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- Desigualdades:
-
4+12x>7+13x
-
x-4x^2/x-1>0
-
-x^2-2x+3>0
-
1/4x>1
-
Expresiones idénticas
- sqrt(x)-(dos /(sqrt(x- dos)))<= tres
- raíz cuadrada de (x) menos (2 dividir por ( raíz cuadrada de (x menos 2))) menos o igual a 3
- raíz cuadrada de (x) menos (dos dividir por ( raíz cuadrada de (x menos dos))) menos o igual a tres
- √(x)-(2/(√(x-2)))<=3
- sqrtx-2/sqrtx-2<=3
- sqrt(x)-(2 dividir por (sqrt(x-2)))<=3
-
Expresiones semejantes
- sqrt(x)-(2/(sqrt(x+2)))<=3
- sqrt(x)-2/(sqrt(x)-2)<3
- sqrt(x)+(2/(sqrt(x-2)))<=3
-
Expresiones con funciones
- Raíz cuadrada sqrt
- sqrt(x^2+4x-5)>x-3
- sqrt(25x^2-10x-195)=>5x-1
- sqrt(x-2)>=0
- sqrt(x^2+3)<x+3
- sqrt(x)>x-2
- Raíz cuadrada sqrt
- sqrt(x^2+4x-5)>x-3
- sqrt(25x^2-10x-195)=>5x-1
- sqrt(x-2)>=0
- sqrt(x^2+3)<x+3
- sqrt(x)>x-2
sqrt(x)-(2/(sqrt(x-2)))<=3 desigualdades
Solución
Ha introducido
[src]
___ 2
\/ x - --------- <= 3
_______
\/ x - 2
$$\sqrt{x} - \frac{2}{\sqrt{x - 2}} \leq 3$$
sqrt(x) - 2/sqrt(x - 2) <= 3
Solución detallada
Se da la desigualdad:
$$\sqrt{x} - \frac{2}{\sqrt{x - 2}} \leq 3$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\sqrt{x} - \frac{2}{\sqrt{x - 2}} = 3$$
Resolvemos:
$$x_{1} = \frac{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + \frac{130}{3} + \frac{288}{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}}}{2} + \frac{11}{2}$$
$$x_{1} = \frac{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + \frac{130}{3} + \frac{288}{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}}}{2} + \frac{11}{2}$$
Las raíces dadas
$$x_{1} = \frac{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + \frac{130}{3} + \frac{288}{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}}}{2} + \frac{11}{2}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \left(\frac{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + \frac{130}{3} + \frac{288}{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}}}{2} + \frac{11}{2}\right)$$
=
$$\frac{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + \frac{130}{3} + \frac{288}{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}}}{2} + \frac{27}{5}$$
lo sustituimos en la expresión
$$\sqrt{x} - \frac{2}{\sqrt{x - 2}} \leq 3$$
$$- \frac{2}{\sqrt{-2 + \left(\frac{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + \frac{130}{3} + \frac{288}{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}}}{2} + \frac{27}{5}\right)}} + \sqrt{\frac{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + \frac{130}{3} + \frac{288}{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}}}{2} + \frac{27}{5}} \leq 3$$
significa que la solución de la desigualdad será con:
$$x \leq \frac{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + \frac{130}{3} + \frac{288}{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}}}{2} + \frac{11}{2}$$
$$\sqrt{x} - \frac{2}{\sqrt{x - 2}} \leq 3$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\sqrt{x} - \frac{2}{\sqrt{x - 2}} = 3$$
Resolvemos:
$$x_{1} = \frac{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + \frac{130}{3} + \frac{288}{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}}}{2} + \frac{11}{2}$$
$$x_{1} = \frac{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + \frac{130}{3} + \frac{288}{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}}}{2} + \frac{11}{2}$$
Las raíces dadas
