Mister Exam
|sin(x)|<1\2 inequation
The solution
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|sin(x)| < 1/2
$$\left|{\sin{\left(x \right)}}\right| < \frac{1}{2}$$
Abs(sin(x)) < 1/2
Detail solution
Given the inequality:
$$\left|{\sin{\left(x \right)}}\right| < \frac{1}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{\sin{\left(x \right)}}\right| = \frac{1}{2}$$
Solve:
Given the equation
$$\left|{\sin{\left(x \right)}}\right| = \frac{1}{2}$$
transform
$$\left|{\sin{\left(x \right)}}\right| - \frac{1}{2} = 0$$
$$\left|{\sin{\left(x \right)}}\right| - \frac{1}{2} = 0$$
Do replacement
$$w = \left|{\sin{\left(x \right)}}\right|$$
Move free summands (without w)
from left part to right part, we given:
$$w = \frac{1}{2}$$
We get the answer: w = 1/2
do backward replacement
$$\left|{\sin{\left(x \right)}}\right| = w$$
substitute w:
$$x_{1} = -12.0427718387609$$
$$x_{2} = 44.5058959258554$$
$$x_{3} = 91.6297857297023$$
$$x_{4} = -21.4675497995303$$
$$x_{5} = -137.706477982353$$
$$x_{6} = 46.6002910282486$$
$$x_{7} = -46.6002910282486$$
$$x_{8} = -34.0339204138894$$
$$x_{9} = 34.0339204138894$$
$$x_{10} = 2.61799387799149$$
$$x_{11} = -5.75958653158129$$
$$x_{12} = 66.497044500984$$
$$x_{13} = -41.3643032722656$$
$$x_{14} = -90.5825881785057$$
$$x_{15} = 5.75958653158129$$
$$x_{16} = 9.94837673636768$$
$$x_{17} = 47.6474885794452$$
$$x_{18} = 90.5825881785057$$
$$x_{19} = 60.2138591938044$$
$$x_{20} = -18.3259571459405$$
$$x_{21} = 71.733032256967$$
$$x_{22} = -60.2138591938044$$
$$x_{23} = -78.0162175641465$$
$$x_{24} = 53.9306738866248$$
$$x_{25} = -19.3731546971371$$
$$x_{26} = -40.317105721069$$
$$x_{27} = 31.9395253114962$$
$$x_{28} = 12.0427718387609$$
$$x_{29} = 84.2994028713261$$
$$x_{30} = 78.0162175641465$$
$$x_{31} = -93.7241808320955$$
$$x_{32} = 22.5147473507269$$
$$x_{33} = -3.66519142918809$$
$$x_{34} = -382.750704962356$$
$$x_{35} = -27.7507351067098$$
$$x_{36} = -66.497044500984$$
$$x_{37} = -65.4498469497874$$
$$x_{38} = -38.2227106186758$$
$$x_{39} = 100.007366139275$$
$$x_{40} = 97.9129710368819$$
$$x_{41} = 49.7418836818384$$
$$x_{42} = -56.025068989018$$
$$x_{43} = -62.3082542961976$$
$$x_{44} = -87.4409955249159$$
$$x_{45} = -85.3466004225227$$
$$x_{46} = 27.7507351067098$$
$$x_{47} = 75.9218224617533$$
$$x_{48} = -49.7418836818384$$
$$x_{49} = -24.60914245312$$
$$x_{50} = -71.733032256967$$
$$x_{51} = 16.2315620435473$$
$$x_{52} = 24.60914245312$$
$$x_{53} = -100.007366139275$$
$$x_{54} = -31.9395253114962$$
$$x_{55} = -9.94837673636768$$
$$x_{56} = -97.9129710368819$$
$$x_{57} = -68.5914396033772$$
$$x_{58} = 38.2227106186758$$
$$x_{59} = 69.6386371545737$$
$$x_{60} = 3.66519142918809$$
$$x_{61} = 0.523598775598299$$
$$x_{62} = -43.4586983746588$$
$$x_{63} = 62.3082542961976$$
$$x_{64} = 82.2050077689329$$
$$x_{65} = 96.8657734856853$$
$$x_{66} = -2.61799387799149$$
