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- Sistemas de desigualdades paso a paso
- Ecuaciones básicas paso a paso
- 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+3x-2<0
-
(x-2)/(x-4)>0
-
x+1>0
- Gráfico de la función y =:
-
(x+1)*(x^2-1)
-
Expresiones idénticas
- (x+ uno)*(x^ dos - uno)>sqrt(dos)/ dos
- (x más 1) multiplicar por (x al cuadrado menos 1) más raíz cuadrada de (2) dividir por 2
- (x más uno) multiplicar por (x en el grado dos menos uno) más raíz cuadrada de (dos) dividir por dos
- (x+1)*(x^2-1)>√(2)/2
- (x+1)*(x2-1)>sqrt(2)/2
- x+1*x2-1>sqrt2/2
- (x+1)*(x²-1)>sqrt(2)/2
- (x+1)*(x en el grado 2-1)>sqrt(2)/2
- (x+1)(x^2-1)>sqrt(2)/2
- (x+1)(x2-1)>sqrt(2)/2
- x+1x2-1>sqrt2/2
- x+1x^2-1>sqrt2/2
- (x+1)*(x^2-1)>sqrt(2) dividir por 2
-
Expresiones semejantes
- sin(x+pi/4)<sqrt(2/2)
- sin2x<=sqrt2/2
- (x-1)*(x^2-1)>sqrt(2)/2
- sin(1/2x)>=sqrt2/2
- (x+1)*(x^2+1)>sqrt(2)/2
- log(3-x)-log((sqrt2/2)/(5-x))>1/2+log(x+7)
-
Expresiones con funciones
- Raíz cuadrada sqrt
- sqrt(x+5)+sqrt(5-x)>4
- sqrt(x+3)-sqrt(x-1)<sqrt(x-2)
- sqrt(x+5)/(x-1)<1
- sqrt(x^3-2*x^2+4*x-2)>=x
- sqrt(x-5)>-2
(x+1)*(x^2-1)>sqrt(2)/2 desigualdades
Solución
Ha introducido
[src]
___
/ 2 \ \/ 2
(x + 1)*\x - 1/ > -----
2
$$\left(x + 1\right) \left(x^{2} - 1\right) > \frac{\sqrt{2}}{2}$$
(x + 1)*(x^2 - 1) > sqrt(2)/2
Solución detallada
Se da la desigualdad:
$$\left(x + 1\right) \left(x^{2} - 1\right) > \frac{\sqrt{2}}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left(x + 1\right) \left(x^{2} - 1\right) = \frac{\sqrt{2}}{2}$$
Resolvemos:
$$x_{1} = - \frac{1}{3} + \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}} + \frac{4}{9 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}}$$
$$x_{2} = - \frac{1}{3} + \frac{4}{9 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}$$
$$x_{3} = - \frac{1}{3} + \frac{4}{9 \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}} + \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}$$
Descartamos las soluciones complejas:
$$x_{1} = - \frac{1}{3} + \frac{4}{9 \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}} + \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}$$
Las raíces dadas
$$x_{1} = - \frac{1}{3} + \frac{4}{9 \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}} + \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \left(- \frac{1}{3} + \frac{4}{9 \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}} + \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}\right)$$
=
$$- \frac{13}{30} + \frac{4}{9 \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}} + \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}$$
lo sustituimos en la expresión
$$\left(x + 1\right) \left(x^{2} - 1\right) > \frac{\sqrt{2}}{2}$$
$$\left(-1 + \left(- \frac{13}{30} + \frac{4}{9 \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}} + \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}\right)^{2}\right) \left(1 + \left(- \frac{13}{30} + \frac{4}{9 \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}} + \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}\right)\right) > \frac{\sqrt{2}}{2}$$
Entonces
$$x < - \frac{1}{3} + \frac{4}{9 \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}} + \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}$$
no se cumple
significa que la solución de la desigualdad será con:
$$x > - \frac{1}{3} + \frac{4}{9 \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}} + \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}$$
$$\left(x + 1\right) \left(x^{2} - 1\right) > \frac{\sqrt{2}}{2}$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\left(x + 1\right) \left(x^{2} - 1\right) = \frac{\sqrt{2}}{2}$$
Resolvemos:
$$x_{1} = - \frac{1}{3} + \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}} + \frac{4}{9 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}}$$
$$x_{2} = - \frac{1}{3} + \frac{4}{9 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}$$
$$x_{3} = - \frac{1}{3} + \frac{4}{9 \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}} + \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}$$
Descartamos las soluciones complejas:
$$x_{1} = - \frac{1}{3} + \frac{4}{9 \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}} + \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}$$
Las raíces dadas
