Sr Examen
Otras calculadoras
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- Sistemas de ecuaciones paso a paso
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-
¿Cómo usar?
- Desigualdades:
-
3x-10<=8x+4
-
-8-7x<8x+13
-
x^2-2x+1>=0
-
(x+7)(x-11)(x+5)<0
-
Expresiones idénticas
- | uno +sinx|<= uno / dos
- módulo de 1 más seno de x| menos o igual a 1 dividir por 2
- módulo de uno más seno de x| menos o igual a uno dividir por dos
- |1+sinx|<=1 dividir por 2
-
Expresiones semejantes
- |1+sin(x)|>=1/2
- |1-sinx|<=1/2
-
Expresiones con funciones
- Módulo |
- |3x+4|>2
- |3x+1|>5-4x
- |2-x|<3
- |2*x^2-9*x+15|>=20
- |4+3x|>=2
|1+sinx|<=1/2 desigualdades
Solución
Ha introducido
[src]
|1 + sin(x)| <= 1/2
$$\left|{\sin{\left(x \right)} + 1}\right| \leq \frac{1}{2}$$
Abs(sin(x) + 1) <= 1/2
Solución detallada
Se da la desigualdad:
$$\left|{\sin{\left(x \right)} + 1}\right| \leq \frac{1}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left|{\sin{\left(x \right)} + 1}\right| = \frac{1}{2}$$
Resolvemos:
Tenemos la ecuación
$$\left|{\sin{\left(x \right)} + 1}\right| = \frac{1}{2}$$
cambiamos
$$\left|{\sin{\left(x \right)} + 1}\right| - \frac{1}{2} = 0$$
$$\left|{\sin{\left(x \right)} + 1}\right| - \frac{1}{2} = 0$$
Sustituimos
$$w = \left|{\sin{\left(x \right)} + 1}\right|$$
Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
$$w = \frac{1}{2}$$
Obtenemos la respuesta: w = 1/2
hacemos cambio inverso
$$\left|{\sin{\left(x \right)} + 1}\right| = w$$
sustituimos w:
$$x_{1} = 66.497044500984$$
$$x_{2} = 100.007366139275$$
$$x_{3} = -90.5825881785057$$
$$x_{4} = -13.0899693899575$$
$$x_{5} = 30.8923277602996$$
$$x_{6} = -63.3554518473942$$
$$x_{7} = -78.0162175641465$$
$$x_{8} = -69.6386371545737$$
$$x_{9} = 97.9129710368819$$
$$x_{10} = 41.3643032722656$$
$$x_{11} = -25.6563400043166$$
$$x_{12} = 79.0634151153431$$
$$x_{13} = 72.7802298081635$$
$$x_{14} = -2.61799387799149$$
$$x_{15} = 22.5147473507269$$
$$x_{16} = 12.0427718387609$$
$$x_{17} = -151.320046147908$$
$$x_{18} = 60.2138591938044$$
$$x_{19} = 18.3259571459405$$
$$x_{20} = 43.4586983746588$$
$$x_{21} = -31.9395253114962$$
$$x_{22} = -6.80678408277789$$
$$x_{23} = -34.0339204138894$$
$$x_{24} = -65.4498469497874$$
$$x_{25} = -46.6002910282486$$
$$x_{26} = 24.60914245312$$
$$x_{27} = 5.75958653158129$$
$$x_{28} = 53.9306738866248$$
$$x_{29} = -84.2994028713261$$
$$x_{30} = -71.733032256967$$
$$x_{31} = -195.302343298165$$
$$x_{32} = 192.160750644576$$
$$x_{33} = -57.0722665402146$$
$$x_{34} = -88.4881930761125$$
$$x_{35} = 35.081117965086$$
$$x_{36} = 66437.8778393414$$
$$x_{37} = 872.839158922364$$
$$x_{38} = 3.66519142918809$$
$$x_{39} = -21.4675497995303$$
$$x_{40} = 9.94837673636768$$
$$x_{41} = -50.789081233035$$
$$x_{42} = 16.2315620435473$$
$$x_{43} = 74.8746249105567$$
$$x_{44} = -59.1666616426078$$
$$x_{45} = 91.6297857297023$$
$$x_{46} = -101.054563690472$$
$$x_{47} = -75.9218224617533$$
$$x_{48} = -19.3731546971371$$
$$x_{49} = -15.1843644923507$$
