Mister Exam
cos(2x-(pi/4))x≤-(√3/2) inequation
The solution
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___ / pi\ -\/ 3 cos|2*x - --|*x <= ------- \ 4 / 2
$$x \cos{\left(2 x - \frac{\pi}{4} \right)} \leq - \frac{\sqrt{3}}{2}$$
x*cos(2*x - pi/4) <= -sqrt(3)/2
Detail solution
Given the inequality:
$$x \cos{\left(2 x - \frac{\pi}{4} \right)} \leq - \frac{\sqrt{3}}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$x \cos{\left(2 x - \frac{\pi}{4} \right)} = - \frac{\sqrt{3}}{2}$$
Solve:
$$x_{1} = 35.7477306320813$$
$$x_{2} = 90.7087141383207$$
$$x_{3} = 92.2889761903236$$
$$x_{4} = 2.57756789869373$$
$$x_{5} = 4.41833218870582$$
$$x_{6} = -41.2228986266421$$
$$x_{7} = -94.6359030607806$$
$$x_{8} = 65.574143045524$$
$$x_{9} = 98.5718624260433$$
$$x_{10} = -88.3523923305876$$
$$x_{11} = -91.4941532834635$$
$$x_{12} = 87.5669502052005$$
$$x_{13} = -69.501507035948$$
$$x_{14} = -17.6959380645662$$
$$x_{15} = -53.7917240434184$$
$$x_{16} = 46.7219223213739$$
$$x_{17} = -80.5086903111165$$
$$x_{18} = -55.3783900020515$$
$$x_{19} = -25.5084617903844$$
$$x_{20} = -14.5596242389401$$
$$x_{21} = -96.2157755104838$$
$$x_{22} = -61.6607785557254$$
$$x_{23} = -99.3572258649787$$
$$x_{24} = -2.16887082755251$$
$$x_{25} = -97.7776427399657$$
$$x_{26} = -77.367316054284$$
$$x_{27} = -66.3596193779403$$
$$x_{28} = 1.48855960517723$$
$$x_{29} = -23.9727106489827$$
$$x_{30} = 64.0167145801539$$
$$x_{31} = -60.0757514718742$$
$$x_{32} = -16.0737101108477$$
$$x_{33} = 5.81575314037901$$
$$x_{34} = -31.7950050422583$$
$$x_{35} = 10.6436031352424$$
$$x_{36} = 68.7160376509593$$
$$x_{37} = 54.5931043261049$$
$$x_{38} = 62.4322180425498$$
$$x_{39} = 12.137966959545$$
$$x_{40} = 18.4333574770289$$
$$x_{41} = -45.9552156096449$$
$$x_{42} = 48.3109505632436$$
$$x_{43} = -82.0688317633173$$
$$x_{44} = -83.6500809710862$$
$$x_{45} = 21.578377129864$$
$$x_{46} = 78.1415757577305$$
$$x_{47} = 43.5796612954408$$
$$x_{48} = -19.2197177647678$$
$$x_{49} = -67.9433144530403$$
$$x_{50} = -11.4262062008546$$
$$x_{51} = 86.0061336494404$$
$$x_{52} = 34.1521398153172$$
$$x_{53} = -39.6735225187304$$
$$x_{54} = 93.850466602772$$
$$x_{55} = 40.4372963456374$$
$$x_{56} = -104.061095462942$$
$$x_{57} = 76.5819752896018$$
$$x_{58} = -8.29895266969519$$
$$x_{59} = -58.5195628953145$$
$$x_{60} = -38.0804389435651$$
$$x_{61} = 49.8640990821956$$
$$x_{62} = 13.7759211770063$$
$$x_{63} = 32.6073049652536$$
$$x_{64} = 79.7233451333755$$
$$x_{65} = -75.7852089599415$$
$$x_{66} = 100.133941444321$$
$$x_{67} = 27.8660932450522$$
$$x_{68} = 56.1482564243358$$
$$x_{69} = -33.3923908048991$$
$$x_{70} = 84.4251735228072$$
$$x_{71} = 7.51899978446191$$
$$x_{72} = -6.61018866689758$$
$$x_{73} = 24.7225236560155$$
$$x_{74} = -47.5074737578972$$
$$x_{75} = 57.7342653899289$$
$$x_{76} = -3.40573797678636$$
$$x_{77} = 70.2992953395131$$
$$x_{78} = -30.2521447011736$$
$$x_{79} = 71.8579058472682$$
$$x_{80} = 26.3272887370679$$
$$x_{81} = -89.9329046251528$$
$$x_{82} = 20.0492573318314$$