$$x_{67} = 25.6563400043166$$
$$x_{68} = 18.3259571459405$$
$$x_{69} = -57.0722665402146$$
$$x_{70} = -53.9306738866248$$
$$x_{71} = -75.9218224617533$$
$$x_{72} = 93.7241808320955$$
$$x_{73} = -91.6297857297023$$
$$x_{74} = -16.2315620435473$$
$$x_{75} = -82.2050077689329$$
$$x_{76} = 40.317105721069$$
$$x_{77} = -69.6386371545737$$
$$x_{78} = -47.6474885794452$$
$$x_{79} = 68.5914396033772$$
$$x_{80} = 88.4881930761125$$
$$x_{81} = -25.6563400043166$$
$$x_{82} = -63.3554518473942$$
$$x_{83} = 56.025068989018$$
$$x_{84} = -84.2994028713261$$
$$x_{1} = -12.0427718387609$$
$$x_{2} = 44.5058959258554$$
$$x_{3} = 91.6297857297023$$
$$x_{4} = -21.4675497995303$$
$$x_{5} = -137.706477982353$$
$$x_{6} = 46.6002910282486$$
$$x_{7} = -46.6002910282486$$
$$x_{8} = -34.0339204138894$$
$$x_{9} = 34.0339204138894$$
$$x_{10} = 2.61799387799149$$
$$x_{11} = -5.75958653158129$$
$$x_{12} = 66.497044500984$$
$$x_{13} = -41.3643032722656$$
$$x_{14} = -90.5825881785057$$
$$x_{15} = 5.75958653158129$$
$$x_{16} = 9.94837673636768$$
$$x_{17} = 47.6474885794452$$
$$x_{18} = 90.5825881785057$$
$$x_{19} = 60.2138591938044$$
$$x_{20} = -18.3259571459405$$
$$x_{21} = 71.733032256967$$
$$x_{22} = -60.2138591938044$$
$$x_{23} = -78.0162175641465$$
$$x_{24} = 53.9306738866248$$
$$x_{25} = -19.3731546971371$$
$$x_{26} = -40.317105721069$$
$$x_{27} = 31.9395253114962$$
$$x_{28} = 12.0427718387609$$
$$x_{29} = 84.2994028713261$$
$$x_{30} = 78.0162175641465$$
$$x_{31} = -93.7241808320955$$
$$x_{32} = 22.5147473507269$$
$$x_{33} = -3.66519142918809$$
$$x_{34} = -382.750704962356$$
$$x_{35} = -27.7507351067098$$
$$x_{36} = -66.497044500984$$
$$x_{37} = -65.4498469497874$$
$$x_{38} = -38.2227106186758$$
$$x_{39} = 100.007366139275$$
$$x_{40} = 97.9129710368819$$
$$x_{41} = 49.7418836818384$$
$$x_{42} = -56.025068989018$$
$$x_{43} = -62.3082542961976$$
$$x_{44} = -87.4409955249159$$
$$x_{45} = -85.3466004225227$$
$$x_{46} = 27.7507351067098$$
$$x_{47} = 75.9218224617533$$
$$x_{48} = -49.7418836818384$$
$$x_{49} = -24.60914245312$$
$$x_{50} = -71.733032256967$$
$$x_{51} = 16.2315620435473$$
$$x_{52} = 24.60914245312$$
$$x_{53} = -100.007366139275$$
$$x_{54} = -31.9395253114962$$
$$x_{55} = -9.94837673636768$$
$$x_{56} = -97.9129710368819$$
$$x_{57} = -68.5914396033772$$
$$x_{58} = 38.2227106186758$$
$$x_{59} = 69.6386371545737$$
$$x_{60} = 3.66519142918809$$
$$x_{61} = 0.523598775598299$$
$$x_{62} = -43.4586983746588$$
$$x_{63} = 62.3082542961976$$
$$x_{64} = 82.2050077689329$$
$$x_{65} = 96.8657734856853$$
$$x_{66} = -2.61799387799149$$
$$x_{67} = 25.6563400043166$$
$$x_{68} = 18.3259571459405$$
$$x_{69} = -57.0722665402146$$
$$x_{70} = -53.9306738866248$$
$$x_{71} = -75.9218224617533$$
$$x_{72} = 93.7241808320955$$
$$x_{73} = -91.6297857297023$$
$$x_{74} = -16.2315620435473$$
$$x_{75} = -82.2050077689329$$
