sin|x|<=1 desigualdades

Se da la desigualdad:
$$\sin{\left(\left|{x}\right| \right)} \leq 1$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\sin{\left(\left|{x}\right| \right)} = 1$$
Resolvemos:
Tenemos la ecuación
$$\sin{\left(\left|{x}\right| \right)} = 1$$
cambiamos
$$\sin{\left(\left|{x}\right| \right)} - 1 = 0$$
$$\sin{\left(\left|{x}\right| \right)} - 1 = 0$$
Sustituimos
$$w = \sin{\left(\left|{x}\right| \right)}$$
Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
$$w = 1$$
Obtenemos la respuesta: w = 1
hacemos cambio inverso
$$\sin{\left(\left|{x}\right| \right)} = w$$
sustituimos w:
$$x_{1} = 89.5353901350773$$
$$x_{2} = 20.420352160156$$
$$x_{3} = -64.4026491641039$$
$$x_{4} = -7.85398205280014$$
$$x_{5} = -39.2699069219675$$
$$x_{6} = 70.6858358251975$$
$$x_{7} = 1.57079582971902$$
$$x_{8} = -89.5353901118113$$
$$x_{9} = 102.101760799573$$
$$x_{10} = -39.2699084145515$$
$$x_{11} = 64.4026499096387$$
$$x_{12} = -20.4203520060805$$
$$x_{13} = -64.4026498988255$$
$$x_{14} = -32.9867232184024$$
$$x_{15} = -32.9867224188086$$
$$x_{16} = 89.535390888605$$
$$x_{17} = -39.2699076683741$$
$$x_{18} = -89.5353906059052$$
$$x_{19} = 7.85398174307326$$
$$x_{20} = 70.6858352127237$$
$$x_{21} = 14.1371671100222$$
$$x_{22} = -45.5530935911043$$
$$x_{23} = 83.2522056907544$$
$$x_{24} = 83.252204888767$$
$$x_{25} = -95.8185763308148$$
$$x_{26} = -51.8362791922783$$
$$x_{27} = 51.8362782775539$$
$$x_{28} = 26.703537322248$$
$$x_{29} = 70.6858344802043$$
$$x_{30} = -83.2522042893833$$
$$x_{31} = 58.1194643979608$$
$$x_{32} = -102.101761026058$$
$$x_{33} = 51.8362789031518$$
$$x_{34} = -45.5530929624673$$
$$x_{35} = 89.5353893728458$$
$$x_{36} = -51.8362786893284$$
$$x_{37} = 95.8185764110282$$
$$x_{38} = 1.57079525114023$$
$$x_{39} = 58.119464520069$$
$$x_{40} = 26.7035380604159$$
$$x_{41} = 39.2699085343272$$
$$x_{42} = -673.871625005449$$
$$x_{43} = 95.8185759975842$$
$$x_{44} = -64.4026502975618$$
$$x_{45} = 32.9867233134552$$
$$x_{46} = 32.9867225164981$$
$$x_{47} = 58.1194636580315$$
$$x_{48} = -26.7035372004893$$
$$x_{49} = -95.818575476176$$
$$x_{50} = -95.8185758680502$$
$$x_{51} = 7.85398177249874$$
$$x_{52} = 76.9690204681432$$
$$x_{53} = -58.1194639046052$$
$$x_{54} = -70.6858351534454$$
$$x_{55} = -7.85398119154045$$
$$x_{56} = -14.1371668370864$$
$$x_{57} = -70.6858331259916$$
$$x_{58} = -20.4203527465087$$
$$x_{59} = -51.8362783335234$$
$$x_{60} = 7.85398112872719$$
$$x_{61} = -1.57079643188553$$
$$x_{62} = -20.420353265929$$
$$x_{63} = 64.4026493072124$$
$$x_{64} = 45.553093730794$$
$$x_{65} = -76.9690203748894$$
$$x_{66} = 45.5530922954328$$
$$x_{67} = -1.57079639503667$$
$$x_{68} = -14.1371667858125$$
$$x_{69} = 20.4203521477756$$
$$x_{70} = 1.57079657289894$$
$$x_{71} = -26.7035379986821$$
$$x_{72} = 51.8362788867584$$
$$x_{73} = -89.535390750197$$
$$x_{74} = 26.7035387715281$$
$$x_{75} = 39.2699077336963$$
$$x_{76} = 64.4026492731727$$
$$x_{77} = -70.6858343571487$$
$$x_{78} = -45.5530935025548$$
$$x_{79} = -76.9690195738024$$
$$x_{80} = -83.2522055723275$$
$$x_{81} = 95.8185760629547$$
$$x_{82} = 95.8185754266891$$
$$x_{83} = 45.5530929823099$$
$$x_{84} = 14.1371673791846$$
$$x_{85} = 76.9690196732095$$
$$x_{86} = 14.1371665172699$$
$$x_{87} = -58.1194639976905$$
$$x_{88} = -7.85398149665124$$