$$x_{50} = 28.7979326579064$$
$$x_{51} = 93.7241808320955$$
$$x_{52} = -44.5058959258554$$
$$x_{53} = -8.90117918517108$$
$$x_{54} = -82.2050077689329$$
$$x_{55} = -0.523598775598299$$
$$x_{56} = 37.1755130674792$$
$$x_{57} = 85.3466004225227$$
$$x_{58} = -40.317105721069$$
$$x_{59} = -94.7713783832921$$
$$x_{60} = 437.20497762458$$
$$x_{61} = 47.6474885794452$$
$$x_{62} = 87.4409955249159$$
$$x_{63} = 62.3082542961976$$
$$x_{64} = 49.7418836818384$$
$$x_{65} = -27.7507351067098$$
$$x_{66} = -52.8834763354282$$
$$x_{67} = 56.025068989018$$
$$x_{68} = -38.2227106186758$$
$$x_{69} = -96.8657734856853$$
$$x_{70} = 81.1578102177363$$
$$x_{71} = 68.5914396033772$$
$$x_{1} = 66.497044500984$$
$$x_{2} = 100.007366139275$$
$$x_{3} = -90.5825881785057$$
$$x_{4} = -13.0899693899575$$
$$x_{5} = 30.8923277602996$$
$$x_{6} = -63.3554518473942$$
$$x_{7} = -78.0162175641465$$
$$x_{8} = -69.6386371545737$$
$$x_{9} = 97.9129710368819$$
$$x_{10} = 41.3643032722656$$
$$x_{11} = -25.6563400043166$$
$$x_{12} = 79.0634151153431$$
$$x_{13} = 72.7802298081635$$
$$x_{14} = -2.61799387799149$$
$$x_{15} = 22.5147473507269$$
$$x_{16} = 12.0427718387609$$
$$x_{17} = -151.320046147908$$
$$x_{18} = 60.2138591938044$$
$$x_{19} = 18.3259571459405$$
$$x_{20} = 43.4586983746588$$
$$x_{21} = -31.9395253114962$$
$$x_{22} = -6.80678408277789$$
$$x_{23} = -34.0339204138894$$
$$x_{24} = -65.4498469497874$$
$$x_{25} = -46.6002910282486$$
$$x_{26} = 24.60914245312$$
$$x_{27} = 5.75958653158129$$
$$x_{28} = 53.9306738866248$$
$$x_{29} = -84.2994028713261$$
$$x_{30} = -71.733032256967$$
$$x_{31} = -195.302343298165$$
$$x_{32} = 192.160750644576$$
$$x_{33} = -57.0722665402146$$
$$x_{34} = -88.4881930761125$$
$$x_{35} = 35.081117965086$$
$$x_{36} = 66437.8778393414$$
$$x_{37} = 872.839158922364$$
$$x_{38} = 3.66519142918809$$
$$x_{39} = -21.4675497995303$$
$$x_{40} = 9.94837673636768$$
$$x_{41} = -50.789081233035$$
$$x_{42} = 16.2315620435473$$
$$x_{43} = 74.8746249105567$$
$$x_{44} = -59.1666616426078$$
$$x_{45} = 91.6297857297023$$
$$x_{46} = -101.054563690472$$
$$x_{47} = -75.9218224617533$$
$$x_{48} = -19.3731546971371$$
$$x_{49} = -15.1843644923507$$
$$x_{50} = 28.7979326579064$$
$$x_{51} = 93.7241808320955$$
$$x_{52} = -44.5058959258554$$
$$x_{53} = -8.90117918517108$$
$$x_{54} = -82.2050077689329$$
$$x_{55} = -0.523598775598299$$
$$x_{56} = 37.1755130674792$$
$$x_{57} = 85.3466004225227$$
$$x_{58} = -40.317105721069$$
$$x_{59} = -94.7713783832921$$
$$x_{60} = 437.20497762458$$
$$x_{61} = 47.6474885794452$$
$$x_{62} = 87.4409955249159$$
$$x_{63} = 62.3082542961976$$
$$x_{64} = 49.7418836818384$$
$$x_{65} = -27.7507351067098$$
$$x_{66} = -52.8834763354282$$
$$x_{67} = 56.025068989018$$
$$x_{68} = -38.2227106186758$$
$$x_{69} = -96.8657734856853$$
$$x_{70} = 81.1578102177363$$
$$x_{71} = 68.5914396033772$$
Las raíces dadas
$$x_{31} = -195.302343298165$$
$$x_{17} = -151.320046147908$$
$$x_{46} = -101.054563690472$$
$$x_{69} = -96.8657734856853$$
