{x \cot{\left(x \right)}}{3} = - \sqrt{3}$$
Resolvemos:
Tenemos la ecuación
$$\frac{x \cot{\left(x \right)}}{3} = - \sqrt{3}$$
cambiamos
$$\frac{x \cot{\left(x \right)}}{3} - 1 + \sqrt{3} = 0$$
$$\frac{x \cot{\left(x \right)}}{3} - 1 + \sqrt{3} = 0$$
Sustituimos
$$w = \cot{\left(x \right)}$$
Abrimos los paréntesis en el miembro izquierdo de la ecuación
-1 + sqrt3 + w*x/3 = 0
Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
$$\frac{w x}{3} + \sqrt{3} = 1$$
Dividamos ambos miembros de la ecuación en (sqrt(3) + w*x/3)/w
w = 1 / ((sqrt(3) + w*x/3)/w)
Obtenemos la respuesta: w = 3*(1 - sqrt(3))/x
hacemos cambio inverso
$$\cot{\left(x \right)} = w$$
sustituimos w:
$$x_{1} = -17.5663599498268$$
$$x_{2} = 58.2084960384468$$
$$x_{3} = -95.8727214040797$$
$$x_{4} = -86.4538289655055$$
$$x_{5} = 17.5663599498268$$
$$x_{6} = 33.1422402299566$$
$$x_{7} = 51.9359961028683$$
$$x_{8} = 5.47194116424952$$
$$x_{9} = 20.6666731902081$$
$$x_{10} = -55.0719450173938$$
$$x_{11} = 61.3455580364299$$
$$x_{12} = 42.5330657930663$$
$$x_{13} = -45.6663912429456$$
$$x_{14} = 89.5933228253095$$
$$x_{15} = -89.5933228253095$$
$$x_{16} = -77.0363686410871$$
$$x_{17} = -70.7591374966673$$
$$x_{18} = -67.6209337051216$$
$$x_{19} = -11.4224997329587$$
$$x_{20} = -83.3144925727197$$
$$x_{21} = -58.2084960384468$$
$$x_{22} = 55.0719450173938$$
$$x_{23} = -64.4830573281395$$
$$x_{24} = 36.2706080271563$$
$$x_{25} = 30.0165410942153$$
$$x_{26} = -42.5330657930663$$
$$x_{27} = -99.0126001978345$$
$$x_{28} = -14.4816677478834$$
$$x_{29} = -20.6666731902081$$
$$x_{30} = 92.7329582495637$$
$$x_{31} = 73.8976273669175$$
$$x_{32} = 14.4816677478834$$
$$x_{33} = 26.8943917751677$$
$$x_{34} = 67.6209337051216$$
$$x_{35} = -30.0165410942153$$
$$x_{36} = 48.8007633260567$$
$$x_{37} = -33.1422402299566$$
$$x_{38} = 99.0126001978345$$
$$x_{39} = -51.9359961028683$$
$$x_{40} = 2.66733838749595$$
$$x_{41} = 45.6663912429456$$
$$x_{42} = 83.3144925727197$$
$$x_{43} = 95.8727214040797$$
$$x_{44} = -80.1753320185758$$
$$x_{45} = 86.4538289655055$$
$$x_{46} = -26.8943917751677$$
$$x_{47} = -61.3455580364299$$
$$x_{48} = -48.8007633260567$$
$$x_{49} = 11.4224997329587$$
$$x_{50} = 39.4010301009269$$
$$x_{51} = 70.7591374966673$$
$$x_{52} = -73.8976273669175$$
$$x_{53} = -36.2706080271563$$
$$x_{54} = 64.4830573281395$$
$$x_{55} = -23.7770983831808$$
$$x_{56} = -39.4010301009269$$
$$x_{57} = -5.47194116424952$$
$$x_{58} = 8.40755575235436$$
$$x_{59} = 77.0363686410871$$
$$x_{60} = -92.7329582495637$$
$$x_{61} = -8.40755575235436$$
$$x_{62} = 80.1753320185758$$
$$x_{63} = 23.7770983831808$$
$$x_{1} = -17.5663599498268$$
$$x_{2} = 58.2084960384468$$
$$x_{3} = -95.8727214040797$$
$$x_{4} = -86.4538289655055$$
$$x_{5} = 17.5663599498268$$
$$x_{6} = 33.1422402299566$$
$$x_{7} = 51.9359961028683$$
$$x_{8} = 5.47194116424952$$
$$x_{9} = 20.6666731902081$$
$$x_{10} = -55.0719450173938$$
$$x_{11} = 61.3455580364299$$
$$x_{12} = 42.5330657930663$$
$$x_{13} = -45.6663912429456$$
$$x_{14} = 89.5933228253095$$
$$x_{15} = -89.5933228253095$$
$$x_{16} = -77.0363686410871$$