$$x_{76} = 40.317105721069$$
$$x_{77} = -69.6386371545737$$
$$x_{78} = -47.6474885794452$$
$$x_{79} = 68.5914396033772$$
$$x_{80} = 88.4881930761125$$
$$x_{81} = -25.6563400043166$$
$$x_{82} = -63.3554518473942$$
$$x_{83} = 56.025068989018$$
$$x_{84} = -84.2994028713261$$
This roots
$$x_{34} = -382.750704962356$$
$$x_{5} = -137.706477982353$$
$$x_{53} = -100.007366139275$$
$$x_{56} = -97.9129710368819$$
$$x_{31} = -93.7241808320955$$
$$x_{73} = -91.6297857297023$$
$$x_{14} = -90.5825881785057$$
$$x_{44} = -87.4409955249159$$
$$x_{45} = -85.3466004225227$$
$$x_{84} = -84.2994028713261$$
$$x_{75} = -82.2050077689329$$
$$x_{23} = -78.0162175641465$$
$$x_{71} = -75.9218224617533$$
$$x_{50} = -71.733032256967$$
$$x_{77} = -69.6386371545737$$
$$x_{57} = -68.5914396033772$$
$$x_{36} = -66.497044500984$$
$$x_{37} = -65.4498469497874$$
$$x_{82} = -63.3554518473942$$
$$x_{43} = -62.3082542961976$$
$$x_{22} = -60.2138591938044$$
$$x_{69} = -57.0722665402146$$
$$x_{42} = -56.025068989018$$
$$x_{70} = -53.9306738866248$$
$$x_{48} = -49.7418836818384$$
$$x_{78} = -47.6474885794452$$
$$x_{7} = -46.6002910282486$$
$$x_{62} = -43.4586983746588$$
$$x_{13} = -41.3643032722656$$
$$x_{26} = -40.317105721069$$
$$x_{38} = -38.2227106186758$$
$$x_{8} = -34.0339204138894$$
$$x_{54} = -31.9395253114962$$
$$x_{35} = -27.7507351067098$$
$$x_{81} = -25.6563400043166$$
$$x_{49} = -24.60914245312$$
$$x_{4} = -21.4675497995303$$
$$x_{25} = -19.3731546971371$$
$$x_{20} = -18.3259571459405$$
$$x_{74} = -16.2315620435473$$
$$x_{1} = -12.0427718387609$$
$$x_{55} = -9.94837673636768$$
$$x_{11} = -5.75958653158129$$
$$x_{33} = -3.66519142918809$$
$$x_{66} = -2.61799387799149$$
$$x_{61} = 0.523598775598299$$
$$x_{10} = 2.61799387799149$$
$$x_{60} = 3.66519142918809$$
$$x_{15} = 5.75958653158129$$
$$x_{16} = 9.94837673636768$$
$$x_{28} = 12.0427718387609$$
$$x_{51} = 16.2315620435473$$
$$x_{68} = 18.3259571459405$$
$$x_{32} = 22.5147473507269$$
$$x_{52} = 24.60914245312$$
$$x_{67} = 25.6563400043166$$
$$x_{46} = 27.7507351067098$$
$$x_{27} = 31.9395253114962$$
$$x_{9} = 34.0339204138894$$
$$x_{58} = 38.2227106186758$$
$$x_{76} = 40.317105721069$$
$$x_{2} = 44.5058959258554$$
$$x_{6} = 46.6002910282486$$
$$x_{17} = 47.6474885794452$$
$$x_{41} = 49.7418836818384$$
$$x_{24} = 53.9306738866248$$
$$x_{83} = 56.025068989018$$
$$x_{19} = 60.2138591938044$$
$$x_{63} = 62.3082542961976$$
$$x_{12} = 66.497044500984$$
$$x_{79} = 68.5914396033772$$
$$x_{59} = 69.6386371545737$$
$$x_{21} = 71.733032256967$$
$$x_{47} = 75.9218224617533$$
$$x_{30} = 78.0162175641465$$
$$x_{64} = 82.2050077689329$$
$$x_{29} = 84.2994028713261$$
$$x_{80} = 88.4881930761125$$
$$x_{18} = 90.5825881785057$$
$$x_{3} = 91.6297857297023$$
$$x_{72} = 93.7241808320955$$
$$x_{65} = 96.8657734856853$$
$$x_{40} = 97.9129710368819$$
$$x_{39} = 100.007366139275$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{34}$$
