Se da la desigualdad:
5cos(7p)−1<−1Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
5cos(7p)−1=−1Resolvemos:
Tenemos la ecuación
5cos(7p)−1=−1cambiamos
5cos(7p)−1=05cos(7p)−1=0Sustituimos
w=cos(7p)Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
5w=1Dividamos ambos miembros de la ecuación en 1/5
w = 1 / (1/5)
Obtenemos la respuesta: w = 5
hacemos cambio inverso
cos(7p)=wsustituimos w:
x1=−7.85398163397448x2=78.3154168644884x3=66.1978452006421x4=41.9627018729494x5=15.9323627432054x6=30.2939291596159x7=−29.845130209103x8=−15.9323627432054x9=85.9449990232065x10=28.0499344070517x11=−47.7970882296161x12=70.2370357552575x13=−81.905808468591x14=58.1194640914112x15=22.215548050385x16=−73.8274273593601x17=−35.6795165657698x18=−2.01959527730772x19=24.0107438524363x20=−21.7667490998721x21=50.0410829821803x22=−50.0410829821803x23=−94.0233801324374x24=−98.0625706870528x25=−24.0107438524363x26=−43.7578976750007x27=−85.9449990232065x28=−41.9627018729494x29=−6.05878583192317x30=72.0322315573088x31=−87.7401948252578x32=37.4747123678211x33=4.26359002987186x34=54.0802735367957x35=−89.9841895778219x36=0.224399475256414x37=67.9930410026934x38=96.2673748850015x39=52.2850777347444x40=44.2066966255135x41=48.245887180129x42=−55.875469338847x43=−39.7187071203852x44=2.01959527730772x45=26.2547386050004x46=88.1889937757706x47=−77.8666179139756x48=−99.8577664891041x49=−65.7490462501292x50=62.1586546460266x51=−72.0322315573088x52=56.7730672398727x53=−3.81479107935903x54=80.1106126665397x55=−13.6883679906412x56=−46.0018924275648x57=46.0018924275648x58=19.9715532978208x59=32.0891249616672x60=−80.5594116170526x61=−67.9930410026934x62=−28.0499344070517x63=98.0625706870528x64=3.81479107935903x65=10.0979763865386x66=−95.8185759344887x67=−91.7793853798732x68=−69.7882368047447x69=18.1763574957695x70=40.1675060708981x71=−83.7010042706423x72=6.05878583192317x73=8.30278058448731x74=−63.9538504480779x75=−61.7098556955138x76=74.276226309873x77=−32.53792391218x78=−76.0714221119243x79=100.306565439617x80=89.9841895778219x81=−33.8843207637185x82=76.0714221119243x83=−51.8362787842316x84=−19.9715532978208x85=−11.8931721885899x86=36.1283155162826x87=84.1498032211552x88=94.0233801324374x89=−25.8059396544876x90=92.2281843303861x91=−59.9146598934625x92=−17.7275585452567x93=−37.9235113183339x94=14.1371669411541x95=63.9538504480779x1=−7.85398163397448x2=78.3154168644884x3=66.1978452006421x4=41.9627018729494x5=15.9323627432054x6=30.2939291596159x7=−29.845130209103x8=−15.9323627432054x9=85.9449990232065x10=28.0499344070517x11=−47.7970882296161x12=70.2370357552575x13=−81.905808468591x14=58.1194640914112x15=22.215548050385x16=−73.8274273593601x17=−35.6795165657698x18=−2.01959527730772x19=24.0107438524363x20=−21.7667490998721x21=50.0410829821803x22=−50.0410829821803x23=−94.0233801324374x24=−98.0625706870528x25=−24.0107438524363x26=−43.7578976750007x27=−85.9449990232065x28=−41.9627018729494x29=−6.05878583192317x30=72.0322315573088x31=−87.7401948252578x32=37.4747123678211x33=4.26359002987186x34