Se da la desigualdad:
∣sin(x)+1∣>23Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
∣sin(x)+1∣=23Resolvemos:
x1=71.733032256967x2=78.0162175641465x3=−81.1578102177363x4=69.6386371545737x5=25.6563400043166x6=−79.0634151153431x7=−12.0427718387609x8=82.2050077689329x9=−60.2138591938044x10=101.054563690472x11=−24.60914245312x12=138.753675533549x13=−53.9306738866248x14=−97.9129710368819x15=−72.7802298081635x16=−18.3259571459405x17=−43.4586983746588x18=90.5825881785057x19=6.80678408277789x20=17438.4572213013x21=84.2994028713261x22=59.1666616426078x23=63.3554518473942x24=0.523598775598299x25=−56.025068989018x26=−16.2315620435473x27=38.2227106186758x28=−22.5147473507269x29=31.9395253114962x30=−37.1755130674792x31=134.564885328763x32=−30.8923277602996x33=40.317105721069x34=88.4881930761125x35=50.789081233035x36=13.0899693899575x37=−62.3082542961976x38=−28.7979326579064x39=−41.3643032722656x40=−74.8746249105567x41=44.5058959258554x42=−85.3466004225227x43=−47.6474885794452x44=94.7713783832921x45=−9.94837673636768x46=−91.6297857297023x47=46.6002910282486x48=−3.66519142918809x49=75.9218224617533x50=96.8657734856853x51=34.0339204138894x52=−5.75958653158129x53=−100.007366139275x54=−66.497044500984x55=−2650.98060085419x56=−627.79493194236x57=−49.7418836818384x58=−35.081117965086x59=52.8834763354282x60=2.61799387799149x61=19.3731546971371x62=−87.4409955249159x63=65.4498469497874x64=8.90117918517108x65=21.4675497995303x66=57.0722665402146x67=−93.7241808320955x68=−4454.25478401473x69=27.7507351067098x70=15.1843644923507x71=−68.5914396033772x1=71.733032256967x2=78.0162175641465x3=−81.1578102177363x4=69.6386371545737x5=25.6563400043166x6=−79.0634151153431x7=−12.0427718387609x8=82.2050077689329x9=−60.2138591938044x10=101.054563690472x11=−24.60914245312x12=138.753675533549x13=−53.9306738866248x14=−97.9129710368819x15=−72.7802298081635x16=−18.3259571459405x17=−43.4586983746588x18=90.5825881785057x19=6.80678408277789x20=17438.4572213013x21=84.2994028713261x22=59.1666616426078x23=63.3554518473942x24=0.523598775598299x25=−56.025068989018x26=−16.2315620435473x27=38.2227106186758x28=−22.5147473507269x29=31.9395253114962x30=−37.1755130674792x31=134.564885328763x32=−30.8923277602996x33=40.317105721069x34=88.4881930761125x35=50.789081233035x36=13.0899693899575x37=−62.3082542961976x38=−28.7979326579064x39=−41.3643032722656x40=−74.8746249105567x41=44.5058959258554x42=−85.3466004225227x43=−47.6474885794452x44=94.7713783832921x45=−9.94837673636768x46=−91.6297857297023x47=46.6002910282486x48=−3.66519142918809x49=75.9218224617533x50=96.8657734856853x51=34.0339204138894x52=−5.75958653158129x53=−100.007366139275x54=−66.497044500984x55=−2650.98060085419x56=−627.79493194236x57=−49.7418836818384x58=−35.081117965086x59=52.8834763354282x60=2.61799387799149x61=19.3731546971371x62=−87.4409955249159x63=65.4498469497874x64=8.90117918517108x65=21.4675497995303x66=57.0722665402146x67=−93.7241808320955x68=−4454.25478401473x69=27.7507351067098x70=15.1843644923507x71=−68.5914396033772Las raíces dadas