$$x_{83} = 42.029105157638$$
$$x_{84} = -22.3644811896294$$
$$x_{85} = -74.2259602830878$$
$$x_{86} = -9.77311232132626$$
$$x_{87} = -52.2372675901086$$
$$x_{88} = -44.3652354315924$$
$$x_{1} = 35.7477306320813$$
$$x_{2} = 90.7087141383207$$
$$x_{3} = 92.2889761903236$$
$$x_{4} = 2.57756789869373$$
$$x_{5} = 4.41833218870582$$
$$x_{6} = -41.2228986266421$$
$$x_{7} = -94.6359030607806$$
$$x_{8} = 65.574143045524$$
$$x_{9} = 98.5718624260433$$
$$x_{10} = -88.3523923305876$$
$$x_{11} = -91.4941532834635$$
$$x_{12} = 87.5669502052005$$
$$x_{13} = -69.501507035948$$
$$x_{14} = -17.6959380645662$$
$$x_{15} = -53.7917240434184$$
$$x_{16} = 46.7219223213739$$
$$x_{17} = -80.5086903111165$$
$$x_{18} = -55.3783900020515$$
$$x_{19} = -25.5084617903844$$
$$x_{20} = -14.5596242389401$$
$$x_{21} = -96.2157755104838$$
$$x_{22} = -61.6607785557254$$
$$x_{23} = -99.3572258649787$$
$$x_{24} = -2.16887082755251$$
$$x_{25} = -97.7776427399657$$
$$x_{26} = -77.367316054284$$
$$x_{27} = -66.3596193779403$$
$$x_{28} = 1.48855960517723$$
$$x_{29} = -23.9727106489827$$
$$x_{30} = 64.0167145801539$$
$$x_{31} = -60.0757514718742$$
$$x_{32} = -16.0737101108477$$
$$x_{33} = 5.81575314037901$$
$$x_{34} = -31.7950050422583$$
$$x_{35} = 10.6436031352424$$
$$x_{36} = 68.7160376509593$$
$$x_{37} = 54.5931043261049$$
$$x_{38} = 62.4322180425498$$
$$x_{39} = 12.137966959545$$
$$x_{40} = 18.4333574770289$$
$$x_{41} = -45.9552156096449$$
$$x_{42} = 48.3109505632436$$
$$x_{43} = -82.0688317633173$$
$$x_{44} = -83.6500809710862$$
$$x_{45} = 21.578377129864$$
$$x_{46} = 78.1415757577305$$
$$x_{47} = 43.5796612954408$$
$$x_{48} = -19.2197177647678$$
$$x_{49} = -67.9433144530403$$
$$x_{50} = -11.4262062008546$$
$$x_{51} = 86.0061336494404$$
$$x_{52} = 34.1521398153172$$
$$x_{53} = -39.6735225187304$$
$$x_{54} = 93.850466602772$$
$$x_{55} = 40.4372963456374$$
$$x_{56} = -104.061095462942$$
$$x_{57} = 76.5819752896018$$
$$x_{58} = -8.29895266969519$$
$$x_{59} = -58.5195628953145$$
$$x_{60} = -38.0804389435651$$
$$x_{61} = 49.8640990821956$$
$$x_{62} = 13.7759211770063$$
$$x_{63} = 32.6073049652536$$
$$x_{64} = 79.7233451333755$$
$$x_{65} = -75.7852089599415$$
$$x_{66} = 100.133941444321$$
$$x_{67} = 27.8660932450522$$
$$x_{68} = 56.1482564243358$$
$$x_{69} = -33.3923908048991$$
$$x_{70} = 84.4251735228072$$
$$x_{71} = 7.51899978446191$$
$$x_{72} = -6.61018866689758$$
$$x_{73} = 24.7225236560155$$
$$x_{74} = -47.5074737578972$$
$$x_{75} = 57.7342653899289$$
$$x_{76} = -3.40573797678636$$
$$x_{77} = 70.2992953395131$$
$$x_{78} = -30.2521447011736$$
$$x_{79} = 71.8579058472682$$
$$x_{80} = 26.3272887370679$$
$$x_{81} = -89.9329046251528$$
$$x_{82} = 20.0492573318314$$
$$x_{83} = 42.029105157638$$
$$x_{84} = -22.3644811896294$$
$$x_{85} = -74.2259602830878$$
$$x_{86} = -9.77311232132626$$
$$x_{87} = -52.2372675901086$$
$$x_{88} = -44.3652354315924$$
This roots
$$x_{56} = -104.061095462942$$
$$x_{23} = -99.3572258649787$$
$$x_{25} = -97.7776427399657$$
$$x_{21} = -96.2157755104838$$
