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¿Cómo usar?
- Desigualdades:
-
x^2+4x-21>0
-
x^2+2x-3<0
-
x^2-25<0
-
(sqrt(3)-1,5)*(3-2x)>0
-
Expresiones idénticas
- (x- dos)*(dos x- uno)/2x^2+7x+ tres > cero
- (x menos 2) multiplicar por (2x menos 1) dividir por 2x al cuadrado más 7x más 3 más 0
- (x menos dos) multiplicar por (dos x menos uno) dividir por 2x al cuadrado más 7x más tres más cero
- (x-2)*(2x-1)/2x2+7x+3>0
- x-2*2x-1/2x2+7x+3>0
- (x-2)*(2x-1)/2x²+7x+3>0
- (x-2)*(2x-1)/2x en el grado 2+7x+3>0
- (x-2)(2x-1)/2x^2+7x+3>0
- (x-2)(2x-1)/2x2+7x+3>0
- x-22x-1/2x2+7x+3>0
- x-22x-1/2x^2+7x+3>0
- (x-2)*(2x-1) dividir por 2x^2+7x+3>0
-
Expresiones semejantes
- (x-2)*(2x-1)/2x^2+7x-3>0
- (x-2)*(2x-1)/2x^2-7x+3>0
- (x+2)*(2x-1)/2x^2+7x+3>0
- (x-2)*(2x+1)/2x^2+7x+3>0
(x-2)*(2x-1)/2x^2+7x+3>0 desigualdades
Solución
Ha introducido
[src]
(x - 2)*(2*x - 1) 2
-----------------*x + 7*x + 3 > 0
2
$$\left(x^{2} \frac{\left(x - 2\right) \left(2 x - 1\right)}{2} + 7 x\right) + 3 > 0$$
x^2*(((x - 2)*(2*x - 1))/2) + 7*x + 3 > 0
Solución detallada
Se da la desigualdad:
$$\left(x^{2} \frac{\left(x - 2\right) \left(2 x - 1\right)}{2} + 7 x\right) + 3 > 0$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left(x^{2} \frac{\left(x - 2\right) \left(2 x - 1\right)}{2} + 7 x\right) + 3 = 0$$
Resolvemos:
$$x_{1} = \frac{5}{8} + \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} - \frac{\sqrt{- \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}} - 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24}}}{2}$$
$$x_{2} = \frac{5}{8} + \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} + \frac{\sqrt{- \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}} - 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24}}}{2}$$
$$x_{3} = - \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{5}{8}$$
$$x_{4} = - \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{5}{8}$$
Descartamos las soluciones complejas:
$$x_{1} = - \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{5}{8}$$
$$x_{2} = - \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{5}{8}$$
Las raíces dadas
$$x_{1} = - \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{5}{8}$$
$$x_{2} = - \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{5}{8}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(- \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{5}{8}\right) + - \frac{1}{10}$$
=
$$- \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{21}{40}$$
lo sustituimos en la expresión
$$\left(x^{2} \frac{\left(x - 2\right) \left(2 x - 1\right)}{2} + 7 x\right) + 3 > 0$$
$$\left(7 \left(- \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{21}{40}\right) + \frac{\left(-2 + \left(- \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{21}{40}\right)\right) \left(2 \left(- \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{21}{40}\right) - 1\right)}{2} \left(- \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{21}{40}\right)^{2}\right) + 3 > 0$$
significa que una de las soluciones de nuestra ecuación será con:
$$x < - \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{5}{8}$$
Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x < - \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{5}{8}$$
$$x > - \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{5}{8}$$
$$\left(x^{2} \frac{\left(x - 2\right) \left(2 x - 1\right)}{2} + 7 x\right) + 3 > 0$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left(x^{2} \frac{\left(x - 2\right) \left(2 x - 1\right)}{2} + 7 x\right) + 3 = 0$$
Resolvemos:
$$x_{1} = \frac{5}{8} + \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} - \frac{\sqrt{- \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}} - 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24}}}{2}$$
$$x_{2} = \frac{5}{8} + \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} + \frac{\sqrt{- \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}} - 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24}}}{2}$$