For example, let's take the point
$$x_{0} = x_{34} - \frac{1}{10}$$
=
$$-382.750704962356 + - \frac{1}{10}$$
=
$$-382.850704962356$$
substitute to the expression
$$\left|{\sin{\left(x \right)}}\right| < \frac{1}{2}$$
$$\left|{\sin{\left(-382.850704962356 \right)}}\right| < \frac{1}{2}$$
one of the solutions of our inequality is:
$$x < -382.750704962356$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < -382.750704962356$$
$$x > -137.706477982353 \wedge x < -100.007366139275$$
$$x > -97.9129710368819 \wedge x < -93.7241808320955$$
$$x > -91.6297857297023 \wedge x < -90.5825881785057$$
$$x > -87.4409955249159 \wedge x < -85.3466004225227$$
$$x > -84.2994028713261 \wedge x < -82.2050077689329$$
$$x > -78.0162175641465 \wedge x < -75.9218224617533$$
$$x > -71.733032256967 \wedge x < -69.6386371545737$$
$$x > -68.5914396033772 \wedge x < -66.497044500984$$
$$x > -65.4498469497874 \wedge x < -63.3554518473942$$
$$x > -62.3082542961976 \wedge x < -60.2138591938044$$
$$x > -57.0722665402146 \wedge x < -56.025068989018$$
$$x > -53.9306738866248 \wedge x < -49.7418836818384$$
$$x > -47.6474885794452 \wedge x < -46.6002910282486$$
$$x > -43.4586983746588 \wedge x < -41.3643032722656$$
$$x > -40.317105721069 \wedge x < -38.2227106186758$$
$$x > -34.0339204138894 \wedge x < -31.9395253114962$$
$$x > -27.7507351067098 \wedge x < -25.6563400043166$$
$$x > -24.60914245312 \wedge x < -21.4675497995303$$
$$x > -19.3731546971371 \wedge x < -18.3259571459405$$
$$x > -16.2315620435473 \wedge x < -12.0427718387609$$
$$x > -9.94837673636768 \wedge x < -5.75958653158129$$
$$x > -3.66519142918809 \wedge x < -2.61799387799149$$
$$x > 0.523598775598299 \wedge x < 2.61799387799149$$
$$x > 3.66519142918809 \wedge x < 5.75958653158129$$
$$x > 9.94837673636768 \wedge x < 12.0427718387609$$
$$x > 16.2315620435473 \wedge x < 18.3259571459405$$
$$x > 22.5147473507269 \wedge x < 24.60914245312$$
$$x > 25.6563400043166 \wedge x < 27.7507351067098$$
$$x > 31.9395253114962 \wedge x < 34.0339204138894$$
$$x > 38.2227106186758 \wedge x < 40.317105721069$$
$$x > 44.5058959258554 \wedge x < 46.6002910282486$$
$$x > 47.6474885794452 \wedge x < 49.7418836818384$$
$$x > 53.9306738866248 \wedge x < 56.025068989018$$
$$x > 60.2138591938044 \wedge x < 62.3082542961976$$
$$x > 66.497044500984 \wedge x < 68.5914396033772$$
$$x > 69.6386371545737 \wedge x < 71.733032256967$$
$$x > 75.9218224617533 \wedge x < 78.0162175641465$$
$$x > 82.2050077689329 \wedge x < 84.2994028713261$$
$$x > 88.4881930761125 \wedge x < 90.5825881785057$$
$$x > 91.6297857297023 \wedge x < 93.7241808320955$$
$$x > 96.8657734856853 \wedge x < 97.9129710368819$$
$$x > 100.007366139275$$
$$\left|{\sin{\left(x \right)}}\right| < \frac{1}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{\sin{\left(x \right)}}\right| = \frac{1}{2}$$