$$x_{59} = -94.7713783832921$$
$$x_{3} = -90.5825881785057$$
$$x_{34} = -88.4881930761125$$
$$x_{29} = -84.2994028713261$$
$$x_{54} = -82.2050077689329$$
$$x_{7} = -78.0162175641465$$
$$x_{47} = -75.9218224617533$$
$$x_{30} = -71.733032256967$$
$$x_{8} = -69.6386371545737$$
$$x_{24} = -65.4498469497874$$
$$x_{6} = -63.3554518473942$$
$$x_{44} = -59.1666616426078$$
$$x_{33} = -57.0722665402146$$
$$x_{66} = -52.8834763354282$$
$$x_{41} = -50.789081233035$$
$$x_{25} = -46.6002910282486$$
$$x_{52} = -44.5058959258554$$
$$x_{58} = -40.317105721069$$
$$x_{68} = -38.2227106186758$$
$$x_{23} = -34.0339204138894$$
$$x_{21} = -31.9395253114962$$
$$x_{65} = -27.7507351067098$$
$$x_{11} = -25.6563400043166$$
$$x_{39} = -21.4675497995303$$
$$x_{48} = -19.3731546971371$$
$$x_{49} = -15.1843644923507$$
$$x_{4} = -13.0899693899575$$
$$x_{53} = -8.90117918517108$$
$$x_{22} = -6.80678408277789$$
$$x_{14} = -2.61799387799149$$
$$x_{55} = -0.523598775598299$$
$$x_{38} = 3.66519142918809$$
$$x_{27} = 5.75958653158129$$
$$x_{40} = 9.94837673636768$$
$$x_{16} = 12.0427718387609$$
$$x_{42} = 16.2315620435473$$
$$x_{19} = 18.3259571459405$$
$$x_{15} = 22.5147473507269$$
$$x_{26} = 24.60914245312$$
$$x_{50} = 28.7979326579064$$
$$x_{5} = 30.8923277602996$$
$$x_{35} = 35.081117965086$$
$$x_{56} = 37.1755130674792$$
$$x_{10} = 41.3643032722656$$
$$x_{20} = 43.4586983746588$$
$$x_{61} = 47.6474885794452$$
$$x_{64} = 49.7418836818384$$
$$x_{28} = 53.9306738866248$$
$$x_{67} = 56.025068989018$$
$$x_{18} = 60.2138591938044$$
$$x_{63} = 62.3082542961976$$
$$x_{1} = 66.497044500984$$
$$x_{71} = 68.5914396033772$$
$$x_{13} = 72.7802298081635$$
$$x_{43} = 74.8746249105567$$
$$x_{12} = 79.0634151153431$$
$$x_{70} = 81.1578102177363$$
$$x_{57} = 85.3466004225227$$
$$x_{62} = 87.4409955249159$$
$$x_{45} = 91.6297857297023$$
$$x_{51} = 93.7241808320955$$
$$x_{9} = 97.9129710368819$$
$$x_{2} = 100.007366139275$$
$$x_{32} = 192.160750644576$$
$$x_{60} = 437.20497762458$$
$$x_{37} = 872.839158922364$$
$$x_{36} = 66437.8778393414$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{31}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{31} - \frac{1}{10}$$
=
$$-195.302343298165 + - \frac{1}{10}$$
=
$$-195.402343298165$$
lo sustituimos en la expresión
$$\left|{\sin{\left(x \right)} + 1}\right| \leq \frac{1}{2}$$
$$\left|{\sin{\left(-195.402343298165 \right)} + 1}\right| \leq \frac{1}{2}$$
significa que una de las soluciones de nuestra ecuación será con:
$$x \leq -195.302343298165$$
Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x \leq -195.302343298165$$
$$x \geq -151.320046147908 \wedge x \leq -101.054563690472$$
$$x \geq -96.8657734856853 \wedge x \leq -94.7713783832921$$
$$x \geq -90.5825881785057 \wedge x \leq -88.4881930761125$$
$$x \geq -84.2994028713261 \wedge x \leq -82.2050077689329$$
$$x \geq -78.0162175641465 \wedge x \leq -75.9218224617533$$
$$x \geq -71.733032256967 \wedge x \leq -69.6386371545737$$
