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|1+sinx|<=1/2 desigualdades

En la desigualdad la incógnita

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
$$\left|{\sin{\left(x \right)} + 1}\right| \leq \frac{1}{2}$$
$$\left|{\sin{\left(-195.402343298165 \right)} + 1}\right| \leq \frac{1}{2}$$
0.416039642398240 <= 1/2

significa que una de las soluciones de nuestra ecuación será con:
$$x \leq -195.302343298165$$
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Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x \leq -195.302343298165$$
$$x \geq -151.320046147908 \wedge x \leq -101.054563690472$$
$$x \geq -96.8657734856853 \wedge x \leq -94.7713783832921$$
$$x \geq -90.5825881785057 \wedge x \leq -88.4881930761125$$
$$x \geq -84.2994028713261 \wedge x \leq -82.2050077689329$$
$$x \geq -78.0162175641465 \wedge x \leq -75.9218224617533$$
$$x \geq -71.733032256967 \wedge x \leq -69.6386371545737$$
$$x \geq -65.4498469497874 \wedge x \leq -63.3554518473942$$
$$x \geq -59.1666616426078 \wedge x \leq -57.0722665402146$$
$$x \geq -52.8834763354282 \wedge x \leq -50.789081233035$$
$$x \geq -46.6002910282486 \wedge x \leq -44.5058959258554$$
$$x \geq -40.317105721069 \wedge x \leq -38.2227106186758$$
$$x \geq -34.0339204138894 \wedge x \leq -31.9395253114962$$
$$x \geq -27.7507351067098 \wedge x \leq -25.6563400043166$$
$$x \geq -21.4675497995303 \wedge x \leq -19.3731546971371$$
$$x \geq -15.1843644923507 \wedge x \leq -13.0899693899575$$
$$x \geq -8.90117918517108 \wedge x \leq -6.80678408277789$$
$$x \geq -2.61799387799149 \wedge x \leq -0.523598775598299$$
$$x \geq 3.66519142918809 \wedge x \leq 5.75958653158129$$
$$x \geq 9.94837673636768 \wedge x \leq 12.0427718387609$$
$$x \geq 16.2315620435473 \wedge x \leq 18.3259571459405$$
$$x \geq 22.5147473507269 \wedge x \leq 24.60914245312$$
$$x \geq 28.7979326579064 \wedge x \leq 30.8923277602996$$
$$x \geq 35.081117965086 \wedge x \leq 37.1755130674792$$
$$x \geq 41.3643032722656 \wedge x \leq 43.4586983746588$$
$$x \geq 47.6474885794452 \wedge x \leq 49.7418836818384$$
$$x \geq 53.9306738866248 \wedge x \leq 56.025068989018$$
$$x \geq 60.2138591938044 \wedge x \leq 62.3082542961976$$
$$x \geq 66.497044500984 \wedge x \leq 68.5914396033772$$
$$x \geq 72.7802298081635 \wedge x \leq 74.8746249105567$$
$$x \geq 79.0634151153431 \wedge x \leq 81.1578102177363$$
$$x \geq 85.3466004225227 \wedge x \leq 87.4409955249159$$
$$x \geq 91.6297857297023 \wedge x \leq 93.7241808320955$$
$$x \geq 97.9129710368819 \wedge x \leq 100.007366139275$$
$$x \geq 192.160750644576 \wedge x \leq 437.20497762458$$
$$x \geq 872.839158922364 \wedge x \leq 66437.8778393414$$
Solución de la desigualdad en el gráfico