Se da la desigualdad:
∣sin(x)∣<21Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
∣sin(x)∣=21Resolvemos:
Tenemos la ecuación
∣sin(x)∣=21cambiamos
∣sin(x)∣−21=0∣sin(x)∣−21=0Sustituimos
w=∣sin(x)∣Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
w=21Obtenemos la respuesta: w = 1/2
hacemos cambio inverso
∣sin(x)∣=wsustituimos w:
x1=−27.7507351067098x2=−47.6474885794452x3=27.7507351067098x4=91.6297857297023x5=−31.9395253114962x6=34.0339204138894x7=−41.3643032722656x8=−5.75958653158129x9=31.9395253114962x10=96.8657734856853x11=40.317105721069x12=44.5058959258554x13=−46.6002910282486x14=16.2315620435473x15=−18.3259571459405x16=−100.007366139275x17=5.75958653158129x18=−57.0722665402146x19=−90.5825881785057x20=−78.0162175641465x21=−91.6297857297023x22=−60.2138591938044x23=−137.706477982353x24=−3.66519142918809x25=46.6002910282486x26=−34.0339204138894x27=71.733032256967x28=−69.6386371545737x29=−97.9129710368819x30=−12.0427718387609x31=60.2138591938044x32=−19.3731546971371x33=69.6386371545737x34=75.9218224617533x35=78.0162175641465x36=−38.2227106186758x37=−75.9218224617533x38=−25.6563400043166x39=3.66519142918809x40=−87.4409955249159x41=53.9306738866248x42=38.2227106186758x43=−21.4675497995303x44=−71.733032256967x45=2.61799387799149x46=12.0427718387609x47=24.60914245312x48=−382.750704962356x49=93.7241808320955x50=−63.3554518473942x51=−9.94837673636768x52=−24.60914245312x53=−16.2315620435473x54=9.94837673636768x55=−62.3082542961976x56=0.523598775598299x57=49.7418836818384x58=18.3259571459405x59=62.3082542961976x60=−56.025068989018x61=−49.7418836818384x62=47.6474885794452x63=97.9129710368819x64=84.2994028713261x65=56.025068989018x66=90.5825881785057x67=66.497044500984x68=−82.2050077689329x69=−40.317105721069x70=−93.7241808320955x71=68.5914396033772x72=−53.9306738866248x73=25.6563400043166x74=−85.3466004225227x75=100.007366139275x76=82.2050077689329x77=−2.61799387799149x78=−43.4586983746588x79=22.5147473507269x80=−84.2994028713261x81=−65.4498469497874x82=−68.5914396033772x83=88.4881930761125x84=−66.497044500984x1=−27.7507351067098x2=−47.6474885794452x3=27.7507351067098x4=91.6297857297023x5=−31.9395253114962x6=34.0339204138894x7=−41.3643032722656x8=−5.75958653158129x9=31.9395253114962x10=96.8657734856853x11=40.317105721069x12=44.5058959258554x13=−46.6002910282486x14=16.2315620435473x15=−18.3259571459405x16=−100.007366139275x17=5.75958653158129x18=−57.0722665402146x19=−90.5825881785057x20=−78.0162175641465x21=−91.6297857297023x22=−60.2138591938044x23=−137.706477982353x24=−3.66519142918809x25=46.6002910282486x26=−34.0339204138894x27=71.733032256967x28=−69.6386371545737x29=−97.9129710368819x30=−12.0427718387609x31=60.2138591938044x32=−19.3731546971371x33=69.6386371545737x34=75.9218224617533x35=78.0162175641465x36=−38.2227106186758x37=−75.9218224617533x38=−25.6563400043166x39=3.66519142918809x40=−87.4409955249159x41=53.9306738866248x42=38.2227106186758x43=−21.4675497995303x44=−71.733032256967x45=2.61799387799149x46=12.0427718387609x47=24.60914245312x48=−382.750704962356x49=93.7241808320955x50=−63.3554518473942x51=−9.94837673636768x52=−24.60914245312x53=−16.2315620435473x54=9.94837673636768x55=−62.3082542961976x56=0.523598775598299x57=49.7418836818384x58=18.3259571459405x59=62.3082542961976x60=−56.025068989018x61=−49.7418836818384x62=47.6474885794452x63=97.9129710368819x64=84.2994028713261x65=56.025068989018x66=90.5825881785057x67=66.497044500984x68=−82.2050077689329x69=−40.317105721069x70=−93.7241808320955x71=68.5914396033772x72=−53.9306738866248x73=25.6563400043166x74=−85.3466004225227x75=100.007366139275x76=82.2050077689329x77=−2.61799387799149x78=−43.4586983746588x79=22.5147473507269x80=−84.2994028713261x81=−65.4498469497874x82=−68.5914396033772x83=88.4881930761125x84=−66.497044500984Las raíces dadas
