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anonymous

  • one year ago

Need the steps to solve this inequality:

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  1. anonymous
    • one year ago
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    \[x^4-x \le 0\]

  2. Michele_Laino
    • one year ago
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    hint: we have to factorize the binomial at the left side

  3. Michele_Laino
    • one year ago
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    for example, at the first step, we can write: \[\Large x\left( {{x^3} - 1} \right) \leqslant 0\]

  4. Michele_Laino
    • one year ago
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    now, we have to factorize x^3-1, do you know how to factorize it?

  5. anonymous
    • one year ago
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    Yes

  6. Michele_Laino
    • one year ago
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    ok! Then rewrite my inequality above, using your factorization

  7. anonymous
    • one year ago
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    would it be (x-1)( x+1)( x -1)?

  8. Michele_Laino
    • one year ago
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    not exactly, we have this: \[\Large x\left( {x - 1} \right)\left( {{x^2} + x + 1} \right) \leqslant 0\]

  9. Michele_Laino
    • one year ago
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    since: \[\Large {x^3} - 1 = \left( {x - 1} \right)\left( {{x^2} + x + 1} \right)\]

  10. anonymous
    • one year ago
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    I see

  11. Michele_Laino
    • one year ago
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    now, we can note that \[\Large {{x^2} + x + 1}\] is always positive

  12. Michele_Laino
    • one year ago
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    so we have to study the sign of x and x-1 only

  13. Michele_Laino
    • one year ago
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    for example, please solve this inequality: \[\Large x - 1 \geqslant 0\]

  14. anonymous
    • one year ago
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    \[x \ge 1\]

  15. Michele_Laino
    • one year ago
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    correct! So we have this drawing: |dw:1440745389311:dw|

  16. Michele_Laino
    • one year ago
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    a continuous line stands for positivity, whereas a dashed line stands for negativity

  17. Michele_Laino
    • one year ago
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    so we have the subsequent drawing: |dw:1440745570772:dw|

  18. Michele_Laino
    • one year ago
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    whereas the signs +, -, + indicate the sign of x^4-x

  19. anonymous
    • one year ago
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    Ok i get that

  20. Michele_Laino
    • one year ago
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    I have used the usual rule: \[\Large \begin{gathered} \left( - \right) \cdot \left( - \right) \cdot \left( + \right) = + \hfill \\ \left( + \right) \cdot \left( - \right) \cdot \left( + \right) = - \hfill \\ \left( + \right) \cdot \left( + \right) \cdot \left( + \right) = + \hfill \\ \end{gathered} \]

  21. Michele_Laino
    • one year ago
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    so, what is your solution?

  22. anonymous
    • one year ago
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    I would like to say Everything less than 1 including 1

  23. Michele_Laino
    • one year ago
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    you have to search for minus sign

  24. anonymous
    • one year ago
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    So the solution would be [0,1]?

  25. Michele_Laino
    • one year ago
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    correct!!

  26. anonymous
    • one year ago
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    I can see it in the lines so thank you for drawing those out

  27. Michele_Laino
    • one year ago
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    :)

  28. anonymous
    • one year ago
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    Ill go over the thread a couple times to study, thank you!

  29. Michele_Laino
    • one year ago
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    :)

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