$$x \geq -65.4498469497874 \wedge x \leq -63.3554518473942$$
$$x \geq -59.1666616426078 \wedge x \leq -57.0722665402146$$
$$x \geq -52.8834763354282 \wedge x \leq -50.789081233035$$
$$x \geq -46.6002910282486 \wedge x \leq -44.5058959258554$$
$$x \geq -40.317105721069 \wedge x \leq -38.2227106186758$$
$$x \geq -34.0339204138894 \wedge x \leq -31.9395253114962$$
$$x \geq -27.7507351067098 \wedge x \leq -25.6563400043166$$
$$x \geq -21.4675497995303 \wedge x \leq -19.3731546971371$$
$$x \geq -15.1843644923507 \wedge x \leq -13.0899693899575$$
$$x \geq -8.90117918517108 \wedge x \leq -6.80678408277789$$
$$x \geq -2.61799387799149 \wedge x \leq -0.523598775598299$$
$$x \geq 3.66519142918809 \wedge x \leq 5.75958653158129$$
$$x \geq 9.94837673636768 \wedge x \leq 12.0427718387609$$
$$x \geq 16.2315620435473 \wedge x \leq 18.3259571459405$$
$$x \geq 22.5147473507269 \wedge x \leq 24.60914245312$$
$$x \geq 28.7979326579064 \wedge x \leq 30.8923277602996$$
$$x \geq 35.081117965086 \wedge x \leq 37.1755130674792$$
$$x \geq 41.3643032722656 \wedge x \leq 43.4586983746588$$
$$x \geq 47.6474885794452 \wedge x \leq 49.7418836818384$$
$$x \geq 53.9306738866248 \wedge x \leq 56.025068989018$$
$$x \geq 60.2138591938044 \wedge x \leq 62.3082542961976$$
$$x \geq 66.497044500984 \wedge x \leq 68.5914396033772$$
$$x \geq 72.7802298081635 \wedge x \leq 74.8746249105567$$
$$x \geq 79.0634151153431 \wedge x \leq 81.1578102177363$$
$$x \geq 85.3466004225227 \wedge x \leq 87.4409955249159$$
$$x \geq 91.6297857297023 \wedge x \leq 93.7241808320955$$
$$x \geq 97.9129710368819 \wedge x \leq 100.007366139275$$
$$x \geq 192.160750644576 \wedge x \leq 437.20497762458$$
$$x \geq 872.839158922364 \wedge x \leq 66437.8778393414$$
$$\left|{\sin{\left(x \right)} + 1}\right| \leq \frac{1}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left|{\sin{\left(x \right)} + 1}\right| = \frac{1}{2}$$
Resolvemos:
Tenemos la ecuación
$$\left|{\sin{\left(x \right)} + 1}\right| = \frac{1}{2}$$
cambiamos
$$\left|{\sin{\left(x \right)} + 1}\right| - \frac{1}{2} = 0$$
$$\left|{\sin{\left(x \right)} + 1}\right| - \frac{1}{2} = 0$$
Sustituimos
$$w = \left|{\sin{\left(x \right)} + 1}\right|$$
Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
$$w = \frac{1}{2}$$
Obtenemos la respuesta: w = 1/2
hacemos cambio inverso
$$\left|{\sin{\left(x \right)} + 1}\right| = w$$
sustituimos w:
$$x_{1} = 66.497044500984$$
$$x_{2} = 100.007366139275$$
$$x_{3} = -90.5825881785057$$
$$x_{4} = -13.0899693899575$$
$$x_{5} = 30.8923277602996$$
$$x_{6} = -63.3554518473942$$
$$x_{7} = -78.0162175641465$$
$$x_{8} = -69.6386371545737$$
$$x_{9} = 97.9129710368819$$
$$x_{10} = 41.3643032722656$$
$$x_{11} = -25.6563400043166$$
$$x_{12} = 79.0634151153431$$
$$x_{13} = 72.7802298081635$$
$$x_{14} = -2.61799387799149$$
$$x_{15} = 22.5147473507269$$
$$x_{16} = 12.0427718387609$$
$$x_{17} = -151.320046147908$$
$$x_{18} = 60.2138591938044$$
$$x_{19} = 18.3259571459405$$
$$x_{20} = 43.4586983746588$$
$$x_{21} = -31.9395253114962$$