$$x_{89} = -83.2522048211133$$
$$x_{90} = 20.4203527610188$$
$$x_{91} = -58.1194645939029$$
$$x_{92} = -14.1371674455661$$
$$x_{1} = 89.5353901350773$$
$$x_{2} = 20.420352160156$$
$$x_{3} = -64.4026491641039$$
$$x_{4} = -7.85398205280014$$
$$x_{5} = -39.2699069219675$$
$$x_{6} = 70.6858358251975$$
$$x_{7} = 1.57079582971902$$
$$x_{8} = -89.5353901118113$$
$$x_{9} = 102.101760799573$$
$$x_{10} = -39.2699084145515$$
$$x_{11} = 64.4026499096387$$
$$x_{12} = -20.4203520060805$$
$$x_{13} = -64.4026498988255$$
$$x_{14} = -32.9867232184024$$
$$x_{15} = -32.9867224188086$$
$$x_{16} = 89.535390888605$$
$$x_{17} = -39.2699076683741$$
$$x_{18} = -89.5353906059052$$
$$x_{19} = 7.85398174307326$$
$$x_{20} = 70.6858352127237$$
$$x_{21} = 14.1371671100222$$
$$x_{22} = -45.5530935911043$$
$$x_{23} = 83.2522056907544$$
$$x_{24} = 83.252204888767$$
$$x_{25} = -95.8185763308148$$
$$x_{26} = -51.8362791922783$$
$$x_{27} = 51.8362782775539$$
$$x_{28} = 26.703537322248$$
$$x_{29} = 70.6858344802043$$
$$x_{30} = -83.2522042893833$$
$$x_{31} = 58.1194643979608$$
$$x_{32} = -102.101761026058$$
$$x_{33} = 51.8362789031518$$
$$x_{34} = -45.5530929624673$$
$$x_{35} = 89.5353893728458$$
$$x_{36} = -51.8362786893284$$
$$x_{37} = 95.8185764110282$$
$$x_{38} = 1.57079525114023$$
$$x_{39} = 58.119464520069$$
$$x_{40} = 26.7035380604159$$
$$x_{41} = 39.2699085343272$$
$$x_{42} = -673.871625005449$$
$$x_{43} = 95.8185759975842$$
$$x_{44} = -64.4026502975618$$
$$x_{45} = 32.9867233134552$$
$$x_{46} = 32.9867225164981$$
$$x_{47} = 58.1194636580315$$
$$x_{48} = -26.7035372004893$$
$$x_{49} = -95.818575476176$$
$$x_{50} = -95.8185758680502$$
$$x_{51} = 7.85398177249874$$
$$x_{52} = 76.9690204681432$$
$$x_{53} = -58.1194639046052$$
$$x_{54} = -70.6858351534454$$
$$x_{55} = -7.85398119154045$$
$$x_{56} = -14.1371668370864$$
$$x_{57} = -70.6858331259916$$
$$x_{58} = -20.4203527465087$$
$$x_{59} = -51.8362783335234$$
$$x_{60} = 7.85398112872719$$
$$x_{61} = -1.57079643188553$$
$$x_{62} = -20.420353265929$$
$$x_{63} = 64.4026493072124$$
$$x_{64} = 45.553093730794$$
$$x_{65} = -76.9690203748894$$
$$x_{66} = 45.5530922954328$$
$$x_{67} = -1.57079639503667$$
$$x_{68} = -14.1371667858125$$
$$x_{69} = 20.4203521477756$$
$$x_{70} = 1.57079657289894$$
$$x_{71} = -26.7035379986821$$
$$x_{72} = 51.8362788867584$$
$$x_{73} = -89.535390750197$$
$$x_{74} = 26.7035387715281$$
$$x_{75} = 39.2699077336963$$
$$x_{76} = 64.4026492731727$$
$$x_{77} = -70.6858343571487$$
$$x_{78} = -45.5530935025548$$
$$x_{79} = -76.9690195738024$$
$$x_{80} = -83.2522055723275$$
$$x_{81} = 95.8185760629547$$
$$x_{82} = 95.8185754266891$$
$$x_{83} = 45.5530929823099$$
$$x_{84} = 14.1371673791846$$
$$x_{85} = 76.9690196732095$$
$$x_{86} = 14.1371665172699$$
$$x_{87} = -58.1194639976905$$
$$x_{88} = -7.85398149665124$$
$$x_{89} = -83.2522048211133$$
$$x_{90} = 20.4203527610188$$
$$x_{91} = -58.1194645939029$$
$$x_{92} = -14.1371674455661$$
Las raíces dadas
$$x_{42} = -673.871625005449$$
$$x_{32} = -102.101761026058$$
$$x_{25} = -95.8185763308148$$
$$x_{50} = -95.8185758680502$$
$$x_{49} = -95.818575476176$$
$$x_{73} = -89.535390750197$$
$$x_{18} = -89.5353906059052$$
$$x_{8} = -89.5353901118113$$
$$x_{80} = -83.2522055723275$$
$$x_{89} = -83.2522048211133$$
$$x_{30} = -83.2522042893833$$
$$x_{65} = -76.9690203748894$$