=54.0802735367957x35=−89.9841895778219x36=0.224399475256414x37=67.9930410026934x38=96.2673748850015x39=52.2850777347444x40=44.2066966255135x41=48.245887180129x42=−55.875469338847x43=−39.7187071203852x44=2.01959527730772x45=26.2547386050004x46=88.1889937757706x47=−77.8666179139756x48=−99.8577664891041x49=−65.7490462501292x50=62.1586546460266x51=−72.0322315573088x52=56.7730672398727x53=−3.81479107935903x54=80.1106126665397x55=−13.6883679906412x56=−46.0018924275648x57=46.0018924275648x58=19.9715532978208x59=32.0891249616672x60=−80.5594116170526x61=−67.9930410026934x62=−28.0499344070517x63=98.0625706870528x64=3.81479107935903x65=10.0979763865386x66=−95.8185759344887x67=−91.7793853798732x68=−69.7882368047447x69=18.1763574957695x70=40.1675060708981x71=−83.7010042706423x72=6.05878583192317x73=8.30278058448731x74=−63.9538504480779x75=−61.7098556955138x76=74.276226309873x77=−32.53792391218x78=−76.0714221119243x79=100.306565439617x80=89.9841895778219x81=−33.8843207637185x82=76.0714221119243x83=−51.8362787842316x84=−19.9715532978208x85=−11.8931721885899x86=36.1283155162826x87=84.1498032211552x88=94.0233801324374x89=−25.8059396544876x90=92.2281843303861x91=−59.9146598934625x92=−17.7275585452567x93=−37.9235113183339x94=14.1371669411541x95=63.9538504480779Las raíces dadas
x48=−99.8577664891041x24=−98.0625706870528x66=−95.8185759344887x23=−94.0233801324374x67=−91.7793853798732x35=−89.9841895778219x31=−87.7401948252578x27=−85.9449990232065x71=−83.7010042706423x13=−81.905808468591x60=−80.5594116170526x47=−77.8666179139756x78=−76.0714221119243x16=−73.8274273593601x51=−72.0322315573088x68=−69.7882368047447x61=−67.9930410026934x49=−65.7490462501292x74=−63.9538504480779x75=−61.7098556955138x91=−59.9146598934625x42=−55.875469338847x83=−51.8362787842316x22=−50.0410829821803x11=−47.7970882296161x56=−46.0018924275648x26=−43.7578976750007x28=−41.9627018729494x43=−39.7187071203852x93=−37.9235113183339x17=−35.6795165657698x81=−33.8843207637185x77=−32.53792391218x7=−29.845130209103x62=−28.0499344070517x89=−25.8059396544876x25=−24.0107438524363x20=−21.7667490998721x84=−19.9715532978208x92=−17.7275585452567x8=−15.9323627432054x55=−13.6883679906412x85=−11.8931721885899x1=−7.85398163397448x29=−6.05878583192317x53=−3.81479107935903x18=−2.01959527730772x36=0.224399475256414x44=2.01959527730772x64=3.81479107935903x33=4.26359002987186x72=6.05878583192317x73=8.30278058448731x65=10.0979763865386x94=14.1371669411541x5=15.9323627432054x69=18.1763574957695x58=19.9715532978208x15=22.215548050385x19=24.0107438524363x45=26.2547386050004x10=28.0499344070517x6=30.2939291596159x59=32.0891249616672x86=36.1283155162826x32=37.4747123678211x70=40.1675060708981x4=41.9627018729494x40=44.2066966255135x57=46.0018924275648x41=48.245887180129x21=50.0410829821803x39=52.2850777347444x34=54.0802735367957x52=56.7730672398727x14=58.1194640914112x50=62.1586546460266x95=63.9538504480779x3=66.1978452006421x37=67.9930410026934x12=70.2370357552575x30=72.0322315573088x76=74.276226309873x82=76.0714221119243x2=78.3154168644884x54=80.1106126665397x87=84.1498032211552x9=85.9449990232065x46=88.1889937757706x80=89.9841895778219x90=92.2281843303861x88=94.0233801324374x38=96.2673748850015x63=98.0625706870528x79=100.306565439617son 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=
