Se da la desigualdad:
∣cos(x)∣<21Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
∣cos(x)∣=21Resolvemos:
Tenemos la ecuación
∣cos(x)∣=21cambiamos
∣cos(x)∣−21=0∣cos(x)∣−21=0Sustituimos
w=∣cos(x)∣Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
w=21Obtenemos la respuesta: w = 1/2
hacemos cambio inverso
∣cos(x)∣=wsustituimos w:
x1=30.3687289847013x2=−83.7758040957278x3=−27.2271363311115x4=76.4454212373516x5=32.4631240870945x6=74.3510261349584x7=60.7374579694027x8=−5.23598775598299x9=4.18879020478639x10=−96.342174710087x11=−76.4454212373516x12=24.0855436775217x13=−55.5014702134197x14=−39.7935069454707x15=2.0943951023932x16=269.129770657526x17=83.7758040957278x18=−61.7846555205993x19=68.0678408277789x20=−35.6047167406843x21=−41.8879020478639x22=−85.870199198121x23=−79.5870138909414x24=3803.42150594604x25=16.7551608191456x26=52.3598775598299x27=77.4926187885482x28=−42.9350995990605x29=11.5191730631626x30=−30.3687289847013x31=54.4542726622231x32=85.870199198121x33=−32.4631240870945x34=−49.2182849062401x35=−24.0855436775217x36=−48.1710873550435x37=−98.4365698124802x38=−673.348025419412x39=−68.0678408277789x40=96.342174710087x41=−11.5191730631626x42=39.7935069454707x43=−52.3598775598299x44=−17.8023583703422x45=82.7286065445312x46=55.5014702134197x47=38.7463093942741x48=99.4837673636768x49=92.1533845053006x50=−19.8967534727354x51=61.7846555205993x52=33.5103216382911x53=−57.5958653158129x54=−2.0943951023932x55=−26.1799387799149x56=−70.162235930172x57=−60.7374579694027x58=46.0766922526503x59=17.8023583703422x60=−90.0589894029074x61=48.1710873550435x62=41.8879020478639x63=10.471975511966x64=70.162235930172x65=−92.1533845053006x66=−54.4542726622231x67=275.412955964705x68=−13.6135681655558x69=98.4365698124802x70=90.0589894029074x71=−99.4837673636768x72=−63.8790506229925x73=19.8967534727354x74=−10.471975511966x75=−93.2005820564972x76=−33.5103216382911x77=−4.18879020478639x78=63.8790506229925x79=−74.3510261349584x80=−46.0766922526503x81=8.37758040957278x82=−71.2094334813686x83=26.1799387799149x84=−77.4926187885482x1=30.3687289847013x2=−83.7758040957278x3=−27.2271363311115x4=76.4454212373516x5=32.4631240870945x6=74.3510261349584x7=60.7374579694027x8=−5.23598775598299x9=4.18879020478639x10=−96.342174710087x11=−76.4454212373516x12=24.0855436775217x13=−55.5014702134197x14=−39.7935069454707x15=2.0943951023932x16=269.129770657526x17=83.7758040957278x18=−61.7846555205993x19=68.0678408277789x20=−35.6047167406843x21=−41.8879020478639x22=−85.870199198121x23=−79.5870138909414x24=3803.42150594604x25=16.7551608191456x26=52.3598775598299x27=77.4926187885482x28=−42.9350995990605x29=11.5191730631626x30=−30.3687289847013x31=54.4542726622231x32=85.870199198121x33=−32.4631240870945x34=−49.2182849062401x35=−24.0855436775217x36=−48.1710873550435x37=−98.4365698124802x38=−673.348025419412x39=−68.0678408277789x40=96.342174710087x41=−11.5191730631626x42=39.7935069454707x43=−52.3598775598299x44=−17.8023583703422x45=82.7286065445312x46=55.5014702134197x47=38.7463093942741x48=99.4837673636768x49=92.1533845053006x50=−19.8967534727354x51=61.7846555205993x52=33.5103216382911x53=−57.5958653158129x54=−2.0943951023932x55=−26.1799387799149x56=−70.162235930172x57=−60.7374579694027x58=46.0766922526503x59=17.8023583703422x60=−90.0589894029074x61=48.1710873550435x62=41.8879020478639x63=10.471975511966x64=70.162235930172x65=−92.1533845053006x66=−54.4542726622231x67=275.412955964705x68=−13.6135681655558x69=98.4365698124802x70=90.0589894029074x71=−99.4837673636768x72=−63.8790506229925x73=19.8967534727354x74=−10.471975511966x75=−93.2005820564972x76=−33.5103216382911x77=−4.18879020478639x78=63.8790506229925x79=−74.3510261349584x80=−46.0766922526503x81=8.37758040957278x82=−71.2094334813686x83=26.1799387799149x84=−77.4926187885482Las raíces dadas