Solve:
Given the equation
$$\left|{\sin{\left(x \right)}}\right| = \frac{1}{2}$$
transform
$$\left|{\sin{\left(x \right)}}\right| - \frac{1}{2} = 0$$
$$\left|{\sin{\left(x \right)}}\right| - \frac{1}{2} = 0$$
Do replacement
$$w = \left|{\sin{\left(x \right)}}\right|$$
Move free summands (without w)
from left part to right part, we given:
$$w = \frac{1}{2}$$
We get the answer: w = 1/2
do backward replacement
$$\left|{\sin{\left(x \right)}}\right| = w$$
substitute w:
$$x_{1} = -12.0427718387609$$
$$x_{2} = 44.5058959258554$$
$$x_{3} = 91.6297857297023$$
$$x_{4} = -21.4675497995303$$
$$x_{5} = -137.706477982353$$
$$x_{6} = 46.6002910282486$$
$$x_{7} = -46.6002910282486$$
$$x_{8} = -34.0339204138894$$
$$x_{9} = 34.0339204138894$$
$$x_{10} = 2.61799387799149$$
$$x_{11} = -5.75958653158129$$
$$x_{12} = 66.497044500984$$
$$x_{13} = -41.3643032722656$$
$$x_{14} = -90.5825881785057$$
$$x_{15} = 5.75958653158129$$
$$x_{16} = 9.94837673636768$$
$$x_{17} = 47.6474885794452$$
$$x_{18} = 90.5825881785057$$
$$x_{19} = 60.2138591938044$$
$$x_{20} = -18.3259571459405$$
$$x_{21} = 71.733032256967$$
$$x_{22} = -60.2138591938044$$
$$x_{23} = -78.0162175641465$$
$$x_{24} = 53.9306738866248$$
$$x_{25} = -19.3731546971371$$
$$x_{26} = -40.317105721069$$
$$x_{27} = 31.9395253114962$$
$$x_{28} = 12.0427718387609$$
$$x_{29} = 84.2994028713261$$
$$x_{30} = 78.0162175641465$$
$$x_{31} = -93.7241808320955$$
$$x_{32} = 22.5147473507269$$
$$x_{33} = -3.66519142918809$$
$$x_{34} = -382.750704962356$$
$$x_{35} = -27.7507351067098$$
$$x_{36} = -66.497044500984$$
$$x_{37} = -65.4498469497874$$
$$x_{38} = -38.2227106186758$$
$$x_{39} = 100.007366139275$$
$$x_{40} = 97.9129710368819$$
$$x_{41} = 49.7418836818384$$
$$x_{42} = -56.025068989018$$
$$x_{43} = -62.3082542961976$$
$$x_{44} = -87.4409955249159$$
$$x_{45} = -85.3466004225227$$
$$x_{46} = 27.7507351067098$$
$$x_{47} = 75.9218224617533$$
$$x_{48} = -49.7418836818384$$
$$x_{49} = -24.60914245312$$
$$x_{50} = -71.733032256967$$
$$x_{51} = 16.2315620435473$$
$$x_{52} = 24.60914245312$$
$$x_{53} = -100.007366139275$$
$$x_{54} = -31.9395253114962$$
$$x_{55} = -9.94837673636768$$
$$x_{56} = -97.9129710368819$$
$$x_{57} = -68.5914396033772$$
$$x_{58} = 38.2227106186758$$
$$x_{59} = 69.6386371545737$$
$$x_{60} = 3.66519142918809$$
$$x_{61} = 0.523598775598299$$
$$x_{62} = -43.4586983746588$$
$$x_{63} = 62.3082542961976$$
$$x_{64} = 82.2050077689329$$
$$x_{65} = 96.8657734856853$$
$$x_{66} = -2.61799387799149$$
$$x_{67} = 25.6563400043166$$
$$x_{68} = 18.3259571459405$$
$$x_{69} = -57.0722665402146$$
$$x_{70} = -53.9306738866248$$
$$x_{71} = -75.9218224617533$$
$$x_{72} = 93.7241808320955$$
$$x_{73} = -91.6297857297023$$
$$x_{74} = -16.2315620435473$$
$$x_{75} = -82.2050077689329$$
$$x_{76} = 40.317105721069$$
$$x_{77} = -69.6386371545737$$
$$x_{78} = -47.6474885794452$$
