Se da la desigualdad:
∣sin(x)∣≤0Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
∣sin(x)∣=0Resolvemos:
Tenemos la ecuación
∣sin(x)∣=0cambiamos
∣sin(x)∣−1=0∣sin(x)∣−1=0Sustituimos
w=∣sin(x)∣Transportamos los términos libres (sin w)
del miembro izquierdo al derecho, obtenemos:
w=1Obtenemos la respuesta: w = 1
hacemos cambio inverso
∣sin(x)∣=wsustituimos w:
x1=72.2566310325652x2=−59.6902604182061x3=81.6814089933346x4=−43.9822971502571x5=−100.530964914873x6=−427.256600888212x7=28.2743338823081x8=65.9734457253857x9=−31.4159265358979x10=−9.42477796076938x11=40.8407044966673x12=56.5486677646163x13=−56.5486677646163x14=12.5663706143592x15=43.9822971502571x16=100.530964914873x17=650.309679293087x18=−3.14159265358979x19=−15.707963267949x20=59.6902604182061x21=6.28318530717959x22=9.42477796076938x23=−53.4070751110265x24=−47.1238898038469x25=−87.9645943005142x26=21.9911485751286x27=87.9645943005142x28=18.8495559215388x29=−25.1327412287183x30=−72.2566310325652x31=37.6991118430775x32=0x33=50.2654824574367x34=−6.28318530717959x35=−65.9734457253857x36=−21.9911485751286x37=75.398223686155x38=84.8230016469244x39=34.5575191894877x40=53.4070751110265x41=15.707963267949x42=−28.2743338823081x43=−285.884931476671x44=−91.106186954104x45=97.3893722612836x46=−69.1150383789755x47=94.2477796076938x48=−50.2654824574367x49=−37.6991118430775x50=−81.6814089933346x51=−3760.48640634698x52=62.8318530717959x53=78.5398163397448x54=31.4159265358979x55=−78.5398163397448x56=−97.3893722612836x57=−75.398223686155x58=−12.5663706143592x59=−94.2477796076938x60=−34.5575191894877x1=72.2566310325652x2=−59.6902604182061x3=81.6814089933346x4=−43.9822971502571x5=−100.530964914873x6=−427.256600888212x7=28.2743338823081x8=65.9734457253857x9=−31.4159265358979x10=−9.42477796076938x11=40.8407044966673x12=56.5486677646163x13=−56.5486677646163x14=12.5663706143592x15=43.9822971502571x16=100.530964914873x17=650.309679293087x18=−3.14159265358979x19=−15.707963267949x20=59.6902604182061x21=6.28318530717959x22=9.42477796076938x23=−53.4070751110265x24=−47.1238898038469x25=−87.9645943005142x26=21.9911485751286x27=87.9645943005142x28=18.8495559215388x29=−25.1327412287183x30=−72.2566310325652x31=37.6991118430775x32=0x33=50.2654824574367x34=−6.28318530717959x35=−65.9734457253857x36=−21.9911485751286x37=75.398223686155x38=84.8230016469244x39=34.5575191894877x40=53.4070751110265x41=15.707963267949x42=−28.2743338823081x43=−285.884931476671x44=−91.106186954104x45=97.3893722612836x46=−69.1150383789755x47=94.2477796076938x48=−50.2654824574367x49=−37.6991118430775x50=−81.6814089933346x51=−3760.48640634698x52=62.8318530717959x53=78.5398163397448x54=31.4159265358979x55=−78.5398163397448x56=−97.3893722612836x57=−75.398223686155x58=−12.5663706143592x59=−94.2477796076938x60=−34.5575191894877Las raíces dadas
