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¿Cómo usar?
- Desigualdades:
-
x^2>=9
- (x-3)^x^2-9>1
- x^2+y^2<=1
-
-x^2+4x-4<=0
-
Expresiones idénticas
- x^ tres +6x^ dos +(dos 8x^ dos +2x- diez)/x- cinco <=2
- x al cubo más 6x al cuadrado más (28x al cuadrado más 2x menos 10) dividir por x menos 5 menos o igual a 2
- x en el grado tres más 6x en el grado dos más (dos 8x en el grado dos más 2x menos diez) dividir por x menos cinco menos o igual a 2
- x3+6x2+(28x2+2x-10)/x-5<=2
- x3+6x2+28x2+2x-10/x-5<=2
- x³+6x²+(28x²+2x-10)/x-5<=2
- x en el grado 3+6x en el grado 2+(28x en el grado 2+2x-10)/x-5<=2
- x^3+6x^2+28x^2+2x-10/x-5<=2
- x^3+6x^2+(28x^2+2x-10) dividir por x-5<=2
-
Expresiones semejantes
- x^3+6x^2+(28x^2+2x+10)/x-5<=2
- x^3-6x^2+(28x^2+2x-10)/x-5<=2
- x^3+6x^2-(28x^2+2x-10)/x-5<=2
- x^3+6x^2+(28x^2+2x-10)/x+5<=2
- x^3+6x^2+(28x^2-2x-10)/x-5<=2
x^3+6x^2+(28x^2+2x-10)/x-5<=2 desigualdades
Solución
Ha introducido
[src]
2
3 2 28*x + 2*x - 10
x + 6*x + ---------------- - 5 <= 2
x
$$\left(\left(x^{3} + 6 x^{2}\right) + \frac{\left(28 x^{2} + 2 x\right) - 10}{x}\right) - 5 \leq 2$$
x^3 + 6*x^2 + (28*x^2 + 2*x - 10)/x - 5 <= 2
Solución detallada
Se da la desigualdad:
$$\left(\left(x^{3} + 6 x^{2}\right) + \frac{\left(28 x^{2} + 2 x\right) - 10}{x}\right) - 5 \leq 2$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left(\left(x^{3} + 6 x^{2}\right) + \frac{\left(28 x^{2} + 2 x\right) - 10}{x}\right) - 5 = 2$$
Resolvemos:
$$x_{1} = - \frac{3}{2} - \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}$$
$$x_{2} = - \frac{3}{2} + \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}$$
$$x_{3} = - \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2} - \frac{3}{2} - \frac{\sqrt{- \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}} - \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}{2}$$
$$x_{4} = - \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2} - \frac{3}{2} + \frac{\sqrt{- \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}} - \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}{2}$$
Descartamos las soluciones complejas:
$$x_{1} = - \frac{3}{2} - \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}$$
$$x_{2} = - \frac{3}{2} + \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}$$
Las raíces dadas
$$x_{1} = - \frac{3}{2} - \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}$$
$$x_{2} = - \frac{3}{2} + \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{1}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(- \frac{3}{2} - \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}\right) + - \frac{1}{10}$$
=
$$- \frac{8}{5} - \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}$$
lo sustituimos en la expresión
$$\left(\left(x^{3} + 6 x^{2}\right) + \frac{\left(28 x^{2} + 2 x\right) - 10}{x}\right) - 5 \leq 2$$
$$-5 + \left(\frac{-10 + \left(2 \left(- \frac{8}{5} - \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}\right) + 28 \left(- \frac{8}{5} - \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}\right)^{2}\right)}{- \frac{8}{5} - \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}} + \left(\left(- \frac{8}{5} - \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}\right)^{3} + 6 \left(- \frac{8}{5} - \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}\right)^{2}\right)\right) \leq 2$$
significa que una de las soluciones de nuestra ecuación será con:
$$x \leq - \frac{3}{2} - \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}$$
Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x \leq - \frac{3}{2} - \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}$$
$$x \geq - \frac{3}{2} + \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}$$
$$\left(\left(x^{3} + 6 x^{2}\right) + \frac{\left(28 x^{2} + 2 x\right) - 10}{x}\right) - 5 \leq 2$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left(\left(x^{3} + 6 x^{2}\right) + \frac{\left(28 x^{2} + 2 x\right) - 10}{x}\right) - 5 = 2$$
Resolvemos:
$$x_{1} = - \frac{3}{2} - \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}$$
$$x_{2} = - \frac{3}{2} + \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}$$
$$x_{3} = - \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2} - \frac{3}{2} - \frac{\sqrt{- \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}} - \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}{2}$$
$$x_{4} = - \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2} - \frac{3}{2} + \frac{\sqrt{- \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}} - \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}{2}$$
Descartamos las soluciones complejas:
$$x_{1} = - \frac{3}{2} - \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}$$