x48=−382.750704962356x23=−137.706477982353x16=−100.007366139275x29=−97.9129710368819x70=−93.7241808320955x21=−91.6297857297023x19=−90.5825881785057x40=−87.4409955249159x74=−85.3466004225227x80=−84.2994028713261x68=−82.2050077689329x20=−78.0162175641465x37=−75.9218224617533x44=−71.733032256967x28=−69.6386371545737x82=−68.5914396033772x84=−66.497044500984x81=−65.4498469497874x50=−63.3554518473942x55=−62.3082542961976x22=−60.2138591938044x18=−57.0722665402146x60=−56.025068989018x72=−53.9306738866248x61=−49.7418836818384x2=−47.6474885794452x13=−46.6002910282486x78=−43.4586983746588x7=−41.3643032722656x69=−40.317105721069x36=−38.2227106186758x26=−34.0339204138894x5=−31.9395253114962x1=−27.7507351067098x38=−25.6563400043166x52=−24.60914245312x43=−21.4675497995303x32=−19.3731546971371x15=−18.3259571459405x53=−16.2315620435473x30=−12.0427718387609x51=−9.94837673636768x8=−5.75958653158129x24=−3.66519142918809x77=−2.61799387799149x56=0.523598775598299x45=2.61799387799149x39=3.66519142918809x17=5.75958653158129x54=9.94837673636768x46=12.0427718387609x14=16.2315620435473x58=18.3259571459405x79=22.5147473507269x47=24.60914245312x73=25.6563400043166x3=27.7507351067098x9=31.9395253114962x6=34.0339204138894x42=38.2227106186758x11=40.317105721069x12=44.5058959258554x25=46.6002910282486x62=47.6474885794452x57=49.7418836818384x41=53.9306738866248x65=56.025068989018x31=60.2138591938044x59=62.3082542961976x67=66.497044500984x71=68.5914396033772x33=69.6386371545737x27=71.733032256967x34=75.9218224617533x35=78.0162175641465x76=82.2050077689329x64=84.2994028713261x83=88.4881930761125x66=90.5825881785057x4=91.6297857297023x49=93.7241808320955x10=96.8657734856853x63=97.9129710368819x75=100.007366139275son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x0<x48Consideremos, por ejemplo, el punto
x0=x48−101=
−382.750704962356+−101=
−382.850704962356lo sustituimos en la expresión
∣sin(x)∣<21∣sin(−382.850704962356)∣<210.411043807676265 < 1/2
significa que una de las soluciones de nuestra ecuación será con:
x<−382.750704962356 _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
-------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-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------
x48 x23 x16 x29 x70 x21 x19 x40 x74 x80 x68 x20 x37 x44 x28 x82 x84 x81 x50 x55 x22 x18 x60 x72 x61 x2 x13 x78 x7 x69 x36 x26 x5 x1 x38 x52 x43 x32 x15 x53 x30 x51 x8 x24 x77 x56 x45 x39 x17 x54 x46 x14 x58 x79 x47 x73 x3 x9 x6 x42 x11 x12 x25 x62 x57 x41 x65 x31 x59 x67 x71 x33 x27 x34 x35 x76 x64 x83 x66 x4 x49 x10 x63 x75Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x<−382.750704962356x>−137.706477982353∧x<−100.007366139275x>−97.9129710368819∧x<−93.7241808320955x>−91.6297857297023∧x<−90.5825881785057x>−87.4409955249159∧x<−85.3466004225227x>−84.2994028713261∧x<−82.2050077689329x>−78.0162175641465∧x<−75.9218224617533x>−71.733032256967∧x<−69.6386371545737x>−68.5914396033772∧x<−66.497044500984x>−65.4498469497874∧x<−63.3554518473942x>−62.3082542961976∧x<−60.2138591938044x>−57.0722665402146∧x<−56.025068989018x>−53.9306738866248∧x<−49.7418836818384x>−47.6474885794452∧x<−46.6002910282486x>−43.4586983746588∧x<−41.3643032722656x>−40.317105721069∧x<−38.2227106186758x>−34.0339204138894∧x<−31.9395253114962x>−27.7507351067098∧x<−25.6563400043166x>−24.60914245312∧x<−21.4675497995303x>−19.3731546971371∧x<−18.3259571459405x>−16.2315620435473∧x<−12.0427718387609x>−9.94837673636768∧x<−5.75958653158129x>−3.66519142918809∧x<−2.61799387799149x>0.523598775598299∧x<2.61799387799149x>3.66519142918809∧x<5.75958653158129x>9.94837673636768∧x<12.0427718387609x>16.2315620435473∧x<18.3259571459405x>22.5147473507269∧x<24.60914245312x>25.6563400043166∧x<27.7507351067098x>31.9395253114962∧x<34.0339204138894x>38.2227106186758∧x<40.317105721069x>44.5058959258554∧x<46.6002910282486x>47.6474885794452∧x<49.7418836818384x>53.9306738866248∧x<56.025068989018x>60.2138591938044∧x<62.3082542961976x>66.497044500984∧x<68.5914396033772x>69.6386371545737∧x<71.733032256967x>75.9218224617533∧x<78.0162175641465x>82.2050077689329∧x<84.2994028713261x>88.4881930761125∧x<90.5825881785057x>91.6297857297023∧x<93.7241808320955x>96.8657734856853∧x<97.9129710368819x>100.007366139275
x^2+15x>0
x^2+3x>0
x^2+15>0
1/2
1/2