$$x_{7} = -94.6359030607806$$
$$x_{11} = -91.4941532834635$$
$$x_{81} = -89.9329046251528$$
$$x_{10} = -88.3523923305876$$
$$x_{44} = -83.6500809710862$$
$$x_{43} = -82.0688317633173$$
$$x_{17} = -80.5086903111165$$
$$x_{26} = -77.367316054284$$
$$x_{65} = -75.7852089599415$$
$$x_{85} = -74.2259602830878$$
$$x_{13} = -69.501507035948$$
$$x_{49} = -67.9433144530403$$
$$x_{27} = -66.3596193779403$$
$$x_{22} = -61.6607785557254$$
$$x_{31} = -60.0757514718742$$
$$x_{59} = -58.5195628953145$$
$$x_{18} = -55.3783900020515$$
$$x_{15} = -53.7917240434184$$
$$x_{87} = -52.2372675901086$$
$$x_{74} = -47.5074737578972$$
$$x_{41} = -45.9552156096449$$
$$x_{88} = -44.3652354315924$$
$$x_{6} = -41.2228986266421$$
$$x_{53} = -39.6735225187304$$
$$x_{60} = -38.0804389435651$$
$$x_{69} = -33.3923908048991$$
$$x_{34} = -31.7950050422583$$
$$x_{78} = -30.2521447011736$$
$$x_{19} = -25.5084617903844$$
$$x_{29} = -23.9727106489827$$
$$x_{84} = -22.3644811896294$$
$$x_{48} = -19.2197177647678$$
$$x_{14} = -17.6959380645662$$
$$x_{32} = -16.0737101108477$$
$$x_{20} = -14.5596242389401$$
$$x_{50} = -11.4262062008546$$
$$x_{86} = -9.77311232132626$$
$$x_{58} = -8.29895266969519$$
$$x_{72} = -6.61018866689758$$
$$x_{76} = -3.40573797678636$$
$$x_{24} = -2.16887082755251$$
$$x_{28} = 1.48855960517723$$
$$x_{4} = 2.57756789869373$$
$$x_{5} = 4.41833218870582$$
$$x_{33} = 5.81575314037901$$
$$x_{71} = 7.51899978446191$$
$$x_{35} = 10.6436031352424$$
$$x_{39} = 12.137966959545$$
$$x_{62} = 13.7759211770063$$
$$x_{40} = 18.4333574770289$$
$$x_{82} = 20.0492573318314$$
$$x_{45} = 21.578377129864$$
$$x_{73} = 24.7225236560155$$
$$x_{80} = 26.3272887370679$$
$$x_{67} = 27.8660932450522$$
$$x_{63} = 32.6073049652536$$
$$x_{52} = 34.1521398153172$$
$$x_{1} = 35.7477306320813$$
$$x_{55} = 40.4372963456374$$
$$x_{83} = 42.029105157638$$
$$x_{47} = 43.5796612954408$$
$$x_{16} = 46.7219223213739$$
$$x_{42} = 48.3109505632436$$
$$x_{61} = 49.8640990821956$$
$$x_{37} = 54.5931043261049$$
$$x_{68} = 56.1482564243358$$
$$x_{75} = 57.7342653899289$$
$$x_{38} = 62.4322180425498$$
$$x_{30} = 64.0167145801539$$
$$x_{8} = 65.574143045524$$
$$x_{36} = 68.7160376509593$$
$$x_{77} = 70.2992953395131$$
$$x_{79} = 71.8579058472682$$
$$x_{57} = 76.5819752896018$$
$$x_{46} = 78.1415757577305$$
$$x_{64} = 79.7233451333755$$
$$x_{70} = 84.4251735228072$$
$$x_{51} = 86.0061336494404$$
$$x_{12} = 87.5669502052005$$
$$x_{2} = 90.7087141383207$$
$$x_{3} = 92.2889761903236$$
$$x_{54} = 93.850466602772$$
$$x_{9} = 98.5718624260433$$
$$x_{66} = 100.133941444321$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{56}$$
For example, let's take the point
$$x_{0} = x_{56} - \frac{1}{10}$$
=
$$-104.061095462942 + - \frac{1}{10}$$
=
$$-104.161095462942$$
substitute to the expression
$$x \cos{\left(2 x - \frac{\pi}{4} \right)} \leq - \frac{\sqrt{3}}{2}$$
$$\left(-104.161095462942\right) \cos{\left(\left(-104.161095462942\right) 2 - \frac{\pi}{4} \right)} \leq - \frac{\sqrt{3}}{2}$$
but
Then
