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Resolver desigualdades/ xsqrt(3x-5)<=5

xsqrt(3x-5)<=5 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src] 
    _________     
x*\/ 3*x - 5  <= 5
x3x−5≤5x \sqrt{3 x - 5} \leq 5 
x*sqrt(3*x - 5) <= 5
Solución detallada
Se da la desigualdad:
x3x−5≤5x \sqrt{3 x - 5} \leq 5
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
x3x−5=5x \sqrt{3 x - 5} = 5
Resolvemos:
x1=258125789162+632514583+59+25789162+632514583x_{1} = \frac{25}{81 \sqrt[3]{\frac{25 \sqrt{789}}{162} + \frac{6325}{1458}}} + \frac{5}{9} + \sqrt[3]{\frac{25 \sqrt{789}}{162} + \frac{6325}{1458}}
x1=258125789162+632514583+59+25789162+632514583x_{1} = \frac{25}{81 \sqrt[3]{\frac{25 \sqrt{789}}{162} + \frac{6325}{1458}}} + \frac{5}{9} + \sqrt[3]{\frac{25 \sqrt{789}}{162} + \frac{6325}{1458}}
Las raíces dadas
x1=258125789162+632514583+59+25789162+632514583x_{1} = \frac{25}{81 \sqrt[3]{\frac{25 \sqrt{789}}{162} + \frac{6325}{1458}}} + \frac{5}{9} + \sqrt[3]{\frac{25 \sqrt{789}}{162} + \frac{6325}{1458}}
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+(258125789162+632514583+59+25789162+632514583)- \frac{1}{10} + \left(\frac{25}{81 \sqrt[3]{\frac{25 \sqrt{789}}{162} + \frac{6325}{1458}}} + \frac{5}{9} + \sqrt[3]{\frac{25 \sqrt{789}}{162} + \frac{6325}{1458}}\right)
=
258125789162+632514583+4190+25789162+632514583\frac{25}{81 \sqrt[3]{\frac{25 \sqrt{789}}{162} + \frac{6325}{1458}}} + \frac{41}{90} + \sqrt[3]{\frac{25 \sqrt{789}}{162} + \frac{6325}{1458}}
lo sustituimos en la expresión
x3x−5≤5x \sqrt{3 x - 5} \leq 5
−5+3(258125789162+632514583+4190+25789162+632514583)(258125789162+632514583+4190+25789162+632514583)≤5\sqrt{-5 + 3 \left(\frac{25}{81 \sqrt[3]{\frac{25 \sqrt{789}}{162} + \frac{6325}{1458}}} + \frac{41}{90} + \sqrt[3]{\frac{25 \sqrt{789}}{162} + \frac{6325}{1458}}\right)} \left(\frac{25}{81 \sqrt[3]{\frac{25 \sqrt{789}}{162} + \frac{6325}{1458}}} + \frac{41}{90} + \sqrt[3]{\frac{25 \sqrt{789}}{162} + \frac{6325}{1458}}\right) \leq 5
          __________________________________________________________________                                                                   
         /                ___________________                                /          ___________________                              \     
        /                /             _____                                 |         /             _____                               |     
       /     109        /  6325   25*\/ 789                  25              |41      /  6325   25*\/ 789                  25            |     
      /    - --- + 3*3 /   ---- + ----------  + --------------------------- *|-- + 3 /   ---- + ----------  + ---------------------------|     
     /        30     \/    1458      162                ___________________  |90   \/    1458      162                ___________________| <= 5
    /                                                  /             _____   |                                       /             _____ |     
   /                                                  /  6325   25*\/ 789    |                                      /  6325   25*\/ 789  |     
  /                                             27*3 /   ---- + ----------   |                                81*3 /   ---- + ---------- |     
\/                                                 \/    1458      162       \                                   \/    1458      162     /

significa que la solución de la desigualdad será con:
x≤258125789162+632514583+59+25789162+632514583x \leq \frac{25}{81 \sqrt[3]{\frac{25 \sqrt{789}}{162} + \frac{6325}{1458}}} + \frac{5}{9} + \sqrt[3]{\frac{25 \sqrt{789}}{162} + \frac{6325}{1458}}
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       x1
Solución de la desigualdad en el gráfico
02468-4-2100100
Respuesta rápida [src] 
   /                            _________________                         \
   |                     2/3 3 /           _____             3 ____       |
   |               5   10   *\/  253 + 9*\/ 789            5*\/ 10        |
And|5/3 <= x, x <= - + -------------------------- + ----------------------|
   |               9               18                    _________________|
   |                                                  3 /           _____ |
   \                                                9*\/  253 + 9*\/ 789  /
53≤x∧x≤510399789+2533+59+10239789+253318\frac{5}{3} \leq x \wedge x \leq \frac{5 \sqrt[3]{10}}{9 \sqrt[3]{9 \sqrt{789} + 253}} + \frac{5}{9} + \frac{10^{\frac{2}{3}} \sqrt[3]{9 \sqrt{789} + 253}}{18} 
(5/3 <= x)∧(x <= 5/9 + 10^(2/3)*(253 + 9*sqrt(789))^(1/3)/18 + 5*10^(1/3)/(9*(253 + 9*sqrt(789))^(1/3)))
Respuesta rápida 2 [src] 
                   _________________                          
            2/3 3 /           _____             3 ____        
      5   10   *\/  253 + 9*\/ 789            5*\/ 10         
[5/3, - + -------------------------- + ----------------------]
      9               18                    _________________ 
                                         3 /           _____  
                                       9*\/  253 + 9*\/ 789
x in [53,510399789+2533+59+10239789+253318]x\ in\ \left[\frac{5}{3}, \frac{5 \sqrt[3]{10}}{9 \sqrt[3]{9 \sqrt{789} + 253}} + \frac{5}{9} + \frac{10^{\frac{2}{3}} \sqrt[3]{9 \sqrt{789} + 253}}{18}\right] 
x in Interval(5/3, 5*10^(1/3)/(9*(9*sqrt(789) + 253)^(1/3)) + 5/9 + 10^(2/3)*(9*sqrt(789) + 253)^(1/3)/18)
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