$$x_{1} = \frac{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + \frac{130}{3} + \frac{288}{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}}}{2} + \frac{11}{2}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \left(\frac{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + \frac{130}{3} + \frac{288}{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}}}{2} + \frac{11}{2}\right)$$
=
$$\frac{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + \frac{130}{3} + \frac{288}{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}}}{2} + \frac{27}{5}$$
lo sustituimos en la expresión
$$\sqrt{x} - \frac{2}{\sqrt{x - 2}} \leq 3$$
$$- \frac{2}{\sqrt{-2 + \left(\frac{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + \frac{130}{3} + \frac{288}{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}}}{2} + \frac{27}{5}\right)}} + \sqrt{\frac{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + \frac{130}{3} + \frac{288}{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}}}{2} + \frac{27}{5}} \leq 3$$
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
/ __________________________________________________________________________________________________________________________________________________
/ / ______________________
/ / 130 / 35119 ______ 288 1823
/ / --- - 2*3 / - ----- + 8*\/ 2442 + --------------------------------------------------------------------------- + -----------------------------
/ ___________________________________________________________________ / 3 \/ 216 ___________________________________________________________________ ______________________
/ / ______________________ / / ______________________ / 35119 ______
/ / 65 / 35119 ______ 1823 / / 65 / 35119 ______ 1823 18*3 / - ----- + 8*\/ 2442
/ / -- + 2*3 / - ----- + 8*\/ 2442 - ----------------------------- / / -- + 2*3 / - ----- + 8*\/ 2442 - ----------------------------- \/ 216
/ / 3 \/ 216 ______________________ / / 3 \/ 216 ______________________
/ / / 35119 ______ / / / 35119 ______
/ / 18*3 / - ----- + 8*\/ 2442 / / 18*3 / - ----- + 8*\/ 2442
/ 27 \/ \/ 216 \/ \/ \/ 216 2
/ -- + --------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\/ 5 2 2 ___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ <= 3
/ __________________________________________________________________________________________________________________________________________________
/ / ______________________
/ / 130 / 35119 ______ 288 1823
/ / --- - 2*3 / - ----- + 8*\/ 2442 + --------------------------------------------------------------------------- + -----------------------------
/ ___________________________________________________________________ / 3 \/ 216 ___________________________________________________________________ ______________________
/ / ______________________ / / ______________________ / 35119 ______
/ / 65 / 35119 ______ 1823 / / 65 / 35119 ______ 1823 18*3 / - ----- + 8*\/ 2442
/ / -- + 2*3 / - ----- + 8*\/ 2442 - ----------------------------- / / -- + 2*3 / - ----- + 8*\/ 2442 - ----------------------------- \/ 216
/ / 3 \/ 216 ______________________ / / 3 \/ 216 ______________________
/ / / 35119 ______ / / / 35119 ______
/ / 18*3 / - ----- + 8*\/ 2442 / / 18*3 / - ----- + 8*\/ 2442
/ 17 \/ \/ 216 \/ \/ \/ 216
/ -- + --------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------------------------
\/ 5 2 2 significa que la solución de la desigualdad será con:
$$x \leq \frac{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + \frac{130}{3} + \frac{288}{\sqrt{- \frac{1823}{18 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}}} + 2 \sqrt[3]{- \frac{35119}{216} + 8 \sqrt{2442}} + \frac{65}{3}}}}}{2} + \frac{11}{2}$$
_____
\
-------•-------
x1
Solución de la desigualdad en el gráfico