$$x \leq -104.061095462942$$
no execute
one of the solutions of our inequality is:
$$x \geq -104.061095462942 \wedge x \leq -99.3572258649787$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \geq -104.061095462942 \wedge x \leq -99.3572258649787$$
$$x \geq -97.7776427399657 \wedge x \leq -96.2157755104838$$
$$x \geq -94.6359030607806 \wedge x \leq -91.4941532834635$$
$$x \geq -89.9329046251528 \wedge x \leq -88.3523923305876$$
$$x \geq -83.6500809710862 \wedge x \leq -82.0688317633173$$
$$x \geq -80.5086903111165 \wedge x \leq -77.367316054284$$
$$x \geq -75.7852089599415 \wedge x \leq -74.2259602830878$$
$$x \geq -69.501507035948 \wedge x \leq -67.9433144530403$$
$$x \geq -66.3596193779403 \wedge x \leq -61.6607785557254$$
$$x \geq -60.0757514718742 \wedge x \leq -58.5195628953145$$
$$x \geq -55.3783900020515 \wedge x \leq -53.7917240434184$$
$$x \geq -52.2372675901086 \wedge x \leq -47.5074737578972$$
$$x \geq -45.9552156096449 \wedge x \leq -44.3652354315924$$
$$x \geq -41.2228986266421 \wedge x \leq -39.6735225187304$$
$$x \geq -38.0804389435651 \wedge x \leq -33.3923908048991$$
$$x \geq -31.7950050422583 \wedge x \leq -30.2521447011736$$
$$x \geq -25.5084617903844 \wedge x \leq -23.9727106489827$$
$$x \geq -22.3644811896294 \wedge x \leq -19.2197177647678$$
$$x \geq -17.6959380645662 \wedge x \leq -16.0737101108477$$
$$x \geq -14.5596242389401 \wedge x \leq -11.4262062008546$$
$$x \geq -9.77311232132626 \wedge x \leq -8.29895266969519$$
$$x \geq -6.61018866689758 \wedge x \leq -3.40573797678636$$
$$x \geq -2.16887082755251 \wedge x \leq 1.48855960517723$$
$$x \geq 2.57756789869373 \wedge x \leq 4.41833218870582$$
$$x \geq 5.81575314037901 \wedge x \leq 7.51899978446191$$
$$x \geq 10.6436031352424 \wedge x \leq 12.137966959545$$
$$x \geq 13.7759211770063 \wedge x \leq 18.4333574770289$$
$$x \geq 20.0492573318314 \wedge x \leq 21.578377129864$$
$$x \geq 24.7225236560155 \wedge x \leq 26.3272887370679$$
$$x \geq 27.8660932450522 \wedge x \leq 32.6073049652536$$
$$x \geq 34.1521398153172 \wedge x \leq 35.7477306320813$$
$$x \geq 40.4372963456374 \wedge x \leq 42.029105157638$$
$$x \geq 43.5796612954408 \wedge x \leq 46.7219223213739$$
$$x \geq 48.3109505632436 \wedge x \leq 49.8640990821956$$
$$x \geq 54.5931043261049 \wedge x \leq 56.1482564243358$$
$$x \geq 57.7342653899289 \wedge x \leq 62.4322180425498$$
$$x \geq 64.0167145801539 \wedge x \leq 65.574143045524$$
$$x \geq 68.7160376509593 \wedge x \leq 70.2992953395131$$
$$x \geq 71.8579058472682 \wedge x \leq 76.5819752896018$$
$$x \geq 78.1415757577305 \wedge x \leq 79.7233451333755$$
$$x \geq 84.4251735228072 \wedge x \leq 86.0061336494404$$
$$x \geq 87.5669502052005 \wedge x \leq 90.7087141383207$$
$$x \geq 92.2889761903236 \wedge x \leq 93.850466602772$$
$$x \geq 98.5718624260433 \wedge x \leq 100.133941444321$$
$$x \cos{\left(2 x - \frac{\pi}{4} \right)} \leq - \frac{\sqrt{3}}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$x \cos{\left(2 x - \frac{\pi}{4} \right)} = - \frac{\sqrt{3}}{2}$$