$$x_{22} = -6.80678408277789$$
$$x_{23} = -34.0339204138894$$
$$x_{24} = -65.4498469497874$$
$$x_{25} = -46.6002910282486$$
$$x_{26} = 24.60914245312$$
$$x_{27} = 5.75958653158129$$
$$x_{28} = 53.9306738866248$$
$$x_{29} = -84.2994028713261$$
$$x_{30} = -71.733032256967$$
$$x_{31} = -195.302343298165$$
$$x_{32} = 192.160750644576$$
$$x_{33} = -57.0722665402146$$
$$x_{34} = -88.4881930761125$$
$$x_{35} = 35.081117965086$$
$$x_{36} = 66437.8778393414$$
$$x_{37} = 872.839158922364$$
$$x_{38} = 3.66519142918809$$
$$x_{39} = -21.4675497995303$$
$$x_{40} = 9.94837673636768$$
$$x_{41} = -50.789081233035$$
$$x_{42} = 16.2315620435473$$
$$x_{43} = 74.8746249105567$$
$$x_{44} = -59.1666616426078$$
$$x_{45} = 91.6297857297023$$
$$x_{46} = -101.054563690472$$
$$x_{47} = -75.9218224617533$$
$$x_{48} = -19.3731546971371$$
$$x_{49} = -15.1843644923507$$
$$x_{50} = 28.7979326579064$$
$$x_{51} = 93.7241808320955$$
$$x_{52} = -44.5058959258554$$
$$x_{53} = -8.90117918517108$$
$$x_{54} = -82.2050077689329$$
$$x_{55} = -0.523598775598299$$
$$x_{56} = 37.1755130674792$$
$$x_{57} = 85.3466004225227$$
$$x_{58} = -40.317105721069$$
$$x_{59} = -94.7713783832921$$
$$x_{60} = 437.20497762458$$
$$x_{61} = 47.6474885794452$$
$$x_{62} = 87.4409955249159$$
$$x_{63} = 62.3082542961976$$
$$x_{64} = 49.7418836818384$$
$$x_{65} = -27.7507351067098$$
$$x_{66} = -52.8834763354282$$
$$x_{67} = 56.025068989018$$
$$x_{68} = -38.2227106186758$$
$$x_{69} = -96.8657734856853$$
$$x_{70} = 81.1578102177363$$
$$x_{71} = 68.5914396033772$$
$$x_{1} = 66.497044500984$$
$$x_{2} = 100.007366139275$$
$$x_{3} = -90.5825881785057$$
$$x_{4} = -13.0899693899575$$
$$x_{5} = 30.8923277602996$$
$$x_{6} = -63.3554518473942$$
$$x_{7} = -78.0162175641465$$
$$x_{8} = -69.6386371545737$$
$$x_{9} = 97.9129710368819$$
$$x_{10} = 41.3643032722656$$
$$x_{11} = -25.6563400043166$$
$$x_{12} = 79.0634151153431$$
$$x_{13} = 72.7802298081635$$
$$x_{14} = -2.61799387799149$$
$$x_{15} = 22.5147473507269$$
$$x_{16} = 12.0427718387609$$
$$x_{17} = -151.320046147908$$
$$x_{18} = 60.2138591938044$$
$$x_{19} = 18.3259571459405$$
$$x_{20} = 43.4586983746588$$
$$x_{21} = -31.9395253114962$$
$$x_{22} = -6.80678408277789$$
$$x_{23} = -34.0339204138894$$
$$x_{24} = -65.4498469497874$$
$$x_{25} = -46.6002910282486$$
$$x_{26} = 24.60914245312$$
$$x_{27} = 5.75958653158129$$
$$x_{28} = 53.9306738866248$$
$$x_{29} = -84.2994028713261$$
$$x_{30} = -71.733032256967$$
$$x_{31} = -195.302343298165$$
$$x_{32} = 192.160750644576$$
$$x_{33} = -57.0722665402146$$
$$x_{34} = -88.4881930761125$$
$$x_{35} = 35.081117965086$$
$$x_{36} = 66437.8778393414$$
$$x_{37} = 872.839158922364$$
$$x_{38} = 3.66519142918809$$
$$x_{39} = -21.4675497995303$$
$$x_{40} = 9.94837673636768$$
$$x_{41} = -50.789081233035$$
$$x_{42} = 16.2315620435473$$
$$x_{43} = 74.8746249105567$$
$$x_{44} = -59.1666616426078$$
$$x_{45} = 91.6297857297023$$
$$x_{46} = -101.054563690472$$
$$x_{47} = -75.9218224617533$$
