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- Sistemas de desigualdades paso a paso
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- Sistemas de ecuaciones paso a paso
- Solución de desigualdades trigonométricas en línea
- Resolución de desigualdades exponenciales en línea
- Resolución de desigualdades con un módulo en línea
-
¿Cómo usar?
- Desigualdades:
-
x^2>1
- (x-2)^(x^2-6x+8)>1
-
(x+1)>0
-
6^x^2-x-1*7^x-2>6
-
Expresiones idénticas
- log dos ^ dos (7x+ cuatro -x^ dos)+5log1/ dos (7x+ cuatro -x^2)+ cuatro > cero
- logaritmo de 2 al cuadrado (7x más 4 menos x al cuadrado ) más 5 logaritmo de 1 dividir por 2(7x más 4 menos x al cuadrado ) más 4 más 0
- logaritmo de dos en el grado dos (7x más cuatro menos x en el grado dos) más 5 logaritmo de 1 dividir por dos (7x más cuatro menos x al cuadrado ) más cuatro más cero
- log22(7x+4-x2)+5log1/2(7x+4-x2)+4>0
- log227x+4-x2+5log1/27x+4-x2+4>0
- log2²(7x+4-x²)+5log1/2(7x+4-x²)+4>0
- log2 en el grado 2(7x+4-x en el grado 2)+5log1/2(7x+4-x en el grado 2)+4>0
- log2^27x+4-x^2+5log1/27x+4-x^2+4>0
- log2^2(7x+4-x^2)+5log1 dividir por 2(7x+4-x^2)+4>0
-
Expresiones semejantes
- log2^2(7x+4+x^2)+5log1/2(7x+4-x^2)+4>0
- log2^2(7x-4-x^2)+5log1/2(7x+4-x^2)+4>0
- log2^2(7x+4-x^2)+5log1/2(7x+4+x^2)+4>0
- log2^2(7x+4-x^2)-5log1/2(7x+4-x^2)+4>0
- log2^2(7x+4-x^2)+5log1/2(7x+4-x^2)-4>0
- log2^2(7x+4-x^2)+5log1/2(7x-4-x^2)+4>0
log2^2(7x+4-x^2)+5log1/2(7x+4-x^2)+4>0 desigualdades
Solución
Ha introducido
[src]
2 / 2\ 5*log(1) / 2\
log (2)*\7*x + 4 - x / + --------*\7*x + 4 - x / + 4 > 0
2
$$\left(\frac{5 \log{\left(1 \right)}}{2} \left(- x^{2} + \left(7 x + 4\right)\right) + \left(- x^{2} + \left(7 x + 4\right)\right) \log{\left(2 \right)}^{2}\right) + 4 > 0$$
((5*log(1))/2)*(-x^2 + 7*x + 4) + (-x^2 + 7*x + 4)*log(2)^2 + 4 > 0
Solución detallada
Se da la desigualdad:
$$\left(\frac{5 \log{\left(1 \right)}}{2} \left(- x^{2} + \left(7 x + 4\right)\right) + \left(- x^{2} + \left(7 x + 4\right)\right) \log{\left(2 \right)}^{2}\right) + 4 > 0$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left(\frac{5 \log{\left(1 \right)}}{2} \left(- x^{2} + \left(7 x + 4\right)\right) + \left(- x^{2} + \left(7 x + 4\right)\right) \log{\left(2 \right)}^{2}\right) + 4 = 0$$
Resolvemos:
Abramos la expresión en la ecuación
$$\left(\frac{5 \log{\left(1 \right)}}{2} \left(- x^{2} + \left(7 x + 4\right)\right) + \left(- x^{2} + \left(7 x + 4\right)\right) \log{\left(2 \right)}^{2}\right) + 4 = 0$$
Obtenemos la ecuación cuadrática
$$- x^{2} \log{\left(2 \right)}^{2} + 7 x \log{\left(2 \right)}^{2} + 4 \log{\left(2 \right)}^{2} + 4 = 0$$
Es la ecuación de la forma
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = - \log{\left(2 \right)}^{2}$$
$$b = 7 \log{\left(2 \right)}^{2}$$
$$c = 4 \log{\left(2 \right)}^{2} + 4$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$x_{1} = - \frac{- 7 \log{\left(2 \right)}^{2} + \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}^{2}}$$
$$x_{2} = - \frac{- \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}} - 7 \log{\left(2 \right)}^{2}}{2 \log{\left(2 \right)}^{2}}$$
$$x_{1} = - \frac{- 7 \log{\left(2 \right)}^{2} + \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}^{2}}$$
$$x_{2} = - \frac{- \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}} - 7 \log{\left(2 \right)}^{2}}{2 \log{\left(2 \right)}^{2}}$$
$$x_{1} = - \frac{- 7 \log{\left(2 \right)}^{2} + \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}^{2}}$$
$$x_{2} = - \frac{- \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}} - 7 \log{\left(2 \right)}^{2}}{2 \log{\left(2 \right)}^{2}}$$
Las raíces dadas