Respuesta rápida 2
[src]
___________________________________________________________________________________________________________________________________________
/ ________________________ ___
___ / 3 / ______ 1823 864*\/ 3
\/ 3 * / 130 - \/ -35119 + 1728*\/ 2442 + --------------------------- + -----------------------------------------------------------------------
________________________________________________________________ / ________________________ ________________________________________________________________
/ ________________________ / 3 / ______ / ________________________
___ / 3 / ______ 1823 / \/ -35119 + 1728*\/ 2442 / 3 / ______ 1823
\/ 3 * / 65 + \/ -35119 + 1728*\/ 2442 - --------------------------- / / 65 + \/ -35119 + 1728*\/ 2442 - ---------------------------
/ ________________________ / / ________________________
/ 3 / ______ / / 3 / ______
11 \/ \/ -35119 + 1728*\/ 2442 \/ \/ \/ -35119 + 1728*\/ 2442
(2, -- + ----------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------]
2 6 6
$$x\ in\ \left(2, \frac{\sqrt{3} \sqrt{- \frac{1823}{\sqrt[3]{-35119 + 1728 \sqrt{2442}}} + \sqrt[3]{-35119 + 1728 \sqrt{2442}} + 65}}{6} + \frac{\sqrt{3} \sqrt{- \sqrt[3]{-35119 + 1728 \sqrt{2442}} + \frac{1823}{\sqrt[3]{-35119 + 1728 \sqrt{2442}}} + 130 + \frac{864 \sqrt{3}}{\sqrt{- \frac{1823}{\sqrt[3]{-35119 + 1728 \sqrt{2442}}} + \sqrt[3]{-35119 + 1728 \sqrt{2442}} + 65}}}}{6} + \frac{11}{2}\right]$$
x in Interval.Lopen(2, sqrt(3)*sqrt(-1823/(-35119 + 1728*sqrt(2442))^(1/3) + (-35119 + 1728*sqrt(2442))^(1/3) + 65)/6 + sqrt(3)*sqrt(-(-35119 + 1728*sqrt(2442))^(1/3) + 1823/(-35119 + 1728*sqrt(2442))^(1/3) + 130 + 864*sqrt(3)/sqrt(-1823/(-35119 + 1728*sqrt(2442))^(1/3) + (-35119 + 1728*sqrt(2442))^(1/3) + 65))/6 + 11/2)
Respuesta rápida
[src]
/ ___________________________________________________________________________________________________________________________________________ \ | / ________________________ ___ | | ___ / 3 / ______ 1823 864*\/ 3 | | \/ 3 * / 130 - \/ -35119 + 1728*\/ 2442 + --------------------------- + ----------------------------------------------------------------------- | | ________________________________________________________________ / ________________________ ________________________________________________________________ | | / ________________________ / 3 / ______ / ________________________ | | ___ / 3 / ______ 1823 / \/ -35119 + 1728*\/ 2442 / 3 / ______ 1823 | | \/ 3 * / 65 + \/ -35119 + 1728*\/ 2442 - --------------------------- / / 65 + \/ -35119 + 1728*\/ 2442 - --------------------------- | | / ________________________ / / ________________________ | | / 3 / ______ / / 3 / ______ | | 11 \/ \/ -35119 + 1728*\/ 2442 \/ \/ \/ -35119 + 1728*\/ 2442 | And|x <= -- + ----------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------, 2 < x| \ 2 6 6 /
$$x \leq \frac{\sqrt{3} \sqrt{- \frac{1823}{\sqrt[3]{-35119 + 1728 \sqrt{2442}}} + \sqrt[3]{-35119 + 1728 \sqrt{2442}} + 65}}{6} + \frac{\sqrt{3} \sqrt{- \sqrt[3]{-35119 + 1728 \sqrt{2442}} + \frac{1823}{\sqrt[3]{-35119 + 1728 \sqrt{2442}}} + 130 + \frac{864 \sqrt{3}}{\sqrt{- \frac{1823}{\sqrt[3]{-35119 + 1728 \sqrt{2442}}} + \sqrt[3]{-35119 + 1728 \sqrt{2442}} + 65}}}}{6} + \frac{11}{2} \wedge 2 < x$$
(2 < x)∧(x <= 11/2 + sqrt(3)*sqrt(65 + (-35119 + 1728*sqrt(2442))^(1/3) - 1823/(-35119 + 1728*sqrt(2442))^(1/3))/6 + sqrt(3)*sqrt(130 - (-35119 + 1728*sqrt(2442))^(1/3) + 1823/(-35119 + 1728*sqrt(2442))^(1/3) + 864*sqrt(3)/sqrt(65 + (-35119 + 1728*sqrt(2442))^(1/3) - 1823/(-35119 + 1728*sqrt(2442))^(1/3)))/6)