$$x_{48} = -19.3731546971371$$
$$x_{49} = -15.1843644923507$$
$$x_{50} = 28.7979326579064$$
$$x_{51} = 93.7241808320955$$
$$x_{52} = -44.5058959258554$$
$$x_{53} = -8.90117918517108$$
$$x_{54} = -82.2050077689329$$
$$x_{55} = -0.523598775598299$$
$$x_{56} = 37.1755130674792$$
$$x_{57} = 85.3466004225227$$
$$x_{58} = -40.317105721069$$
$$x_{59} = -94.7713783832921$$
$$x_{60} = 437.20497762458$$
$$x_{61} = 47.6474885794452$$
$$x_{62} = 87.4409955249159$$
$$x_{63} = 62.3082542961976$$
$$x_{64} = 49.7418836818384$$
$$x_{65} = -27.7507351067098$$
$$x_{66} = -52.8834763354282$$
$$x_{67} = 56.025068989018$$
$$x_{68} = -38.2227106186758$$
$$x_{69} = -96.8657734856853$$
$$x_{70} = 81.1578102177363$$
$$x_{71} = 68.5914396033772$$
Las raíces dadas
$$x_{31} = -195.302343298165$$
$$x_{17} = -151.320046147908$$
$$x_{46} = -101.054563690472$$
$$x_{69} = -96.8657734856853$$
$$x_{59} = -94.7713783832921$$
$$x_{3} = -90.5825881785057$$
$$x_{34} = -88.4881930761125$$
$$x_{29} = -84.2994028713261$$
$$x_{54} = -82.2050077689329$$
$$x_{7} = -78.0162175641465$$
$$x_{47} = -75.9218224617533$$
$$x_{30} = -71.733032256967$$
$$x_{8} = -69.6386371545737$$
$$x_{24} = -65.4498469497874$$
$$x_{6} = -63.3554518473942$$
$$x_{44} = -59.1666616426078$$
$$x_{33} = -57.0722665402146$$
$$x_{66} = -52.8834763354282$$
$$x_{41} = -50.789081233035$$
$$x_{25} = -46.6002910282486$$
$$x_{52} = -44.5058959258554$$
$$x_{58} = -40.317105721069$$
$$x_{68} = -38.2227106186758$$
$$x_{23} = -34.0339204138894$$
$$x_{21} = -31.9395253114962$$
$$x_{65} = -27.7507351067098$$
$$x_{11} = -25.6563400043166$$
$$x_{39} = -21.4675497995303$$
$$x_{48} = -19.3731546971371$$
$$x_{49} = -15.1843644923507$$
$$x_{4} = -13.0899693899575$$
$$x_{53} = -8.90117918517108$$
$$x_{22} = -6.80678408277789$$
$$x_{14} = -2.61799387799149$$
$$x_{55} = -0.523598775598299$$
$$x_{38} = 3.66519142918809$$
$$x_{27} = 5.75958653158129$$
$$x_{40} = 9.94837673636768$$
$$x_{16} = 12.0427718387609$$
$$x_{42} = 16.2315620435473$$
$$x_{19} = 18.3259571459405$$
$$x_{15} = 22.5147473507269$$
$$x_{26} = 24.60914245312$$
$$x_{50} = 28.7979326579064$$
$$x_{5} = 30.8923277602996$$
$$x_{35} = 35.081117965086$$
$$x_{56} = 37.1755130674792$$
$$x_{10} = 41.3643032722656$$
$$x_{20} = 43.4586983746588$$
$$x_{61} = 47.6474885794452$$
$$x_{64} = 49.7418836818384$$
$$x_{28} = 53.9306738866248$$
$$x_{67} = 56.025068989018$$
$$x_{18} = 60.2138591938044$$
$$x_{63} = 62.3082542961976$$
$$x_{1} = 66.497044500984$$
$$x_{71} = 68.5914396033772$$
$$x_{13} = 72.7802298081635$$
$$x_{43} = 74.8746249105567$$
$$x_{12} = 79.0634151153431$$
$$x_{70} = 81.1578102177363$$
$$x_{57} = 85.3466004225227$$
$$x_{62} = 87.4409955249159$$
$$x_{45} = 91.6297857297023$$
$$x_{51} = 93.7241808320955$$
$$x_{9} = 97.9129710368819$$
$$x_{2} = 100.007366139275$$
$$x_{32} = 192.160750644576$$
$$x_{60} = 437.20497762458$$
$$x_{37} = 872.839158922364$$
$$x_{36} = 66437.8778393414$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{31}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{31} - \frac{1}{10}$$