x68=−4454.25478401473x55=−2650.98060085419x56=−627.79493194236x53=−100.007366139275x14=−97.9129710368819x67=−93.7241808320955x46=−91.6297857297023x62=−87.4409955249159x42=−85.3466004225227x3=−81.1578102177363x6=−79.0634151153431x40=−74.8746249105567x15=−72.7802298081635x71=−68.5914396033772x54=−66.497044500984x37=−62.3082542961976x9=−60.2138591938044x25=−56.025068989018x13=−53.9306738866248x57=−49.7418836818384x43=−47.6474885794452x17=−43.4586983746588x39=−41.3643032722656x30=−37.1755130674792x58=−35.081117965086x32=−30.8923277602996x38=−28.7979326579064x11=−24.60914245312x28=−22.5147473507269x16=−18.3259571459405x26=−16.2315620435473x7=−12.0427718387609x45=−9.94837673636768x52=−5.75958653158129x48=−3.66519142918809x24=0.523598775598299x60=2.61799387799149x19=6.80678408277789x64=8.90117918517108x36=13.0899693899575x70=15.1843644923507x61=19.3731546971371x65=21.4675497995303x5=25.6563400043166x69=27.7507351067098x29=31.9395253114962x51=34.0339204138894x27=38.2227106186758x33=40.317105721069x41=44.5058959258554x47=46.6002910282486x35=50.789081233035x59=52.8834763354282x66=57.0722665402146x22=59.1666616426078x23=63.3554518473942x63=65.4498469497874x4=69.6386371545737x1=71.733032256967x49=75.9218224617533x2=78.0162175641465x8=82.2050077689329x21=84.2994028713261x34=88.4881930761125x18=90.5825881785057x44=94.7713783832921x50=96.8657734856853x10=101.054563690472x31=134.564885328763x12=138.753675533549x20=17438.4572213013son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x0<x68Consideremos, por ejemplo, el punto
x0=x68−101=
−4454.25478401473+−101=
−4454.35478401473lo sustituimos en la expresión
∣sin(x)+1∣>23∣sin(−4454.35478401473)+1∣>231.41104380767631 > 3/2
Entonces
x<−4454.25478401473no se cumple
significa que una de las soluciones de nuestra ecuación será con:
x>−4454.25478401473∧x<−2650.98060085419 _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------
x68 x55 x56 x53 x14 x67 x46 x62 x42 x3 x6 x40 x15 x71 x54 x37 x9 x25 x13 x57 x43 x17 x39 x30 x58 x32 x38 x11 x28 x16 x26 x7 x45 x52 x48 x24 x60 x19 x64 x36 x70 x61 x65 x5 x69 x29 x51 x27 x33 x41 x47 x35 x59 x66 x22 x23 x63 x4 x1 x49 x2 x8 x21 x34 x18 x44 x50 x10 x31 x12 x20Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x>−4454.25478401473∧x<−2650.98060085419x>−627.79493194236∧x<−100.007366139275x>−97.9129710368819∧x<−93.7241808320955x>−91.6297857297023∧x<−87.4409955249159x>−85.3466004225227∧x<−81.1578102177363x>−79.0634151153431∧x<−74.8746249105567x>−72.7802298081635∧x<−68.5914396033772x>−66.497044500984∧x<−62.3082542961976x>−60.2138591938044∧x<−56.025068989018x>−53.9306738866248∧x<−49.7418836818384x>−47.6474885794452∧x<−43.4586983746588x>−41.3643032722656∧x<−37.1755130674792x>−35.081117965086∧x<−30.8923277602996x>−28.7979326579064∧x<−24.60914245312x>−22.5147473507269∧x<−18.3259571459405x>−16.2315620435473∧x<−12.0427718387609x>−9.94837673636768∧x<−5.75958653158129x>−3.66519142918809∧x<0.523598775598299x>2.61799387799149∧x<6.80678408277789x>8.90117918517108∧x<13.0899693899575x>15.1843644923507∧x<19.3731546971371x>21.4675497995303∧x<25.6563400043166x>27.7507351067098∧x<31.9395253114962x>34.0339204138894∧x<38.2227106186758x>40.317105721069∧x<44.5058959258554x>46.6002910282486∧x<50.789081233035x>52.8834763354282∧x<57.0722665402146x>59.1666616426078∧x<63.3554518473942x>65.4498469497874∧x<69.6386371545737x>71.733032256967∧x<75.9218224617533x>78.0162175641465∧x<82.2050077689329x>84.2994028713261∧x<88.4881930761125x>90.5825881785057∧x<94.7713783832921x>96.8657734856853∧x<101.054563690472x>134.564885328763∧x<138.753675533549x>17438.4572213013
-x^2-2x+3>0
x^2>9
x^2-10x<0
(x^2-x-1)*(x^2-x-7)<-5