$$x_{17} = -70.7591374966673$$
$$x_{18} = -67.6209337051216$$
$$x_{19} = -11.4224997329587$$
$$x_{20} = -83.3144925727197$$
$$x_{21} = -58.2084960384468$$
$$x_{22} = 55.0719450173938$$
$$x_{23} = -64.4830573281395$$
$$x_{24} = 36.2706080271563$$
$$x_{25} = 30.0165410942153$$
$$x_{26} = -42.5330657930663$$
$$x_{27} = -99.0126001978345$$
$$x_{28} = -14.4816677478834$$
$$x_{29} = -20.6666731902081$$
$$x_{30} = 92.7329582495637$$
$$x_{31} = 73.8976273669175$$
$$x_{32} = 14.4816677478834$$
$$x_{33} = 26.8943917751677$$
$$x_{34} = 67.6209337051216$$
$$x_{35} = -30.0165410942153$$
$$x_{36} = 48.8007633260567$$
$$x_{37} = -33.1422402299566$$
$$x_{38} = 99.0126001978345$$
$$x_{39} = -51.9359961028683$$
$$x_{40} = 2.66733838749595$$
$$x_{41} = 45.6663912429456$$
$$x_{42} = 83.3144925727197$$
$$x_{43} = 95.8727214040797$$
$$x_{44} = -80.1753320185758$$
$$x_{45} = 86.4538289655055$$
$$x_{46} = -26.8943917751677$$
$$x_{47} = -61.3455580364299$$
$$x_{48} = -48.8007633260567$$
$$x_{49} = 11.4224997329587$$
$$x_{50} = 39.4010301009269$$
$$x_{51} = 70.7591374966673$$
$$x_{52} = -73.8976273669175$$
$$x_{53} = -36.2706080271563$$
$$x_{54} = 64.4830573281395$$
$$x_{55} = -23.7770983831808$$
$$x_{56} = -39.4010301009269$$
$$x_{57} = -5.47194116424952$$
$$x_{58} = 8.40755575235436$$
$$x_{59} = 77.0363686410871$$
$$x_{60} = -92.7329582495637$$
$$x_{61} = -8.40755575235436$$
$$x_{62} = 80.1753320185758$$
$$x_{63} = 23.7770983831808$$
Las raíces dadas
$$x_{27} = -99.0126001978345$$
$$x_{3} = -95.8727214040797$$
$$x_{60} = -92.7329582495637$$
$$x_{15} = -89.5933228253095$$
$$x_{4} = -86.4538289655055$$
$$x_{20} = -83.3144925727197$$
$$x_{44} = -80.1753320185758$$
$$x_{16} = -77.0363686410871$$
$$x_{52} = -73.8976273669175$$
$$x_{17} = -70.7591374966673$$
$$x_{18} = -67.6209337051216$$
$$x_{23} = -64.4830573281395$$
$$x_{47} = -61.3455580364299$$
$$x_{21} = -58.2084960384468$$
$$x_{10} = -55.0719450173938$$
$$x_{39} = -51.9359961028683$$
$$x_{48} = -48.8007633260567$$
$$x_{13} = -45.6663912429456$$
$$x_{26} = -42.5330657930663$$
$$x_{56} = -39.4010301009269$$
$$x_{53} = -36.2706080271563$$
$$x_{37} = -33.1422402299566$$
$$x_{35} = -30.0165410942153$$
$$x_{46} = -26.8943917751677$$
$$x_{55} = -23.7770983831808$$
$$x_{29} = -20.6666731902081$$
$$x_{1} = -17.5663599498268$$
$$x_{28} = -14.4816677478834$$
$$x_{19} = -11.4224997329587$$
$$x_{61} = -8.40755575235436$$
$$x_{57} = -5.47194116424952$$
$$x_{40} = 2.66733838749595$$
$$x_{8} = 5.47194116424952$$
$$x_{58} = 8.40755575235436$$
$$x_{49} = 11.4224997329587$$
$$x_{32} = 14.4816677478834$$
$$x_{5} = 17.5663599498268$$
$$x_{9} = 20.6666731902081$$
$$x_{63} = 23.7770983831808$$
$$x_{33} = 26.8943917751677$$
$$x_{25} = 30.0165410942153$$
$$x_{6} = 33.1422402299566$$
$$x_{24} = 36.2706080271563$$
$$x_{50} = 39.4010301009269$$
$$x_{12} = 42.5330657930663$$
$$x_{41} = 45.6663912429456$$
$$x_{36} = 48.8007633260567$$
$$x_{7} = 51.9359961028683$$
$$x_{22} = 55.0719450173938$$
$$x_{2} = 58.2084960384468$$
$$x_{11} = 61.3455580364299$$
$$x_{54} = 64.4830573281395$$