$$x_{2} = - \frac{3}{2} + \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}$$
Las raíces dadas
$$x_{1} = - \frac{3}{2} - \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}$$
$$x_{2} = - \frac{3}{2} + \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{1}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(- \frac{3}{2} - \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}\right) + - \frac{1}{10}$$
=
$$- \frac{8}{5} - \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}$$
lo sustituimos en la expresión
$$\left(\left(x^{3} + 6 x^{2}\right) + \frac{\left(28 x^{2} + 2 x\right) - 10}{x}\right) - 5 \leq 2$$
$$-5 + \left(\frac{-10 + \left(2 \left(- \frac{8}{5} - \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}\right) + 28 \left(- \frac{8}{5} - \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}\right)^{2}\right)}{- \frac{8}{5} - \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}} + \left(\left(- \frac{8}{5} - \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}\right)^{3} + 6 \left(- \frac{8}{5} - \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}\right)^{2}\right)\right) \leq 2$$
2
/ _____________________________________________________________________________________________________________________________________________________________\
| / _______________________ |
| / / ___________ |
| / 58 / 62579 \/ 244609665 124 377 |
| / - -- - 2*3 / ----- + ------------- + ---------------------------------------------------------------------------------- - ------------------------------ |
| ________________________________________________________________________ / 3 \/ 432 144 ________________________________________________________________________ _______________________ |
| / _______________________ / / _______________________ / ___________ |
| / / ___________ / / / ___________ / 62579 \/ 244609665 |
| / 29 / 62579 \/ 244609665 377 / / 29 / 62579 \/ 244609665 377 9*3 / ----- + ------------- |
| / - -- + 2*3 / ----- + ------------- + ------------------------------ / / - -- + 2*3 / ----- + ------------- + ------------------------------ \/ 432 144 |
3 2 | / 3 \/ 432 144 _______________________ / / 3 \/ 432 144 _______________________ |
/ _____________________________________________________________________________________________________________________________________________________________\ / _____________________________________________________________________________________________________________________________________________________________\ ________________________________________________________________________ _____________________________________________________________________________________________________________________________________________________________ | / / ___________ / / / ___________ |
| / _______________________ | | / _______________________ | / _______________________ / _______________________ | / / 62579 \/ 244609665 / / / 62579 \/ 244609665 |
| / / ___________ | | / / ___________ | / / ___________ / / ___________ | / 9*3 / ----- + ------------- / / 9*3 / ----- + ------------- |
| / 58 / 62579 \/ 244609665 124 377 | | / 58 / 62579 \/ 244609665 124 377 | 66 / 29 / 62579 \/ 244609665 377 / 58 / 62579 \/ 244609665 124 377 | 8 \/ \/ 432 144 \/ \/ \/ 432 144 |
| / - -- - 2*3 / ----- + ------------- + ---------------------------------------------------------------------------------- - ------------------------------ | | / - -- - 2*3 / ----- + ------------- + ---------------------------------------------------------------------------------- - ------------------------------ | - -- + / - -- + 2*3 / ----- + ------------- + ------------------------------ - / - -- - 2*3 / ----- + ------------- + ---------------------------------------------------------------------------------- - ------------------------------ + 28*|- - + ---------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
| ________________________________________________________________________ / 3 \/ 432 144 ________________________________________________________________________ _______________________ | | ________________________________________________________________________ / 3 \/ 432 144 ________________________________________________________________________ _______________________ | 5 / 3 \/ 432 144 _______________________ / 3 \/ 432 144 ________________________________________________________________________ _______________________ \ 5 2 2 /
| / _______________________ / / _______________________ / ___________ | | / _______________________ / / _______________________ / ___________ | / / ___________ / / _______________________ / ___________
| / / ___________ / / / ___________ / 62579 \/ 244609665 | | / / ___________ / / / ___________ / 62579 \/ 244609665 | / / 62579 \/ 244609665 / / / ___________ / 62579 \/ 244609665
