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Resolver desigualdades/ sqrt(x)-(2/(sqrt(x-2)))<=3

sqrt(x)-(2/(sqrt(x-2)))<=3 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src] 
  ___       2         
\/ x  - --------- <= 3
          _______     
        \/ x - 2
x−2x−2≤3\sqrt{x} - \frac{2}{\sqrt{x - 2}} \leq 3 
sqrt(x) - 2/sqrt(x - 2) <= 3
Solución detallada
Se da la desigualdad:
x−2x−2≤3\sqrt{x} - \frac{2}{\sqrt{x - 2}} \leq 3
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
x−2x−2=3\sqrt{x} - \frac{2}{\sqrt{x - 2}} = 3
Resolvemos:
x1=−182318−35119216+824423+2−35119216+824423+6532+−2−35119216+824423+182318−35119216+824423+1303+288−182318−35119216+824423+2−35119216+824423+6532+112x_{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}
x1=−182318−35119216+824423+2−35119216+824423+6532+−2−35119216+824423+182318−35119216+824423+1303+288−182318−35119216+824423+2−35119216+824423+6532+112x_{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
x1=−182318−35119216+824423+2−35119216+824423+6532+−2−35119216+824423+182318−35119216+824423+1303+288−182318−35119216+824423+2−35119216+824423+6532+112x_{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:
x0≤x1x_{0} \leq x_{1}
Consideremos, por ejemplo, el punto
x0=x1−110x_{0} = x_{1} - \frac{1}{10}
=
−110+(−182318−35119216+824423+2−35119216+824423+6532+−2−35119216+824423+182318−35119216+824423+1303+288−182318−35119216+824423+2−35119216+824423+6532+112)- \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)
=
−182318−35119216+824423+2−35119216+824423+6532+−2−35119216+824423+182318−35119216+824423+1303+288−182318−35119216+824423+2−35119216+824423+6532+275\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
x−2x−2≤3\sqrt{x} - \frac{2}{\sqrt{x - 2}} \leq 3
−2−2+(−182318−35119216+824423+2−35119216+824423+6532+−2−35119216+824423+182318−35119216+824423+1303+288−182318−35119216+824423+2−35119216+824423+6532+275)+−182318−35119216+824423+2−35119216+824423+6532+−2−35119216+824423+182318−35119216+824423+1303+288−182318−35119216+824423+2−35119216+824423+6532+275≤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
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             /                                                                                                /             ______________________                                                                                                                                                                                                                                                                                                                                                                                          
            /                                                                                                /  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≤−182318−35119216+824423+2−35119216+824423+6532+−2−35119216+824423+182318−35119216+824423+1303+288−182318−35119216+824423+2−35119216+824423+6532+112x \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
05-25-20-15-10-5101520253035404550-5050
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 (2,3−1823−35119+172824423+−35119+172824423+656+3−−35119+172824423+1823−35119+172824423+130+8643−1823−35119+172824423+−35119+172824423+656+112]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≤3−1823−35119+172824423+−35119+172824423+656+3−−35119+172824423+1823−35119+172824423+130+8643−1823−35119+172824423+−35119+172824423+656+112∧2<xx \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)
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