$$x_{79} = 68.5914396033772$$
$$x_{80} = 88.4881930761125$$
$$x_{81} = -25.6563400043166$$
$$x_{82} = -63.3554518473942$$
$$x_{83} = 56.025068989018$$
$$x_{84} = -84.2994028713261$$
$$x_{1} = -12.0427718387609$$
$$x_{2} = 44.5058959258554$$
$$x_{3} = 91.6297857297023$$
$$x_{4} = -21.4675497995303$$
$$x_{5} = -137.706477982353$$
$$x_{6} = 46.6002910282486$$
$$x_{7} = -46.6002910282486$$
$$x_{8} = -34.0339204138894$$
$$x_{9} = 34.0339204138894$$
$$x_{10} = 2.61799387799149$$
$$x_{11} = -5.75958653158129$$
$$x_{12} = 66.497044500984$$
$$x_{13} = -41.3643032722656$$
$$x_{14} = -90.5825881785057$$
$$x_{15} = 5.75958653158129$$
$$x_{16} = 9.94837673636768$$
$$x_{17} = 47.6474885794452$$
$$x_{18} = 90.5825881785057$$
$$x_{19} = 60.2138591938044$$
$$x_{20} = -18.3259571459405$$
$$x_{21} = 71.733032256967$$
$$x_{22} = -60.2138591938044$$
$$x_{23} = -78.0162175641465$$
$$x_{24} = 53.9306738866248$$
$$x_{25} = -19.3731546971371$$
$$x_{26} = -40.317105721069$$
$$x_{27} = 31.9395253114962$$
$$x_{28} = 12.0427718387609$$
$$x_{29} = 84.2994028713261$$
$$x_{30} = 78.0162175641465$$
$$x_{31} = -93.7241808320955$$
$$x_{32} = 22.5147473507269$$
$$x_{33} = -3.66519142918809$$
$$x_{34} = -382.750704962356$$
$$x_{35} = -27.7507351067098$$
$$x_{36} = -66.497044500984$$
$$x_{37} = -65.4498469497874$$
$$x_{38} = -38.2227106186758$$
$$x_{39} = 100.007366139275$$
$$x_{40} = 97.9129710368819$$
$$x_{41} = 49.7418836818384$$
$$x_{42} = -56.025068989018$$
$$x_{43} = -62.3082542961976$$
$$x_{44} = -87.4409955249159$$
$$x_{45} = -85.3466004225227$$
$$x_{46} = 27.7507351067098$$
$$x_{47} = 75.9218224617533$$
$$x_{48} = -49.7418836818384$$
$$x_{49} = -24.60914245312$$
$$x_{50} = -71.733032256967$$
$$x_{51} = 16.2315620435473$$
$$x_{52} = 24.60914245312$$
$$x_{53} = -100.007366139275$$
$$x_{54} = -31.9395253114962$$
$$x_{55} = -9.94837673636768$$
$$x_{56} = -97.9129710368819$$
$$x_{57} = -68.5914396033772$$
$$x_{58} = 38.2227106186758$$
$$x_{59} = 69.6386371545737$$
$$x_{60} = 3.66519142918809$$
$$x_{61} = 0.523598775598299$$
$$x_{62} = -43.4586983746588$$
$$x_{63} = 62.3082542961976$$
$$x_{64} = 82.2050077689329$$
$$x_{65} = 96.8657734856853$$
$$x_{66} = -2.61799387799149$$
$$x_{67} = 25.6563400043166$$
$$x_{68} = 18.3259571459405$$
$$x_{69} = -57.0722665402146$$
$$x_{70} = -53.9306738866248$$
$$x_{71} = -75.9218224617533$$
$$x_{72} = 93.7241808320955$$
$$x_{73} = -91.6297857297023$$
$$x_{74} = -16.2315620435473$$
$$x_{75} = -82.2050077689329$$
$$x_{76} = 40.317105721069$$
$$x_{77} = -69.6386371545737$$
$$x_{78} = -47.6474885794452$$
$$x_{79} = 68.5914396033772$$
$$x_{80} = 88.4881930761125$$
$$x_{81} = -25.6563400043166$$
$$x_{82} = -63.3554518473942$$
$$x_{83} = 56.025068989018$$
$$x_{84} = -84.2994028713261$$
This roots
$$x_{34} = -382.750704962356$$
$$x_{5} = -137.706477982353$$
$$x_{53} = -100.007366139275$$