$$x_{79} = -76.9690195738024$$
$$x_{54} = -70.6858351534454$$
$$x_{77} = -70.6858343571487$$
$$x_{57} = -70.6858331259916$$
$$x_{44} = -64.4026502975618$$
$$x_{13} = -64.4026498988255$$
$$x_{3} = -64.4026491641039$$
$$x_{91} = -58.1194645939029$$
$$x_{87} = -58.1194639976905$$
$$x_{53} = -58.1194639046052$$
$$x_{26} = -51.8362791922783$$
$$x_{36} = -51.8362786893284$$
$$x_{59} = -51.8362783335234$$
$$x_{22} = -45.5530935911043$$
$$x_{78} = -45.5530935025548$$
$$x_{34} = -45.5530929624673$$
$$x_{10} = -39.2699084145515$$
$$x_{17} = -39.2699076683741$$
$$x_{5} = -39.2699069219675$$
$$x_{14} = -32.9867232184024$$
$$x_{15} = -32.9867224188086$$
$$x_{71} = -26.7035379986821$$
$$x_{48} = -26.7035372004893$$
$$x_{62} = -20.420353265929$$
$$x_{58} = -20.4203527465087$$
$$x_{12} = -20.4203520060805$$
$$x_{92} = -14.1371674455661$$
$$x_{56} = -14.1371668370864$$
$$x_{68} = -14.1371667858125$$
$$x_{4} = -7.85398205280014$$
$$x_{88} = -7.85398149665124$$
$$x_{55} = -7.85398119154045$$
$$x_{61} = -1.57079643188553$$
$$x_{67} = -1.57079639503667$$
$$x_{38} = 1.57079525114023$$
$$x_{7} = 1.57079582971902$$
$$x_{70} = 1.57079657289894$$
$$x_{60} = 7.85398112872719$$
$$x_{19} = 7.85398174307326$$
$$x_{51} = 7.85398177249874$$
$$x_{86} = 14.1371665172699$$
$$x_{21} = 14.1371671100222$$
$$x_{84} = 14.1371673791846$$
$$x_{69} = 20.4203521477756$$
$$x_{2} = 20.420352160156$$
$$x_{90} = 20.4203527610188$$
$$x_{28} = 26.703537322248$$
$$x_{40} = 26.7035380604159$$
$$x_{74} = 26.7035387715281$$
$$x_{46} = 32.9867225164981$$
$$x_{45} = 32.9867233134552$$
$$x_{75} = 39.2699077336963$$
$$x_{41} = 39.2699085343272$$
$$x_{66} = 45.5530922954328$$
$$x_{83} = 45.5530929823099$$
$$x_{64} = 45.553093730794$$
$$x_{27} = 51.8362782775539$$
$$x_{72} = 51.8362788867584$$
$$x_{33} = 51.8362789031518$$
$$x_{47} = 58.1194636580315$$
$$x_{31} = 58.1194643979608$$
$$x_{39} = 58.119464520069$$
$$x_{76} = 64.4026492731727$$
$$x_{63} = 64.4026493072124$$
$$x_{11} = 64.4026499096387$$
$$x_{29} = 70.6858344802043$$
$$x_{20} = 70.6858352127237$$
$$x_{6} = 70.6858358251975$$
$$x_{85} = 76.9690196732095$$
$$x_{52} = 76.9690204681432$$
$$x_{24} = 83.252204888767$$
$$x_{23} = 83.2522056907544$$
$$x_{35} = 89.5353893728458$$
$$x_{1} = 89.5353901350773$$
$$x_{16} = 89.535390888605$$
$$x_{82} = 95.8185754266891$$
$$x_{43} = 95.8185759975842$$
$$x_{81} = 95.8185760629547$$
$$x_{37} = 95.8185764110282$$
$$x_{9} = 102.101760799573$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{42}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{42} - \frac{1}{10}$$
=
$$-673.871625005449 + - \frac{1}{10}$$
=
$$-673.971625005449$$
lo sustituimos en la expresión
$$\sin{\left(\left|{x}\right| \right)} \leq 1$$
$$\sin{\left(\left|{-673.971625005449}\right| \right)} \leq 1$$

0.995004084368889 <= 1

significa que una de las soluciones de nuestra ecuación será con:
$$x \leq -673.871625005449$$