$$x_{3} = - \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{5}{8}$$
$$x_{4} = - \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{5}{8}$$
Descartamos las soluciones complejas:
$$x_{1} = - \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{5}{8}$$
$$x_{2} = - \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{5}{8}$$
Las raíces dadas
$$x_{1} = - \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{5}{8}$$
$$x_{2} = - \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{5}{8}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(- \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{5}{8}\right) + - \frac{1}{10}$$
=
$$- \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{21}{40}$$
lo sustituimos en la expresión
$$\left(x^{2} \frac{\left(x - 2\right) \left(2 x - 1\right)}{2} + 7 x\right) + 3 > 0$$
$$\left(7 \left(- \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{21}{40}\right) + \frac{\left(-2 + \left(- \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{21}{40}\right)\right) \left(2 \left(- \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{21}{40}\right) - 1\right)}{2} \left(- \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{21}{40}\right)^{2}\right) + 3 > 0$$
2
/ ______________________________________________________________________________________________________________________________________________\ / ______________________________________________________________________________________________________________________________________________\
| / ___________________ | | / ___________________ |
| / / ________ | | / / ________ |
| / 43 / 7091 \/ 488841 179 403 | | / 43 / 7091 \/ 488841 179 403 |
| / -- - 2*3 / ---- + ---------- - --------------------------- + ---------------------------------------------------------------------------- | | / -- - 2*3 / ---- + ---------- - --------------------------- + ---------------------------------------------------------------------------- |
| _______________________________________________________________ / 24 \/ 1728 576 ___________________ _______________________________________________________________ | | _______________________________________________________________ / 24 \/ 1728 576 ___________________ _______________________________________________________________ |
| / ___________________ / / ________ / ___________________ | | / ___________________ / / ________ / ___________________ |
| / / ________ / / 7091 \/ 488841 / / ________ | | / / ________ / / 7091 \/ 488841 / / ________ |
| / 43 / 7091 \/ 488841 179 / 36*3 / ---- + ---------- / 43 / 7091 \/ 488841 179 | | / 43 / 7091 \/ 488841 179 / 36*3 / ---- + ---------- / 43 / 7091 \/ 488841 179 |
| / -- + 2*3 / ---- + ---------- + --------------------------- / \/ 1728 576 32* / -- + 2*3 / ---- + ---------- + --------------------------- | | / -- + 2*3 / ---- + ---------- + --------------------------- / \/ 1728 576 32* / -- + 2*3 / ---- + ---------- + --------------------------- |
| / 48 \/ 1728 576 ___________________ / / 48 \/ 1728 576 ___________________ | | / 48 \/ 1728 576 ___________________ / / 48 \/ 1728 576 ___________________ |
______________________________________________________________________________________________________________________________________________ | / / ________ / / / ________ | | / / ________ / / / ________ | / _______________________________________________________________ ______________________________________________________________________________________________________________________________________________\
/ ___________________ | / / 7091 \/ 488841 / / / 7091 \/ 488841 | | / / 7091 \/ 488841 / / / 7091 \/ 488841 | | / ___________________ / ___________________ |
/ / ________ | / 36*3 / ---- + ---------- / / 36*3 / ---- + ---------- | | / 36*3 / ---- + ---------- / / 36*3 / ---- + ---------- | | / / ________ / / ________ | > 0