Solve:
$$x_{1} = 35.7477306320813$$
$$x_{2} = 90.7087141383207$$
$$x_{3} = 92.2889761903236$$
$$x_{4} = 2.57756789869373$$
$$x_{5} = 4.41833218870582$$
$$x_{6} = -41.2228986266421$$
$$x_{7} = -94.6359030607806$$
$$x_{8} = 65.574143045524$$
$$x_{9} = 98.5718624260433$$
$$x_{10} = -88.3523923305876$$
$$x_{11} = -91.4941532834635$$
$$x_{12} = 87.5669502052005$$
$$x_{13} = -69.501507035948$$
$$x_{14} = -17.6959380645662$$
$$x_{15} = -53.7917240434184$$
$$x_{16} = 46.7219223213739$$
$$x_{17} = -80.5086903111165$$
$$x_{18} = -55.3783900020515$$
$$x_{19} = -25.5084617903844$$
$$x_{20} = -14.5596242389401$$
$$x_{21} = -96.2157755104838$$
$$x_{22} = -61.6607785557254$$
$$x_{23} = -99.3572258649787$$
$$x_{24} = -2.16887082755251$$
$$x_{25} = -97.7776427399657$$
$$x_{26} = -77.367316054284$$
$$x_{27} = -66.3596193779403$$
$$x_{28} = 1.48855960517723$$
$$x_{29} = -23.9727106489827$$
$$x_{30} = 64.0167145801539$$
$$x_{31} = -60.0757514718742$$
$$x_{32} = -16.0737101108477$$
$$x_{33} = 5.81575314037901$$
$$x_{34} = -31.7950050422583$$
$$x_{35} = 10.6436031352424$$
$$x_{36} = 68.7160376509593$$
$$x_{37} = 54.5931043261049$$
$$x_{38} = 62.4322180425498$$
$$x_{39} = 12.137966959545$$
$$x_{40} = 18.4333574770289$$
$$x_{41} = -45.9552156096449$$
$$x_{42} = 48.3109505632436$$
$$x_{43} = -82.0688317633173$$
$$x_{44} = -83.6500809710862$$
$$x_{45} = 21.578377129864$$
$$x_{46} = 78.1415757577305$$
$$x_{47} = 43.5796612954408$$
$$x_{48} = -19.2197177647678$$
$$x_{49} = -67.9433144530403$$
$$x_{50} = -11.4262062008546$$
$$x_{51} = 86.0061336494404$$
$$x_{52} = 34.1521398153172$$
$$x_{53} = -39.6735225187304$$
$$x_{54} = 93.850466602772$$
$$x_{55} = 40.4372963456374$$
$$x_{56} = -104.061095462942$$
$$x_{57} = 76.5819752896018$$
$$x_{58} = -8.29895266969519$$
$$x_{59} = -58.5195628953145$$
$$x_{60} = -38.0804389435651$$
$$x_{61} = 49.8640990821956$$
$$x_{62} = 13.7759211770063$$
$$x_{63} = 32.6073049652536$$
$$x_{64} = 79.7233451333755$$
$$x_{65} = -75.7852089599415$$
$$x_{66} = 100.133941444321$$
$$x_{67} = 27.8660932450522$$
$$x_{68} = 56.1482564243358$$
$$x_{69} = -33.3923908048991$$
$$x_{70} = 84.4251735228072$$
$$x_{71} = 7.51899978446191$$
$$x_{72} = -6.61018866689758$$
$$x_{73} = 24.7225236560155$$
$$x_{74} = -47.5074737578972$$
$$x_{75} = 57.7342653899289$$
$$x_{76} = -3.40573797678636$$
$$x_{77} = 70.2992953395131$$
$$x_{78} = -30.2521447011736$$
$$x_{79} = 71.8579058472682$$
$$x_{80} = 26.3272887370679$$
$$x_{81} = -89.9329046251528$$
$$x_{82} = 20.0492573318314$$
$$x_{83} = 42.029105157638$$
$$x_{84} = -22.3644811896294$$
$$x_{85} = -74.2259602830878$$
$$x_{86} = -9.77311232132626$$
$$x_{87} = -52.2372675901086$$
$$x_{88} = -44.3652354315924$$
$$x_{1} = 35.7477306320813$$
$$x_{2} = 90.7087141383207$$
$$x_{3} = 92.2889761903236$$
$$x_{4} = 2.57756789869373$$
$$x_{5} = 4.41833218870582$$
$$x_{6} = -41.2228986266421$$
$$x_{7} = -94.6359030607806$$
$$x_{8} = 65.574143045524$$
$$x_{9} = 98.5718624260433$$
$$x_{10} = -88.3523923305876$$