$$x_{1} = - \frac{- 7 \log{\left(2 \right)}^{2} + \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}^{2}}$$
$$x_{2} = - \frac{- \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}} - 7 \log{\left(2 \right)}^{2}}{2 \log{\left(2 \right)}^{2}}$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{1}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{- 7 \log{\left(2 \right)}^{2} + \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}^{2}} + - \frac{1}{10}$$
=
$$- \frac{- 7 \log{\left(2 \right)}^{2} + \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}^{2}} - \frac{1}{10}$$
lo sustituimos en la expresión
$$\left(\frac{5 \log{\left(1 \right)}}{2} \left(- x^{2} + \left(7 x + 4\right)\right) + \left(- x^{2} + \left(7 x + 4\right)\right) \log{\left(2 \right)}^{2}\right) + 4 > 0$$
$$\left(\left(\left(7 \left(- \frac{- 7 \log{\left(2 \right)}^{2} + \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}^{2}} - \frac{1}{10}\right) + 4\right) - \left(- \frac{- 7 \log{\left(2 \right)}^{2} + \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}^{2}} - \frac{1}{10}\right)^{2}\right) \log{\left(2 \right)}^{2} + \frac{5 \log{\left(1 \right)}}{2} \left(\left(7 \left(- \frac{- 7 \log{\left(2 \right)}^{2} + \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}^{2}} - \frac{1}{10}\right) + 4\right) - \left(- \frac{- 7 \log{\left(2 \right)}^{2} + \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}^{2}} - \frac{1}{10}\right)^{2}\right)\right) + 4 > 0$$
Entonces
$$x < - \frac{- 7 \log{\left(2 \right)}^{2} + \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}^{2}}$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x > - \frac{- 7 \log{\left(2 \right)}^{2} + \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}^{2}} \wedge x < - \frac{- \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}} - 7 \log{\left(2 \right)}^{2}}{2 \log{\left(2 \right)}^{2}}$$
$$\left(\frac{5 \log{\left(1 \right)}}{2} \left(- x^{2} + \left(7 x + 4\right)\right) + \left(- x^{2} + \left(7 x + 4\right)\right) \log{\left(2 \right)}^{2}\right) + 4 > 0$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left(\frac{5 \log{\left(1 \right)}}{2} \left(- x^{2} + \left(7 x + 4\right)\right) + \left(- x^{2} + \left(7 x + 4\right)\right) \log{\left(2 \right)}^{2}\right) + 4 = 0$$
Resolvemos:
Abramos la expresión en la ecuación
$$\left(\frac{5 \log{\left(1 \right)}}{2} \left(- x^{2} + \left(7 x + 4\right)\right) + \left(- x^{2} + \left(7 x + 4\right)\right) \log{\left(2 \right)}^{2}\right) + 4 = 0$$
Obtenemos la ecuación cuadrática
$$- x^{2} \log{\left(2 \right)}^{2} + 7 x \log{\left(2 \right)}^{2} + 4 \log{\left(2 \right)}^{2} + 4 = 0$$
Es la ecuación de la forma
a*x^2 + b*x + c = 0
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = - \log{\left(2 \right)}^{2}$$
$$b = 7 \log{\left(2 \right)}^{2}$$
$$c = 4 \log{\left(2 \right)}^{2} + 4$$
, entonces
D = b^2 - 4 * a * c =
(7*log(2)^2)^2 - 4 * (-log(2)^2) * (4 + 4*log(2)^2) = 49*log(2)^4 + 4*log(2)^2*(4 + 4*log(2)^2)
Como D > 0 la ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = - \frac{- 7 \log{\left(2 \right)}^{2} + \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}^{2}}$$
$$x_{2} = - \frac{- \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}} - 7 \log{\left(2 \right)}^{2}}{2 \log{\left(2 \right)}^{2}}$$
$$x_{1} = - \frac{- 7 \log{\left(2 \right)}^{2} + \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}^{2}}$$
$$x_{2} = - \frac{- \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}} - 7 \log{\left(2 \right)}^{2}}{2 \log{\left(2 \right)}^{2}}$$
$$x_{1} = - \frac{- 7 \log{\left(2 \right)}^{2} + \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}^{2}}$$