$$x_{56} = -97.9129710368819$$
$$x_{31} = -93.7241808320955$$
$$x_{73} = -91.6297857297023$$
$$x_{14} = -90.5825881785057$$
$$x_{44} = -87.4409955249159$$
$$x_{45} = -85.3466004225227$$
$$x_{84} = -84.2994028713261$$
$$x_{75} = -82.2050077689329$$
$$x_{23} = -78.0162175641465$$
$$x_{71} = -75.9218224617533$$
$$x_{50} = -71.733032256967$$
$$x_{77} = -69.6386371545737$$
$$x_{57} = -68.5914396033772$$
$$x_{36} = -66.497044500984$$
$$x_{37} = -65.4498469497874$$
$$x_{82} = -63.3554518473942$$
$$x_{43} = -62.3082542961976$$
$$x_{22} = -60.2138591938044$$
$$x_{69} = -57.0722665402146$$
$$x_{42} = -56.025068989018$$
$$x_{70} = -53.9306738866248$$
$$x_{48} = -49.7418836818384$$
$$x_{78} = -47.6474885794452$$
$$x_{7} = -46.6002910282486$$
$$x_{62} = -43.4586983746588$$
$$x_{13} = -41.3643032722656$$
$$x_{26} = -40.317105721069$$
$$x_{38} = -38.2227106186758$$
$$x_{8} = -34.0339204138894$$
$$x_{54} = -31.9395253114962$$
$$x_{35} = -27.7507351067098$$
$$x_{81} = -25.6563400043166$$
$$x_{49} = -24.60914245312$$
$$x_{4} = -21.4675497995303$$
$$x_{25} = -19.3731546971371$$
$$x_{20} = -18.3259571459405$$
$$x_{74} = -16.2315620435473$$
$$x_{1} = -12.0427718387609$$
$$x_{55} = -9.94837673636768$$
$$x_{11} = -5.75958653158129$$
$$x_{33} = -3.66519142918809$$
$$x_{66} = -2.61799387799149$$
$$x_{61} = 0.523598775598299$$
$$x_{10} = 2.61799387799149$$
$$x_{60} = 3.66519142918809$$
$$x_{15} = 5.75958653158129$$
$$x_{16} = 9.94837673636768$$
$$x_{28} = 12.0427718387609$$
$$x_{51} = 16.2315620435473$$
$$x_{68} = 18.3259571459405$$
$$x_{32} = 22.5147473507269$$
$$x_{52} = 24.60914245312$$
$$x_{67} = 25.6563400043166$$
$$x_{46} = 27.7507351067098$$
$$x_{27} = 31.9395253114962$$
$$x_{9} = 34.0339204138894$$
$$x_{58} = 38.2227106186758$$
$$x_{76} = 40.317105721069$$
$$x_{2} = 44.5058959258554$$
$$x_{6} = 46.6002910282486$$
$$x_{17} = 47.6474885794452$$
$$x_{41} = 49.7418836818384$$
$$x_{24} = 53.9306738866248$$
$$x_{83} = 56.025068989018$$
$$x_{19} = 60.2138591938044$$
$$x_{63} = 62.3082542961976$$
$$x_{12} = 66.497044500984$$
$$x_{79} = 68.5914396033772$$
$$x_{59} = 69.6386371545737$$
$$x_{21} = 71.733032256967$$
$$x_{47} = 75.9218224617533$$
$$x_{30} = 78.0162175641465$$
$$x_{64} = 82.2050077689329$$
$$x_{29} = 84.2994028713261$$
$$x_{80} = 88.4881930761125$$
$$x_{18} = 90.5825881785057$$
$$x_{3} = 91.6297857297023$$
$$x_{72} = 93.7241808320955$$
$$x_{65} = 96.8657734856853$$
$$x_{40} = 97.9129710368819$$
$$x_{39} = 100.007366139275$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{34}$$
For example, let's take the point
$$x_{0} = x_{34} - \frac{1}{10}$$
=
$$-382.750704962356 + - \frac{1}{10}$$
=
$$-382.850704962356$$
substitute to the expression
$$\left|{\sin{\left(x \right)}}\right| < \frac{1}{2}$$
$$\left|{\sin{\left(-382.850704962356 \right)}}\right| < \frac{1}{2}$$
0.411043807676265 < 1/2