$$x_{11} = -91.4941532834635$$
$$x_{12} = 87.5669502052005$$
$$x_{13} = -69.501507035948$$
$$x_{14} = -17.6959380645662$$
$$x_{15} = -53.7917240434184$$
$$x_{16} = 46.7219223213739$$
$$x_{17} = -80.5086903111165$$
$$x_{18} = -55.3783900020515$$
$$x_{19} = -25.5084617903844$$
$$x_{20} = -14.5596242389401$$
$$x_{21} = -96.2157755104838$$
$$x_{22} = -61.6607785557254$$
$$x_{23} = -99.3572258649787$$
$$x_{24} = -2.16887082755251$$
$$x_{25} = -97.7776427399657$$
$$x_{26} = -77.367316054284$$
$$x_{27} = -66.3596193779403$$
$$x_{28} = 1.48855960517723$$
$$x_{29} = -23.9727106489827$$
$$x_{30} = 64.0167145801539$$
$$x_{31} = -60.0757514718742$$
$$x_{32} = -16.0737101108477$$
$$x_{33} = 5.81575314037901$$
$$x_{34} = -31.7950050422583$$
$$x_{35} = 10.6436031352424$$
$$x_{36} = 68.7160376509593$$
$$x_{37} = 54.5931043261049$$
$$x_{38} = 62.4322180425498$$
$$x_{39} = 12.137966959545$$
$$x_{40} = 18.4333574770289$$
$$x_{41} = -45.9552156096449$$
$$x_{42} = 48.3109505632436$$
$$x_{43} = -82.0688317633173$$
$$x_{44} = -83.6500809710862$$
$$x_{45} = 21.578377129864$$
$$x_{46} = 78.1415757577305$$
$$x_{47} = 43.5796612954408$$
$$x_{48} = -19.2197177647678$$
$$x_{49} = -67.9433144530403$$
$$x_{50} = -11.4262062008546$$
$$x_{51} = 86.0061336494404$$
$$x_{52} = 34.1521398153172$$
$$x_{53} = -39.6735225187304$$
$$x_{54} = 93.850466602772$$
$$x_{55} = 40.4372963456374$$
$$x_{56} = -104.061095462942$$
$$x_{57} = 76.5819752896018$$
$$x_{58} = -8.29895266969519$$
$$x_{59} = -58.5195628953145$$
$$x_{60} = -38.0804389435651$$
$$x_{61} = 49.8640990821956$$
$$x_{62} = 13.7759211770063$$
$$x_{63} = 32.6073049652536$$
$$x_{64} = 79.7233451333755$$
$$x_{65} = -75.7852089599415$$
$$x_{66} = 100.133941444321$$
$$x_{67} = 27.8660932450522$$
$$x_{68} = 56.1482564243358$$
$$x_{69} = -33.3923908048991$$
$$x_{70} = 84.4251735228072$$
$$x_{71} = 7.51899978446191$$
$$x_{72} = -6.61018866689758$$
$$x_{73} = 24.7225236560155$$
$$x_{74} = -47.5074737578972$$
$$x_{75} = 57.7342653899289$$
$$x_{76} = -3.40573797678636$$
$$x_{77} = 70.2992953395131$$
$$x_{78} = -30.2521447011736$$
$$x_{79} = 71.8579058472682$$
$$x_{80} = 26.3272887370679$$
$$x_{81} = -89.9329046251528$$
$$x_{82} = 20.0492573318314$$
$$x_{83} = 42.029105157638$$
$$x_{84} = -22.3644811896294$$
$$x_{85} = -74.2259602830878$$
$$x_{86} = -9.77311232132626$$
$$x_{87} = -52.2372675901086$$
$$x_{88} = -44.3652354315924$$
This roots
$$x_{56} = -104.061095462942$$
$$x_{23} = -99.3572258649787$$
$$x_{25} = -97.7776427399657$$
$$x_{21} = -96.2157755104838$$
$$x_{7} = -94.6359030607806$$
$$x_{11} = -91.4941532834635$$
$$x_{81} = -89.9329046251528$$
$$x_{10} = -88.3523923305876$$
$$x_{44} = -83.6500809710862$$
$$x_{43} = -82.0688317633173$$
$$x_{17} = -80.5086903111165$$
$$x_{26} = -77.367316054284$$
$$x_{65} = -75.7852089599415$$
$$x_{85} = -74.2259602830878$$
$$x_{13} = -69.501507035948$$
$$x_{49} = -67.9433144530403$$
$$x_{27} = -66.3596193779403$$
$$x_{22} = -61.6607785557254$$
$$x_{31} = -60.0757514718742$$