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       x42      x32      x25      x50      x49      x73      x18      x8      x80      x89      x30      x65      x79      x54      x77      x57      x44      x13      x3      x91      x87      x53      x26      x36      x59      x22      x78      x34      x10      x17      x5      x14      x15      x71      x48      x62      x58      x12      x92      x56      x68      x4      x88      x55      x61      x67      x38      x7      x70      x60      x19      x51      x86      x21      x84      x69      x2      x90      x28      x40      x74      x46      x45      x75      x41      x66      x83      x64      x27      x72      x33      x47      x31      x39      x76      x63      x11      x29      x20      x6      x85      x52      x24      x23      x35      x1      x16      x82      x43      x81      x37      x9

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x \leq -673.871625005449$$
$$x \geq -102.101761026058 \wedge x \leq -95.8185763308148$$
$$x \geq -95.8185758680502 \wedge x \leq -95.818575476176$$
$$x \geq -89.535390750197 \wedge x \leq -89.5353906059052$$
$$x \geq -89.5353901118113 \wedge x \leq -83.2522055723275$$
$$x \geq -83.2522048211133 \wedge x \leq -83.2522042893833$$
$$x \geq -76.9690203748894 \wedge x \leq -76.9690195738024$$
$$x \geq -70.6858351534454 \wedge x \leq -70.6858343571487$$
$$x \geq -70.6858331259916 \wedge x \leq -64.4026502975618$$
$$x \geq -64.4026498988255 \wedge x \leq -64.4026491641039$$
$$x \geq -58.1194645939029 \wedge x \leq -58.1194639976905$$
$$x \geq -58.1194639046052 \wedge x \leq -51.8362791922783$$
$$x \geq -51.8362786893284 \wedge x \leq -51.8362783335234$$
$$x \geq -45.5530935911043 \wedge x \leq -45.5530935025548$$
$$x \geq -45.5530929624673 \wedge x \leq -39.2699084145515$$
$$x \geq -39.2699076683741 \wedge x \leq -39.2699069219675$$
$$x \geq -32.9867232184024 \wedge x \leq -32.9867224188086$$
$$x \geq -26.7035379986821 \wedge x \leq -26.7035372004893$$
$$x \geq -20.420353265929 \wedge x \leq -20.4203527465087$$
$$x \geq -20.4203520060805 \wedge x \leq -14.1371674455661$$
$$x \geq -14.1371668370864 \wedge x \leq -14.1371667858125$$
$$x \geq -7.85398205280014 \wedge x \leq -7.85398149665124$$
$$x \geq -7.85398119154045 \wedge x \leq -1.57079643188553$$
$$x \geq -1.57079639503667 \wedge x \leq 1.57079525114023$$
$$x \geq 1.57079582971902 \wedge x \leq 1.57079657289894$$
$$x \geq 7.85398112872719 \wedge x \leq 7.85398174307326$$
$$x \geq 7.85398177249874 \wedge x \leq 14.1371665172699$$
$$x \geq 14.1371671100222 \wedge x \leq 14.1371673791846$$
$$x \geq 20.4203521477756 \wedge x \leq 20.420352160156$$
$$x \geq 20.4203527610188 \wedge x \leq 26.703537322248$$
$$x \geq 26.7035380604159 \wedge x \leq 26.7035387715281$$
$$x \geq 32.9867225164981 \wedge x \leq 32.9867233134552$$
$$x \geq 39.2699077336963 \wedge x \leq 39.2699085343272$$
$$x \geq 45.5530922954328 \wedge x \leq 45.5530929823099$$
$$x \geq 45.553093730794 \wedge x \leq 51.8362782775539$$
$$x \geq 51.8362788867584 \wedge x \leq 51.8362789031518$$
$$x \geq 58.1194636580315 \wedge x \leq 58.1194643979608$$
$$x \geq 58.119464520069 \wedge x \leq 64.4026492731727$$
$$x \geq 64.4026493072124 \wedge x \leq 64.4026499096387$$
$$x \geq 70.6858344802043 \wedge x \leq 70.6858352127237$$
$$x \geq 70.6858358251975 \wedge x \leq 76.9690196732095$$
$$x \geq 76.9690204681432 \wedge x \leq 83.252204888767$$
$$x \geq 83.2522056907544 \wedge x \leq 89.5353893728458$$
$$x \geq 89.5353901350773 \wedge x \leq 89.535390888605$$
$$x \geq 95.8185754266891 \wedge x \leq 95.8185759975842$$
$$x \geq 95.8185760629547 \wedge x \leq 95.8185764110282$$
$$x \geq 102.101760799573$$