| / 29 / 62579 \/ 244609665 377 / / 29 / 62579 \/ 244609665 377 9*3 / ----- + ------------- | | / 29 / 62579 \/ 244609665 377 / / 29 / 62579 \/ 244609665 377 9*3 / ----- + ------------- | / 9*3 / ----- + ------------- / / 29 / 62579 \/ 244609665 377 9*3 / ----- + -------------
| / - -- + 2*3 / ----- + ------------- + ------------------------------ / / - -- + 2*3 / ----- + ------------- + ------------------------------ \/ 432 144 | | / - -- + 2*3 / ----- + ------------- + ------------------------------ / / - -- + 2*3 / ----- + ------------- + ------------------------------ \/ 432 144 | \/ \/ 432 144 / / - -- + 2*3 / ----- + ------------- + ------------------------------ \/ 432 144
| / 3 \/ 432 144 _______________________ / / 3 \/ 432 144 _______________________ | | / 3 \/ 432 144 _______________________ / / 3 \/ 432 144 _______________________ | / / 3 \/ 432 144 _______________________ <= 2
| / / ___________ / / / ___________ | | / / ___________ / / / ___________ | / / / ___________
| / / 62579 \/ 244609665 / / / 62579 \/ 244609665 | | / / 62579 \/ 244609665 / / / 62579 \/ 244609665 | / / / 62579 \/ 244609665
| / 9*3 / ----- + ------------- / / 9*3 / ----- + ------------- | | / 9*3 / ----- + ------------- / / 9*3 / ----- + ------------- | / / 9*3 / ----- + -------------
| 8 \/ \/ 432 144 \/ \/ \/ 432 144 | | 8 \/ \/ 432 144 \/ \/ \/ 432 144 | \/ \/ \/ 432 144
-5 + |- - + ---------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + 6*|- - + ---------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\ 5 2 2 / \ 5 2 2 / _____________________________________________________________________________________________________________________________________________________________
/ _______________________
/ / ___________
/ 58 / 62579 \/ 244609665 124 377
/ - -- - 2*3 / ----- + ------------- + ---------------------------------------------------------------------------------- - ------------------------------
________________________________________________________________________ / 3 \/ 432 144 ________________________________________________________________________ _______________________
/ _______________________ / / _______________________ / ___________
/ / ___________ / / / ___________ / 62579 \/ 244609665
/ 29 / 62579 \/ 244609665 377 / / 29 / 62579 \/ 244609665 377 9*3 / ----- + -------------
/ - -- + 2*3 / ----- + ------------- + ------------------------------ / / - -- + 2*3 / ----- + ------------- + ------------------------------ \/ 432 144
/ 3 \/ 432 144 _______________________ / / 3 \/ 432 144 _______________________
/ / ___________ / / / ___________
/ / 62579 \/ 244609665 / / / 62579 \/ 244609665
/ 9*3 / ----- + ------------- / / 9*3 / ----- + -------------
8 \/ \/ 432 144 \/ \/ \/ 432 144
- - + ---------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------
5 2 2 significa que una de las soluciones de nuestra ecuación será con:
$$x \leq - \frac{3}{2} - \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}$$
_____ _____
\ /
-------•-------•-------
x1 x2Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x \leq - \frac{3}{2} - \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}$$
$$x \geq - \frac{3}{2} + \frac{\sqrt{- \frac{58}{3} - 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}} - \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + \frac{124}{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}}}{2} + \frac{\sqrt{- \frac{29}{3} + \frac{377}{9 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}} + 2 \sqrt[3]{\frac{\sqrt{244609665}}{144} + \frac{62579}{432}}}}{2}$$
Solución de la desigualdad en el gráfico
Respuesta rápida 2
[src]
/ 4 3 2 \ / 4 3 2 \ (-oo, CRootOf\x + 6*x + 28*x - 5*x - 10, 0/] U (0, CRootOf\x + 6*x + 28*x - 5*x - 10, 1/]
$$x\ in\ \left(-\infty, \operatorname{CRootOf} {\left(x^{4} + 6 x^{3} + 28 x^{2} - 5 x - 10, 0\right)}\right] \cup \left(0, \operatorname{CRootOf} {\left(x^{4} + 6 x^{3} + 28 x^{2} - 5 x - 10, 1\right)}\right]$$
x in Union(Interval(-oo, CRootOf(x^4 + 6*x^3 + 28*x^2 - 5*x - 10, 0)), Interval.Lopen(0, CRootOf(x^4 + 6*x^3 + 28*x^2 - 5*x - 10, 1)))
Respuesta rápida
[src]
/ / / 4 3 2 \ \ / / 4 3 2 \ \\ Or\And\x <= CRootOf\x + 6*x + 28*x - 5*x - 10, 0/, -oo < x/, And\x <= CRootOf\x + 6*x + 28*x - 5*x - 10, 1/, 0 < x//
$$\left(x \leq \operatorname{CRootOf} {\left(x^{4} + 6 x^{3} + 28 x^{2} - 5 x - 10, 0\right)} \wedge -\infty < x\right) \vee \left(x \leq \operatorname{CRootOf} {\left(x^{4} + 6 x^{3} + 28 x^{2} - 5 x - 10, 1\right)} \wedge 0 < x\right)$$
((-oo < x)∧(x <= CRootOf(x^4 + 6*x^3 + 28*x^2 - 5*x - 10, 0)))∨((0 < x)∧(x <= CRootOf(x^4 + 6*x^3 + 28*x^2 - 5*x - 10, 1)))
Gráfico