$$x_{2} = - \frac{- \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}} - 7 \log{\left(2 \right)}^{2}}{2 \log{\left(2 \right)}^{2}}$$
Las raíces dadas
$$x_{1} = - \frac{- 7 \log{\left(2 \right)}^{2} + \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}^{2}}$$
$$x_{2} = - \frac{- \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}} - 7 \log{\left(2 \right)}^{2}}{2 \log{\left(2 \right)}^{2}}$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} < x_{1}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{- 7 \log{\left(2 \right)}^{2} + \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}^{2}} + - \frac{1}{10}$$
=
$$- \frac{- 7 \log{\left(2 \right)}^{2} + \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}^{2}} - \frac{1}{10}$$
lo sustituimos en la expresión
$$\left(\frac{5 \log{\left(1 \right)}}{2} \left(- x^{2} + \left(7 x + 4\right)\right) + \left(- x^{2} + \left(7 x + 4\right)\right) \log{\left(2 \right)}^{2}\right) + 4 > 0$$
$$\left(\left(\left(7 \left(- \frac{- 7 \log{\left(2 \right)}^{2} + \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}^{2}} - \frac{1}{10}\right) + 4\right) - \left(- \frac{- 7 \log{\left(2 \right)}^{2} + \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}^{2}} - \frac{1}{10}\right)^{2}\right) \log{\left(2 \right)}^{2} + \frac{5 \log{\left(1 \right)}}{2} \left(\left(7 \left(- \frac{- 7 \log{\left(2 \right)}^{2} + \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}^{2}} - \frac{1}{10}\right) + 4\right) - \left(- \frac{- 7 \log{\left(2 \right)}^{2} + \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}^{2}} - \frac{1}{10}\right)^{2}\right)\right) + 4 > 0$$
/ 2 \
| / ________________________________________ \ / ________________________________________ \|
| | / 4 2 / 2 \ 2 | | / 4 2 / 2 \ 2 ||
2 |33 | 1 \/ 49*log (2) + 4*log (2)*\4 + 4*log (2)/ - 7*log (2)| 7*\\/ 49*log (2) + 4*log (2)*\4 + 4*log (2)/ - 7*log (2)/| > 0
4 + log (2)*|-- - |- -- - -------------------------------------------------------| - -----------------------------------------------------------|
|10 | 10 2 | 2 |
\ \ 2*log (2) / 2*log (2) / Entonces
$$x < - \frac{- 7 \log{\left(2 \right)}^{2} + \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}^{2}}$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x > - \frac{- 7 \log{\left(2 \right)}^{2} + \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}^{2}} \wedge x < - \frac{- \sqrt{49 \log{\left(2 \right)}^{4} + 4 \left(4 \log{\left(2 \right)}^{2} + 4\right) \log{\left(2 \right)}^{2}} - 7 \log{\left(2 \right)}^{2}}{2 \log{\left(2 \right)}^{2}}$$
_____
/ \
-------ο-------ο-------
x1 x2
Solución de la desigualdad en el gráfico
Respuesta rápida 2
[src]
_________________ _________________
/ 2 / 2
7 \/ 16 + 65*log (2) 7 \/ 16 + 65*log (2)
(- - --------------------, - + --------------------)
2 2*log(2) 2 2*log(2)
$$x\ in\ \left(- \frac{\sqrt{16 + 65 \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}} + \frac{7}{2}, \frac{7}{2} + \frac{\sqrt{16 + 65 \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}}\right)$$
x in Interval.open(-sqrt(16 + 65*log(2)^2)/(2*log(2)) + 7/2, 7/2 + sqrt(16 + 65*log(2)^2)/(2*log(2)))
Respuesta rápida
[src]
/ _________________ _________________ \ | / 2 / 2 | | 7 \/ 16 + 65*log (2) 7 \/ 16 + 65*log (2) | And|x < - + --------------------, - - -------------------- < x| \ 2 2*log(2) 2 2*log(2) /
$$x < \frac{7}{2} + \frac{\sqrt{16 + 65 \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}} \wedge - \frac{\sqrt{16 + 65 \log{\left(2 \right)}^{2}}}{2 \log{\left(2 \right)}} + \frac{7}{2} < x$$
(x < 7/2 + sqrt(16 + 65*log(2)^2)/(2*log(2)))∧(7/2 - sqrt(16 + 65*log(2)^2)/(2*log(2)) < x)
Gráfico