$$x_{1} = - \frac{1}{3} + \frac{4}{9 \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}} + \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \left(- \frac{1}{3} + \frac{4}{9 \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}} + \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}\right)$$
=
$$- \frac{13}{30} + \frac{4}{9 \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}} + \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}$$
lo sustituimos en la expresión
$$\left(x + 1\right) \left(x^{2} - 1\right) > \frac{\sqrt{2}}{2}$$
$$\left(-1 + \left(- \frac{13}{30} + \frac{4}{9 \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}} + \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}\right)^{2}\right) \left(1 + \left(- \frac{13}{30} + \frac{4}{9 \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}} + \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}\right)\right) > \frac{\sqrt{2}}{2}$$
/ 2\
| / ________________________________________________ \ | / ________________________________________________ \
| | / _________________________ | | | / _________________________ |
| | / / 2 | | | / / 2 |
| | / / / ___\ | | | / / / ___\ |
| | / / | 16 \/ 2 | | | | / / | 16 \/ 2 | |
| | / / |- -- - -----| ___ | | | / / |- -- - -----| ___ |
| | 13 / 8 / 64 \ 27 2 / \/ 2 4 | | |17 / 8 / 64 \ 27 2 / \/ 2 4 | ___
|-1 + |- -- + 3 / -- + / - --- + --------------- + ----- + -----------------------------------------------------------| |*|-- + 3 / -- + / - --- + --------------- + ----- + -----------------------------------------------------------| \/ 2
| | 30 \/ 27 \/ 729 4 4 ________________________________________________| | |30 \/ 27 \/ 729 4 4 ________________________________________________| > -----
| | / _________________________ | | | / _________________________ | 2
| | / / 2 | | | / / 2 |
| | / / / ___\ | | | / / / ___\ |
| | / / | 16 \/ 2 | | | | / / | 16 \/ 2 | |
| | / / |- -- - -----| ___ | | | / / |- -- - -----| ___ |
| | / 8 / 64 \ 27 2 / \/ 2 | | | / 8 / 64 \ 27 2 / \/ 2 |
| | 9*3 / -- + / - --- + --------------- + ----- | | | 9*3 / -- + / - --- + --------------- + ----- |
\ \ \/ 27 \/ 729 4 4 / / \ \/ 27 \/ 729 4 4 /
Entonces
$$x < - \frac{1}{3} + \frac{4}{9 \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}} + \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}$$
no se cumple
significa que la solución de la desigualdad será con:
$$x > - \frac{1}{3} + \frac{4}{9 \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}} + \sqrt[3]{\frac{8}{27} + \frac{\sqrt{2}}{4} + \sqrt{- \frac{64}{729} + \frac{\left(- \frac{\sqrt{2}}{2} - \frac{16}{27}\right)^{2}}{4}}}$$
_____
/
-------ο-------
x1
Solución de la desigualdad en el gráfico
Respuesta rápida
[src]
/ _________________________________________________ \ | / ___________________________ | | / / 2 | | 3 ___ 3 / / / ___\ ___ 2/3 | | 1 \/ 2 *\/ 32 + \/ -1024 + \-32 - 27*\/ 2 / + 27*\/ 2 4*2 | And|x < oo, - - + ------------------------------------------------------------ + -------------------------------------------------------- < x| | 3 6 _________________________________________________ | | / ___________________________ | | / / 2 | | 3 / / / ___\ ___ | \ 3*\/ 32 + \/ -1024 + \-32 - 27*\/ 2 / + 27*\/ 2 /
$$x < \infty \wedge - \frac{1}{3} + \frac{4 \cdot 2^{\frac{2}{3}}}{3 \sqrt[3]{32 + 27 \sqrt{2} + \sqrt{-1024 + \left(- 27 \sqrt{2} - 32\right)^{2}}}} + \frac{\sqrt[3]{2} \sqrt[3]{32 + 27 \sqrt{2} + \sqrt{-1024 + \left(- 27 \sqrt{2} - 32\right)^{2}}}}{6} < x$$
(x < oo)∧(-1/3 + 2^(1/3)*(32 + sqrt(-1024 + (-32 - 27*sqrt(2))^2) + 27*sqrt(2))^(1/3)/6 + 4*2^(2/3)/(3*(32 + sqrt(-1024 + (-32 - 27*sqrt(2))^2) + 27*sqrt(2))^(1/3)) < x)
Respuesta rápida 2
[src]
_________________________________________________
/ ___________________________
/ / 2
3 ___ 3 / / / ___\ ___ 2/3
1 \/ 2 *\/ 32 + \/ -1024 + \-32 - 27*\/ 2 / + 27*\/ 2 4*2
(- - + ------------------------------------------------------------ + --------------------------------------------------------, oo)
3 6 _________________________________________________
/ ___________________________
/ / 2
3 / / / ___\ ___
3*\/ 32 + \/ -1024 + \-32 - 27*\/ 2 / + 27*\/ 2
$$x\ in\ \left(- \frac{1}{3} + \frac{4 \cdot 2^{\frac{2}{3}}}{3 \sqrt[3]{32 + 27 \sqrt{2} + \sqrt{-1024 + \left(- 27 \sqrt{2} - 32\right)^{2}}}} + \frac{\sqrt[3]{2} \sqrt[3]{32 + 27 \sqrt{2} + \sqrt{-1024 + \left(- 27 \sqrt{2} - 32\right)^{2}}}}{6}, \infty\right)$$
x in Interval.open(-1/3 + 4*2^(2/3)/(3*(32 + 27*sqrt(2) + sqrt(-1024 + (-27*sqrt(2) - 32)^2))^(1/3)) + 2^(1/3)*(32 + 27*sqrt(2) + sqrt(-1024 + (-27*sqrt(2) - 32)^2))^(1/3)/6, oo)