=
$$-195.302343298165 + - \frac{1}{10}$$
=
$$-195.402343298165$$
lo sustituimos en la expresión
$$\left|{\sin{\left(x \right)} + 1}\right| \leq \frac{1}{2}$$
$$\left|{\sin{\left(-195.402343298165 \right)} + 1}\right| \leq \frac{1}{2}$$
0.416039642398240 <= 1/2
significa que una de las soluciones de nuestra ecuación será con:
$$x \leq -195.302343298165$$
_____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------
x31 x17 x46 x69 x59 x3 x34 x29 x54 x7 x47 x30 x8 x24 x6 x44 x33 x66 x41 x25 x52 x58 x68 x23 x21 x65 x11 x39 x48 x49 x4 x53 x22 x14 x55 x38 x27 x40 x16 x42 x19 x15 x26 x50 x5 x35 x56 x10 x20 x61 x64 x28 x67 x18 x63 x1 x71 x13 x43 x12 x70 x57 x62 x45 x51 x9 x2 x32 x60 x37 x36Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x \leq -195.302343298165$$
$$x \geq -151.320046147908 \wedge x \leq -101.054563690472$$
$$x \geq -96.8657734856853 \wedge x \leq -94.7713783832921$$
$$x \geq -90.5825881785057 \wedge x \leq -88.4881930761125$$
$$x \geq -84.2994028713261 \wedge x \leq -82.2050077689329$$
$$x \geq -78.0162175641465 \wedge x \leq -75.9218224617533$$
$$x \geq -71.733032256967 \wedge x \leq -69.6386371545737$$
$$x \geq -65.4498469497874 \wedge x \leq -63.3554518473942$$
$$x \geq -59.1666616426078 \wedge x \leq -57.0722665402146$$
$$x \geq -52.8834763354282 \wedge x \leq -50.789081233035$$
$$x \geq -46.6002910282486 \wedge x \leq -44.5058959258554$$
$$x \geq -40.317105721069 \wedge x \leq -38.2227106186758$$
$$x \geq -34.0339204138894 \wedge x \leq -31.9395253114962$$
$$x \geq -27.7507351067098 \wedge x \leq -25.6563400043166$$
$$x \geq -21.4675497995303 \wedge x \leq -19.3731546971371$$
$$x \geq -15.1843644923507 \wedge x \leq -13.0899693899575$$
$$x \geq -8.90117918517108 \wedge x \leq -6.80678408277789$$
$$x \geq -2.61799387799149 \wedge x \leq -0.523598775598299$$
$$x \geq 3.66519142918809 \wedge x \leq 5.75958653158129$$
$$x \geq 9.94837673636768 \wedge x \leq 12.0427718387609$$
$$x \geq 16.2315620435473 \wedge x \leq 18.3259571459405$$
$$x \geq 22.5147473507269 \wedge x \leq 24.60914245312$$
$$x \geq 28.7979326579064 \wedge x \leq 30.8923277602996$$
$$x \geq 35.081117965086 \wedge x \leq 37.1755130674792$$
$$x \geq 41.3643032722656 \wedge x \leq 43.4586983746588$$
$$x \geq 47.6474885794452 \wedge x \leq 49.7418836818384$$
$$x \geq 53.9306738866248 \wedge x \leq 56.025068989018$$
$$x \geq 60.2138591938044 \wedge x \leq 62.3082542961976$$
$$x \geq 66.497044500984 \wedge x \leq 68.5914396033772$$
$$x \geq 72.7802298081635 \wedge x \leq 74.8746249105567$$
$$x \geq 79.0634151153431 \wedge x \leq 81.1578102177363$$
$$x \geq 85.3466004225227 \wedge x \leq 87.4409955249159$$
$$x \geq 91.6297857297023 \wedge x \leq 93.7241808320955$$
$$x \geq 97.9129710368819 \wedge x \leq 100.007366139275$$
$$x \geq 192.160750644576 \wedge x \leq 437.20497762458$$
$$x \geq 872.839158922364 \wedge x \leq 66437.8778393414$$
Solución de la desigualdad en el gráfico