$$x_{59} = -58.5195628953145$$
$$x_{18} = -55.3783900020515$$
$$x_{15} = -53.7917240434184$$
$$x_{87} = -52.2372675901086$$
$$x_{74} = -47.5074737578972$$
$$x_{41} = -45.9552156096449$$
$$x_{88} = -44.3652354315924$$
$$x_{6} = -41.2228986266421$$
$$x_{53} = -39.6735225187304$$
$$x_{60} = -38.0804389435651$$
$$x_{69} = -33.3923908048991$$
$$x_{34} = -31.7950050422583$$
$$x_{78} = -30.2521447011736$$
$$x_{19} = -25.5084617903844$$
$$x_{29} = -23.9727106489827$$
$$x_{84} = -22.3644811896294$$
$$x_{48} = -19.2197177647678$$
$$x_{14} = -17.6959380645662$$
$$x_{32} = -16.0737101108477$$
$$x_{20} = -14.5596242389401$$
$$x_{50} = -11.4262062008546$$
$$x_{86} = -9.77311232132626$$
$$x_{58} = -8.29895266969519$$
$$x_{72} = -6.61018866689758$$
$$x_{76} = -3.40573797678636$$
$$x_{24} = -2.16887082755251$$
$$x_{28} = 1.48855960517723$$
$$x_{4} = 2.57756789869373$$
$$x_{5} = 4.41833218870582$$
$$x_{33} = 5.81575314037901$$
$$x_{71} = 7.51899978446191$$
$$x_{35} = 10.6436031352424$$
$$x_{39} = 12.137966959545$$
$$x_{62} = 13.7759211770063$$
$$x_{40} = 18.4333574770289$$
$$x_{82} = 20.0492573318314$$
$$x_{45} = 21.578377129864$$
$$x_{73} = 24.7225236560155$$
$$x_{80} = 26.3272887370679$$
$$x_{67} = 27.8660932450522$$
$$x_{63} = 32.6073049652536$$
$$x_{52} = 34.1521398153172$$
$$x_{1} = 35.7477306320813$$
$$x_{55} = 40.4372963456374$$
$$x_{83} = 42.029105157638$$
$$x_{47} = 43.5796612954408$$
$$x_{16} = 46.7219223213739$$
$$x_{42} = 48.3109505632436$$
$$x_{61} = 49.8640990821956$$
$$x_{37} = 54.5931043261049$$
$$x_{68} = 56.1482564243358$$
$$x_{75} = 57.7342653899289$$
$$x_{38} = 62.4322180425498$$
$$x_{30} = 64.0167145801539$$
$$x_{8} = 65.574143045524$$
$$x_{36} = 68.7160376509593$$
$$x_{77} = 70.2992953395131$$
$$x_{79} = 71.8579058472682$$
$$x_{57} = 76.5819752896018$$
$$x_{46} = 78.1415757577305$$
$$x_{64} = 79.7233451333755$$
$$x_{70} = 84.4251735228072$$
$$x_{51} = 86.0061336494404$$
$$x_{12} = 87.5669502052005$$
$$x_{2} = 90.7087141383207$$
$$x_{3} = 92.2889761903236$$
$$x_{54} = 93.850466602772$$
$$x_{9} = 98.5718624260433$$
$$x_{66} = 100.133941444321$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{56}$$
For example, let's take the point
$$x_{0} = x_{56} - \frac{1}{10}$$
=
$$-104.061095462942 + - \frac{1}{10}$$
=
$$-104.161095462942$$
substitute to the expression
$$x \cos{\left(2 x - \frac{\pi}{4} \right)} \leq - \frac{\sqrt{3}}{2}$$
$$\left(-104.161095462942\right) \cos{\left(\left(-104.161095462942\right) 2 - \frac{\pi}{4} \right)} \leq - \frac{\sqrt{3}}{2}$$
___
/ pi\ -\/ 3
-104.161095462942*cos|208.322190925885 + --| <= -------
\ 4 / 2
but
___
/ pi\ -\/ 3
-104.161095462942*cos|208.322190925885 + --| >= -------
\ 4 / 2
Then
$$x \leq -104.061095462942$$
no execute
one of the solutions of our inequality is:
$$x \geq -104.061095462942 \wedge x \leq -99.3572258649787$$