/ 43 / 7091 \/ 488841 179 403 |21 \/ \/ 1728 576 \/ \/ \/ 1728 576 | | 59 \/ \/ 1728 576 \/ \/ \/ 1728 576 | |1 / 43 / 7091 \/ 488841 179 / 43 / 7091 \/ 488841 179 403 |
7* / -- - 2*3 / ---- + ---------- - --------------------------- + ---------------------------------------------------------------------------- |-- - ------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------------| *|- -- - ------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------------|*|-- - / -- + 2*3 / ---- + ---------- + --------------------------- - / -- - 2*3 / ---- + ---------- - --------------------------- + ---------------------------------------------------------------------------- |
_______________________________________________________________ / 24 \/ 1728 576 ___________________ _______________________________________________________________ \40 2 2 / \ 40 2 2 / |20 / 48 \/ 1728 576 ___________________ / 24 \/ 1728 576 ___________________ _______________________________________________________________ |
/ ___________________ / / ________ / ___________________ | / / ________ / / ________ / ___________________ |
/ / ________ / / 7091 \/ 488841 / / ________ | / / 7091 \/ 488841 / / 7091 \/ 488841 / / ________ |
/ 43 / 7091 \/ 488841 179 / 36*3 / ---- + ---------- / 43 / 7091 \/ 488841 179 | / 36*3 / ---- + ---------- / 36*3 / ---- + ---------- / 43 / 7091 \/ 488841 179 |
7* / -- + 2*3 / ---- + ---------- + --------------------------- / \/ 1728 576 32* / -- + 2*3 / ---- + ---------- + --------------------------- | \/ \/ 1728 576 / \/ 1728 576 32* / -- + 2*3 / ---- + ---------- + --------------------------- |
/ 48 \/ 1728 576 ___________________ / / 48 \/ 1728 576 ___________________ | / / 48 \/ 1728 576 ___________________ |
/ / ________ / / / ________ | / / / ________ |
/ / 7091 \/ 488841 / / / 7091 \/ 488841 | / / / 7091 \/ 488841 |
/ 36*3 / ---- + ---------- / / 36*3 / ---- + ---------- | / / 36*3 / ---- + ---------- |
267 \/ \/ 1728 576 \/ \/ \/ 1728 576 \ \/ \/ \/ 1728 576 /
--- - --------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------------------- + -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
40 2 2 2 significa que una de las soluciones de nuestra ecuación será con:
$$x < - \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{5}{8}$$
_____ _____
\ /
-------ο-------ο-------
x1 x2Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x < - \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{5}{8}$$
$$x > - \frac{\sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}} - \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + \frac{43}{24} + \frac{403}{32 \sqrt{\frac{43}{48} + \frac{179}{36 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}} + 2 \sqrt[3]{\frac{\sqrt{488841}}{576} + \frac{7091}{1728}}}}}}{2} + \frac{5}{8}$$
Solución de la desigualdad en el gráfico
Respuesta rápida
[src]
/ / / 4 3 2 \\ / / 4 3 2 \ \\ Or\And\-oo < x, x < CRootOf\2*x - 5*x + 2*x + 14*x + 6, 0//, And\x < oo, CRootOf\2*x - 5*x + 2*x + 14*x + 6, 1/ < x//
$$\left(-\infty < x \wedge x < \operatorname{CRootOf} {\left(2 x^{4} - 5 x^{3} + 2 x^{2} + 14 x + 6, 0\right)}\right) \vee \left(x < \infty \wedge \operatorname{CRootOf} {\left(2 x^{4} - 5 x^{3} + 2 x^{2} + 14 x + 6, 1\right)} < x\right)$$
((-oo < x)∧(x < CRootOf(2*x^4 - 5*x^3 + 2*x^2 + 14*x + 6, 0)))∨((x < oo)∧(CRootOf(2*x^4 - 5*x^3 + 2*x^2 + 14*x + 6, 1) < x))
Respuesta rápida 2
[src]
/ 4 3 2 \ / 4 3 2 \ (-oo, CRootOf\2*x - 5*x + 2*x + 14*x + 6, 0/) U (CRootOf\2*x - 5*x + 2*x + 14*x + 6, 1/, oo)
$$x\ in\ \left(-\infty, \operatorname{CRootOf} {\left(2 x^{4} - 5 x^{3} + 2 x^{2} + 14 x + 6, 0\right)}\right) \cup \left(\operatorname{CRootOf} {\left(2 x^{4} - 5 x^{3} + 2 x^{2} + 14 x + 6, 1\right)}, \infty\right)$$
x in Union(Interval.open(-oo, CRootOf(2*x^4 - 5*x^3 + 2*x^2 + 14*x + 6, 0)), Interval.open(CRootOf(2*x^4 - 5*x^3 + 2*x^2 + 14*x + 6, 1), oo))
Gráfico