$$x_{34} = 67.6209337051216$$
$$x_{51} = 70.7591374966673$$
$$x_{31} = 73.8976273669175$$
$$x_{59} = 77.0363686410871$$
$$x_{62} = 80.1753320185758$$
$$x_{42} = 83.3144925727197$$
$$x_{45} = 86.4538289655055$$
$$x_{14} = 89.5933228253095$$
$$x_{30} = 92.7329582495637$$
$$x_{43} = 95.8727214040797$$
$$x_{38} = 99.0126001978345$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{27}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{27} - \frac{1}{10}$$
=
$$-99.0126001978345 + - \frac{1}{10}$$
=
$$-99.1126001978345$$
lo sustituimos en la expresión
$$\frac{x \cot{\left(x \right)}}{3} \leq - \sqrt{3}$$
$$\frac{\left(-99.1126001978345\right) \cot{\left(-99.1126001978345 \right)}}{3} \leq - \sqrt{3}$$
___
-5.07533455988926 <= -\/ 3
significa que una de las soluciones de nuestra ecuación será con:
$$x \leq -99.0126001978345$$
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x27 x3 x60 x15 x4 x20 x44 x16 x52 x17 x18 x23 x47 x21 x10 x39 x48 x13 x26 x56 x53 x37 x35 x46 x55 x29 x1 x28 x19 x61 x57 x40 x8 x58 x49 x32 x5 x9 x63 x33 x25 x6 x24 x50 x12 x41 x36 x7 x22 x2 x11 x54 x34 x51 x31 x59 x62 x42 x45 x14 x30 x43 x38
Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x \leq -99.0126001978345$$
$$x \geq -95.8727214040797 \wedge x \leq -92.7329582495637$$
$$x \geq -89.5933228253095 \wedge x \leq -86.4538289655055$$
$$x \geq -83.3144925727197 \wedge x \leq -80.1753320185758$$
$$x \geq -77.0363686410871 \wedge x \leq -73.8976273669175$$
$$x \geq -70.7591374966673 \wedge x \leq -67.6209337051216$$
$$x \geq -64.4830573281395 \wedge x \leq -61.3455580364299$$
$$x \geq -58.2084960384468 \wedge x \leq -55.0719450173938$$
$$x \geq -51.9359961028683 \wedge x \leq -48.8007633260567$$
$$x \geq -45.6663912429456 \wedge x \leq -42.5330657930663$$
$$x \geq -39.4010301009269 \wedge x \leq -36.2706080271563$$
$$x \geq -33.1422402299566 \wedge x \leq -30.0165410942153$$
$$x \geq -26.8943917751677 \wedge x \leq -23.7770983831808$$
$$x \geq -20.6666731902081 \wedge x \leq -17.5663599498268$$
$$x \geq -14.4816677478834 \wedge x \leq -11.4224997329587$$
$$x \geq -8.40755575235436 \wedge x \leq -5.47194116424952$$
$$x \geq 2.66733838749595 \wedge x \leq 5.47194116424952$$
$$x \geq 8.40755575235436 \wedge x \leq 11.4224997329587$$
$$x \geq 14.4816677478834 \wedge x \leq 17.5663599498268$$
$$x \geq 20.6666731902081 \wedge x \leq 23.7770983831808$$
$$x \geq 26.8943917751677 \wedge x \leq 30.0165410942153$$
$$x \geq 33.1422402299566 \wedge x \leq 36.2706080271563$$
$$x \geq 39.4010301009269 \wedge x \leq 42.5330657930663$$
$$x \geq 45.6663912429456 \wedge x \leq 48.8007633260567$$
$$x \geq 51.9359961028683 \wedge x \leq 55.0719450173938$$
$$x \geq 58.2084960384468 \wedge x \leq 61.3455580364299$$
$$x \geq 64.4830573281395 \wedge x \leq 67.6209337051216$$
$$x \geq 70.7591374966673 \wedge x \leq 73.8976273669175$$
$$x \geq 77.0363686410871 \wedge x \leq 80.1753320185758$$
$$x \geq 83.3144925727197 \wedge x \leq 86.4538289655055$$
$$x \geq 89.5933228253095 \wedge x \leq 92.7329582495637$$
$$x \geq 95.8727214040797 \wedge x \leq 99.0126001978345$$