_____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
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x56 x23 x25 x21 x7 x11 x81 x10 x44 x43 x17 x26 x65 x85 x13 x49 x27 x22 x31 x59 x18 x15 x87 x74 x41 x88 x6 x53 x60 x69 x34 x78 x19 x29 x84 x48 x14 x32 x20 x50 x86 x58 x72 x76 x24 x28 x4 x5 x33 x71 x35 x39 x62 x40 x82 x45 x73 x80 x67 x63 x52 x1 x55 x83 x47 x16 x42 x61 x37 x68 x75 x38 x30 x8 x36 x77 x79 x57 x46 x64 x70 x51 x12 x2 x3 x54 x9 x66Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \geq -104.061095462942 \wedge x \leq -99.3572258649787$$
$$x \geq -97.7776427399657 \wedge x \leq -96.2157755104838$$
$$x \geq -94.6359030607806 \wedge x \leq -91.4941532834635$$
$$x \geq -89.9329046251528 \wedge x \leq -88.3523923305876$$
$$x \geq -83.6500809710862 \wedge x \leq -82.0688317633173$$
$$x \geq -80.5086903111165 \wedge x \leq -77.367316054284$$
$$x \geq -75.7852089599415 \wedge x \leq -74.2259602830878$$
$$x \geq -69.501507035948 \wedge x \leq -67.9433144530403$$
$$x \geq -66.3596193779403 \wedge x \leq -61.6607785557254$$
$$x \geq -60.0757514718742 \wedge x \leq -58.5195628953145$$
$$x \geq -55.3783900020515 \wedge x \leq -53.7917240434184$$
$$x \geq -52.2372675901086 \wedge x \leq -47.5074737578972$$
$$x \geq -45.9552156096449 \wedge x \leq -44.3652354315924$$
$$x \geq -41.2228986266421 \wedge x \leq -39.6735225187304$$
$$x \geq -38.0804389435651 \wedge x \leq -33.3923908048991$$
$$x \geq -31.7950050422583 \wedge x \leq -30.2521447011736$$
$$x \geq -25.5084617903844 \wedge x \leq -23.9727106489827$$
$$x \geq -22.3644811896294 \wedge x \leq -19.2197177647678$$
$$x \geq -17.6959380645662 \wedge x \leq -16.0737101108477$$
$$x \geq -14.5596242389401 \wedge x \leq -11.4262062008546$$
$$x \geq -9.77311232132626 \wedge x \leq -8.29895266969519$$
$$x \geq -6.61018866689758 \wedge x \leq -3.40573797678636$$
$$x \geq -2.16887082755251 \wedge x \leq 1.48855960517723$$
$$x \geq 2.57756789869373 \wedge x \leq 4.41833218870582$$
$$x \geq 5.81575314037901 \wedge x \leq 7.51899978446191$$
$$x \geq 10.6436031352424 \wedge x \leq 12.137966959545$$
$$x \geq 13.7759211770063 \wedge x \leq 18.4333574770289$$
$$x \geq 20.0492573318314 \wedge x \leq 21.578377129864$$
$$x \geq 24.7225236560155 \wedge x \leq 26.3272887370679$$
$$x \geq 27.8660932450522 \wedge x \leq 32.6073049652536$$
$$x \geq 34.1521398153172 \wedge x \leq 35.7477306320813$$
$$x \geq 40.4372963456374 \wedge x \leq 42.029105157638$$
$$x \geq 43.5796612954408 \wedge x \leq 46.7219223213739$$
$$x \geq 48.3109505632436 \wedge x \leq 49.8640990821956$$
$$x \geq 54.5931043261049 \wedge x \leq 56.1482564243358$$
$$x \geq 57.7342653899289 \wedge x \leq 62.4322180425498$$
$$x \geq 64.0167145801539 \wedge x \leq 65.574143045524$$
$$x \geq 68.7160376509593 \wedge x \leq 70.2992953395131$$
$$x \geq 71.8579058472682 \wedge x \leq 76.5819752896018$$
$$x \geq 78.1415757577305 \wedge x \leq 79.7233451333755$$
$$x \geq 84.4251735228072 \wedge x \leq 86.0061336494404$$
$$x \geq 87.5669502052005 \wedge x \leq 90.7087141383207$$
$$x \geq 92.2889761903236 \wedge x \leq 93.850466602772$$
$$x \geq 98.5718624260433 \wedge x \leq 100.133941444321$$
Solving inequality on a graph