x38=−673.348025419412x71=−99.4837673636768x37=−98.4365698124802x10=−96.342174710087x75=−93.2005820564972x65=−92.1533845053006x60=−90.0589894029074x22=−85.870199198121x2=−83.7758040957278x23=−79.5870138909414x84=−77.4926187885482x11=−76.4454212373516x79=−74.3510261349584x82=−71.2094334813686x56=−70.162235930172x39=−68.0678408277789x72=−63.8790506229925x18=−61.7846555205993x57=−60.7374579694027x53=−57.5958653158129x13=−55.5014702134197x66=−54.4542726622231x43=−52.3598775598299x34=−49.2182849062401x36=−48.1710873550435x80=−46.0766922526503x28=−42.9350995990605x21=−41.8879020478639x14=−39.7935069454707x20=−35.6047167406843x76=−33.5103216382911x33=−32.4631240870945x30=−30.3687289847013x3=−27.2271363311115x55=−26.1799387799149x35=−24.0855436775217x50=−19.8967534727354x44=−17.8023583703422x68=−13.6135681655558x41=−11.5191730631626x74=−10.471975511966x8=−5.23598775598299x77=−4.18879020478639x54=−2.0943951023932x15=2.0943951023932x9=4.18879020478639x81=8.37758040957278x63=10.471975511966x29=11.5191730631626x25=16.7551608191456x59=17.8023583703422x73=19.8967534727354x12=24.0855436775217x83=26.1799387799149x1=30.3687289847013x5=32.4631240870945x52=33.5103216382911x47=38.7463093942741x42=39.7935069454707x62=41.8879020478639x58=46.0766922526503x61=48.1710873550435x26=52.3598775598299x31=54.4542726622231x46=55.5014702134197x7=60.7374579694027x51=61.7846555205993x78=63.8790506229925x19=68.0678408277789x64=70.162235930172x6=74.3510261349584x4=76.4454212373516x27=77.4926187885482x45=82.7286065445312x17=83.7758040957278x32=85.870199198121x70=90.0589894029074x49=92.1533845053006x40=96.342174710087x69=98.4365698124802x48=99.4837673636768x16=269.129770657526x67=275.412955964705x24=3803.42150594604son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x0<x38Consideremos, por ejemplo, el punto
x0=x38−101=
−673.348025419412+−101=
−673.448025419412lo sustituimos en la expresión
∣cos(x)∣<21∣cos(−673.448025419412)∣<210.411043807676242 < 1/2
significa que una de las soluciones de nuestra ecuación será con:
x<−673.348025419412 _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
-------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-------
x38 x71 x37 x10 x75 x65 x60 x22 x2 x23 x84 x11 x79 x82 x56 x39 x72 x18 x57 x53 x13 x66 x43 x34 x36 x80 x28 x21 x14 x20 x76 x33 x30 x3 x55 x35 x50 x44 x68 x41 x74 x8 x77 x54 x15 x9 x81 x63 x29 x25 x59 x73 x12 x83 x1 x5 x52 x47 x42 x62 x58 x61 x26 x31 x46 x7 x51 x78 x19 x64 x6 x4 x27 x45 x17 x32 x70 x49 x40 x69 x48 x16 x67 x24Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x<−673.348025419412x>−99.4837673636768∧x<−98.4365698124802x>−96.342174710087∧x<−93.2005820564972x>−92.1533845053006∧x<−90.0589894029074x>−85.870199198121∧x<−83.7758040957278x>−79.5870138909414∧x<−77.4926187885482x>−76.4454212373516∧x<−74.3510261349584x>−71.2094334813686∧x<−70.162235930172x>−68.0678408277789∧x<−63.8790506229925x>−61.7846555205993∧x<−60.7374579694027x>−57.5958653158129∧x<−55.5014702134197x>−54.4542726622231∧x<−52.3598775598299x>−49.2182849062401∧x<−48.1710873550435x>−46.0766922526503∧x<−42.9350995990605x>−41.8879020478639∧x<−39.7935069454707x>−35.6047167406843∧x<−33.5103216382911x>−32.4631240870945∧x<−30.3687289847013x>−27.2271363311115∧x<−26.1799387799149x>−24.0855436775217∧x<−19.8967534727354x>−17.8023583703422∧x<−13.6135681655558x>−11.5191730631626∧x<−10.471975511966x>−5.23598775598299∧x<−4.18879020478639x>−2.0943951023932∧x<2.0943951023932x>4.18879020478639∧x<8.37758040957278x>10.471975511966∧x<11.5191730631626x>16.7551608191456∧x<17.8023583703422x>19.8967534727354∧x<24.0855436775217x>26.1799387799149∧x<30.3687289847013x>32.4631240870945∧x<33.5103216382911x>38.7463093942741∧x<39.7935069454707x>41.8879020478639∧x<46.0766922526503x>48.1710873550435∧x<52.3598775598299x>54.4542726622231∧x<55.5014702134197x>60.7374579694027∧x<61.7846555205993x>63.8790506229925∧x<68.0678408277789x>70.162235930172∧x<74.3510261349584x>76.4454212373516∧x<77.4926187885482x>82.7286065445312∧x<83.7758040957278x>85.870199198121∧x<90.0589894029074x>92.1533845053006∧x<96.342174710087x>98.4365698124802∧x<99.4837673636768x>269.129770657526∧x<275.412955964705x>3803.42150594604
-x^2+3x-2<0
x^2+4x-5<0
x+1>0
x^2-6x-27>0
(1/2)
(1/2)