x51=−3760.48640634698x6=−427.256600888212x43=−285.884931476671x5=−100.530964914873x56=−97.3893722612836x59=−94.2477796076938x44=−91.106186954104x25=−87.9645943005142x50=−81.6814089933346x55=−78.5398163397448x57=−75.398223686155x30=−72.2566310325652x46=−69.1150383789755x35=−65.9734457253857x2=−59.6902604182061x13=−56.5486677646163x23=−53.4070751110265x48=−50.2654824574367x24=−47.1238898038469x4=−43.9822971502571x49=−37.6991118430775x60=−34.5575191894877x9=−31.4159265358979x42=−28.2743338823081x29=−25.1327412287183x36=−21.9911485751286x19=−15.707963267949x58=−12.5663706143592x10=−9.42477796076938x34=−6.28318530717959x18=−3.14159265358979x32=0x21=6.28318530717959x22=9.42477796076938x14=12.5663706143592x41=15.707963267949x28=18.8495559215388x26=21.9911485751286x7=28.2743338823081x54=31.4159265358979x39=34.5575191894877x31=37.6991118430775x11=40.8407044966673x15=43.9822971502571x33=50.2654824574367x40=53.4070751110265x12=56.5486677646163x20=59.6902604182061x52=62.8318530717959x8=65.9734457253857x1=72.2566310325652x37=75.398223686155x53=78.5398163397448x3=81.6814089933346x38=84.8230016469244x27=87.9645943005142x47=94.2477796076938x45=97.3893722612836x16=100.530964914873x17=650.309679293087son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x0≤x51Consideremos, por ejemplo, el punto
x0=x51−101=
−3760.48640634698+−101=
−3760.58640634698lo sustituimos en la expresión
∣sin(x)∣≤0∣sin(−3760.58640634698)∣≤00.0998334166469064 <= 0
pero
0.0998334166469064 >= 0
Entonces
x≤−3760.48640634698no se cumple
significa que una de las soluciones de nuestra ecuación será con:
x≥−3760.48640634698∧x≤−427.256600888212 _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------
x51 x6 x43 x5 x56 x59 x44 x25 x50 x55 x57 x30 x46 x35 x2 x13 x23 x48 x24 x4 x49 x60 x9 x42 x29 x36 x19 x58 x10 x34 x18 x32 x21 x22 x14 x41 x28 x26 x7 x54 x39 x31 x11 x15 x33 x40 x12 x20 x52 x8 x1 x37 x53 x3 x38 x27 x47 x45 x16 x17Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x≥−3760.48640634698∧x≤−427.256600888212x≥−285.884931476671∧x≤−100.530964914873x≥−97.3893722612836∧x≤−94.2477796076938x≥−91.106186954104∧x≤−87.9645943005142x≥−81.6814089933346∧x≤−78.5398163397448x≥−75.398223686155∧x≤−72.2566310325652x≥−69.1150383789755∧x≤−65.9734457253857x≥−59.6902604182061∧x≤−56.5486677646163x≥−53.4070751110265∧x≤−50.2654824574367x≥−47.1238898038469∧x≤−43.9822971502571x≥−37.6991118430775∧x≤−34.5575191894877x≥−31.4159265358979∧x≤−28.2743338823081x≥−25.1327412287183∧x≤−21.9911485751286x≥−15.707963267949∧x≤−12.5663706143592x≥−9.42477796076938∧x≤−6.28318530717959x≥−3.14159265358979∧x≤0x≥6.28318530717959∧x≤9.42477796076938x≥12.5663706143592∧x≤15.707963267949x≥18.8495559215388∧x≤21.9911485751286x≥28.2743338823081∧x≤31.4159265358979x≥34.5575191894877∧x≤37.6991118430775x≥40.8407044966673∧x≤43.9822971502571x≥50.2654824574367∧x≤53.4070751110265x≥56.5486677646163∧x≤59.6902604182061x≥62.8318530717959∧x≤65.9734457253857x≥72.2566310325652∧x≤75.398223686155x≥78.5398163397448∧x≤81.6814089933346x≥84.8230016469244∧x≤87.9645943005142x≥94.2477796076938∧x≤97.3893722612836x≥100.530964914873∧x≤650.309679293087
-3-5x<=x+3
(x-5,7)*(x-7,2)>0
(x-4)^2/(x-1)>0
x-4x^2/x-1>0