−99.8577664891041+−101=
−99.9577664891041lo sustituimos en la expresión
5cos(7p)−1<−15cos(7p)−1<−1 cos(7*p)
-1 + -------- < -1
5 Entonces
x<−99.8577664891041no se cumple
significa que una de las soluciones de nuestra ecuación será con:
x>−99.8577664891041∧x<−98.0625706870528 _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
-------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-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------
x48 x24 x66 x23 x67 x35 x31 x27 x71 x13 x60 x47 x78 x16 x51 x68 x61 x49 x74 x75 x91 x42 x83 x22 x11 x56 x26 x28 x43 x93 x17 x81 x77 x7 x62 x89 x25 x20 x84 x92 x8 x55 x85 x1 x29 x53 x18 x36 x44 x64 x33 x72 x73 x65 x94 x5 x69 x58 x15 x19 x45 x10 x6 x59 x86 x32 x70 x4 x40 x57 x41 x21 x39 x34 x52 x14 x50 x95 x3 x37 x12 x30 x76 x82 x2 x54 x87 x9 x46 x80 x90 x88 x38 x63 x79Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x>−99.8577664891041∧x<−98.0625706870528x>−95.8185759344887∧x<−94.0233801324374x>−91.7793853798732∧x<−89.9841895778219x>−87.7401948252578∧x<−85.9449990232065x>−83.7010042706423∧x<−81.905808468591x>−80.5594116170526∧x<−77.8666179139756x>−76.0714221119243∧x<−73.8274273593601x>−72.0322315573088∧x<−69.7882368047447x>−67.9930410026934∧x<−65.7490462501292x>−63.9538504480779∧x<−61.7098556955138x>−59.9146598934625∧x<−55.875469338847x>−51.8362787842316∧x<−50.0410829821803x>−47.7970882296161∧x<−46.0018924275648x>−43.7578976750007∧x<−41.9627018729494x>−39.7187071203852∧x<−37.9235113183339x>−35.6795165657698∧x<−33.8843207637185x>−32.53792391218∧x<−29.845130209103x>−28.0499344070517∧x<−25.8059396544876x>−24.0107438524363∧x<−21.7667490998721x>−19.9715532978208∧x<−17.7275585452567x>−15.9323627432054∧x<−13.6883679906412x>−11.8931721885899∧x<−7.85398163397448x>−6.05878583192317∧x<−3.81479107935903x>−2.01959527730772∧x<0.224399475256414x>2.01959527730772∧x<3.81479107935903x>4.26359002987186∧x<6.05878583192317x>8.30278058448731∧x<10.0979763865386x>14.1371669411541∧x<15.9323627432054x>18.1763574957695∧x<19.9715532978208x>22.215548050385∧x<24.0107438524363x>26.2547386050004∧x<28.0499344070517x>30.2939291596159∧x<32.0891249616672x>36.1283155162826∧x<37.4747123678211x>40.1675060708981∧x<41.9627018729494x>44.2066966255135∧x<46.0018924275648x>48.245887180129∧x<50.0410829821803x>52.2850777347444∧x<54.0802735367957x>56.7730672398727∧x<58.1194640914112x>62.1586546460266∧x<63.9538504480779x>66.1978452006421∧x<67.9930410026934x>70.2370357552575∧x<72.0322315573088x>74.276226309873∧x<76.0714221119243x>78.3154168644884∧x<80.1106126665397x>84.1498032211552∧x<85.9449990232065x>88.1889937757706∧x<89.9841895778219x>92.2281843303861∧x<94.0233801324374x>96.2673748850015∧x<98.0625706870528x>100.306565439617
2x^2-6x+4<=0
18(x-2)+33>-7+16x
(x-1)*(x+1)<0
-1
-1