one of the solutions of our inequality is:
$$x < -382.750704962356$$
_____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
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x34 x5 x53 x56 x31 x73 x14 x44 x45 x84 x75 x23 x71 x50 x77 x57 x36 x37 x82 x43 x22 x69 x42 x70 x48 x78 x7 x62 x13 x26 x38 x8 x54 x35 x81 x49 x4 x25 x20 x74 x1 x55 x11 x33 x66 x61 x10 x60 x15 x16 x28 x51 x68 x32 x52 x67 x46 x27 x9 x58 x76 x2 x6 x17 x41 x24 x83 x19 x63 x12 x79 x59 x21 x47 x30 x64 x29 x80 x18 x3 x72 x65 x40 x39Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < -382.750704962356$$
$$x > -137.706477982353 \wedge x < -100.007366139275$$
$$x > -97.9129710368819 \wedge x < -93.7241808320955$$
$$x > -91.6297857297023 \wedge x < -90.5825881785057$$
$$x > -87.4409955249159 \wedge x < -85.3466004225227$$
$$x > -84.2994028713261 \wedge x < -82.2050077689329$$
$$x > -78.0162175641465 \wedge x < -75.9218224617533$$
$$x > -71.733032256967 \wedge x < -69.6386371545737$$
$$x > -68.5914396033772 \wedge x < -66.497044500984$$
$$x > -65.4498469497874 \wedge x < -63.3554518473942$$
$$x > -62.3082542961976 \wedge x < -60.2138591938044$$
$$x > -57.0722665402146 \wedge x < -56.025068989018$$
$$x > -53.9306738866248 \wedge x < -49.7418836818384$$
$$x > -47.6474885794452 \wedge x < -46.6002910282486$$
$$x > -43.4586983746588 \wedge x < -41.3643032722656$$
$$x > -40.317105721069 \wedge x < -38.2227106186758$$
$$x > -34.0339204138894 \wedge x < -31.9395253114962$$
$$x > -27.7507351067098 \wedge x < -25.6563400043166$$
$$x > -24.60914245312 \wedge x < -21.4675497995303$$
$$x > -19.3731546971371 \wedge x < -18.3259571459405$$
$$x > -16.2315620435473 \wedge x < -12.0427718387609$$
$$x > -9.94837673636768 \wedge x < -5.75958653158129$$
$$x > -3.66519142918809 \wedge x < -2.61799387799149$$
$$x > 0.523598775598299 \wedge x < 2.61799387799149$$
$$x > 3.66519142918809 \wedge x < 5.75958653158129$$
$$x > 9.94837673636768 \wedge x < 12.0427718387609$$
$$x > 16.2315620435473 \wedge x < 18.3259571459405$$
$$x > 22.5147473507269 \wedge x < 24.60914245312$$
$$x > 25.6563400043166 \wedge x < 27.7507351067098$$
$$x > 31.9395253114962 \wedge x < 34.0339204138894$$
$$x > 38.2227106186758 \wedge x < 40.317105721069$$
$$x > 44.5058959258554 \wedge x < 46.6002910282486$$
$$x > 47.6474885794452 \wedge x < 49.7418836818384$$
$$x > 53.9306738866248 \wedge x < 56.025068989018$$
$$x > 60.2138591938044 \wedge x < 62.3082542961976$$
$$x > 66.497044500984 \wedge x < 68.5914396033772$$
$$x > 69.6386371545737 \wedge x < 71.733032256967$$
$$x > 75.9218224617533 \wedge x < 78.0162175641465$$
$$x > 82.2050077689329 \wedge x < 84.2994028713261$$
$$x > 88.4881930761125 \wedge x < 90.5825881785057$$
$$x > 91.6297857297023 \wedge x < 93.7241808320955$$
$$x > 96.8657734856853 \wedge x < 97.9129710368819$$
$$x > 100.007366139275$$
Solving inequality on a graph