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(3-log2(x))/1+cosx>=0 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src]
    log(x)              
3 - ------              
    log(2)              
---------- + cos(x) >= 0
    1                   
log(x)log(2)+31+cos(x)0\frac{- \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + 3}{1} + \cos{\left(x \right)} \geq 0
(-log(x)/log(2) + 3)/1 + cos(x) >= 0
Solución detallada
Se da la desigualdad:
log(x)log(2)+31+cos(x)0\frac{- \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + 3}{1} + \cos{\left(x \right)} \geq 0
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
log(x)log(2)+31+cos(x)=0\frac{- \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + 3}{1} + \cos{\left(x \right)} = 0
Resolvemos:
x1=20.1295538002964+2.22342128297618ix_{1} = -20.1295538002964 + 2.22342128297618 i
x2=82.6160711927159+2.42061188114759ix_{2} = -82.6160711927159 + 2.42061188114759 i
x3=63.8207892240727+2.38083269168861ix_{3} = -63.8207892240727 + 2.38083269168861 i
x4=13.312007139712x_{4} = 13.312007139712
x5=7.87643984391822x_{5} = 7.87643984391822
x6=13.9536940007645+2.18341941731657ix_{6} = -13.9536940007645 + 2.18341941731657 i
x7=51.3034446770698+2.34812577532967ix_{7} = -51.3034446770698 + 2.34812577532967 i
x8=1.975702615194242.01480012789324ix_{8} = -1.97570261519424 - 2.01480012789324 i
x9=1.97570261519424+2.01480012789324ix_{9} = -1.97570261519424 + 2.01480012789324 i
x10=11.5546483277425x_{10} = 11.5546483277425
x11=13.95369400076452.18341941731657ix_{11} = -13.9536940007645 - 2.18341941731657 i
x12=7.848431922648562.13046917744445ix_{12} = -7.84843192264856 - 2.13046917744445 i
x13=7.84843192264856+2.13046917744445ix_{13} = -7.84843192264856 + 2.13046917744445 i
x14=3.52493788165728+0.687640014989386ix_{14} = 3.52493788165728 + 0.687640014989386 i
Descartamos las soluciones complejas:
x1=13.312007139712x_{1} = 13.312007139712
x2=7.87643984391822x_{2} = 7.87643984391822
x3=11.5546483277425x_{3} = 11.5546483277425
Las raíces dadas
x2=7.87643984391822x_{2} = 7.87643984391822
x3=11.5546483277425x_{3} = 11.5546483277425
x1=13.312007139712x_{1} = 13.312007139712
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x0x2x_{0} \leq x_{2}
Consideremos, por ejemplo, el punto
x0=x2110x_{0} = x_{2} - \frac{1}{10}
=
110+7.87643984391822- \frac{1}{10} + 7.87643984391822
=
7.776439843918227.77643984391822
lo sustituimos en la expresión
log(x)log(2)+31+cos(x)0\frac{- \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + 3}{1} + \cos{\left(x \right)} \geq 0
log(7.77643984391822)log(2)+31+cos(7.77643984391822)0\frac{- \frac{\log{\left(7.77643984391822 \right)}}{\log{\left(2 \right)}} + 3}{1} + \cos{\left(7.77643984391822 \right)} \geq 0
                   2.05109862984833     
3.07746410711665 - ---------------- >= 0
                        log(2)          

significa que una de las soluciones de nuestra ecuación será con:
x7.87643984391822x \leq 7.87643984391822
 _____           _____          
      \         /     \    
-------•-------•-------•-------
       x2      x3      x1

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x7.87643984391822x \leq 7.87643984391822
x11.5546483277425x13.312007139712x \geq 11.5546483277425 \wedge x \leq 13.312007139712
Solución de la desigualdad en el gráfico
02468-8-6-4-2-1010-2020