anonymous
  • anonymous
Need the steps to solve this inequality:
Mathematics
jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
\[x^4-x \le 0\]
Michele_Laino
  • Michele_Laino
hint: we have to factorize the binomial at the left side
Michele_Laino
  • Michele_Laino
for example, at the first step, we can write: \[\Large x\left( {{x^3} - 1} \right) \leqslant 0\]

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Michele_Laino
  • Michele_Laino
now, we have to factorize x^3-1, do you know how to factorize it?
anonymous
  • anonymous
Yes
Michele_Laino
  • Michele_Laino
ok! Then rewrite my inequality above, using your factorization
anonymous
  • anonymous
would it be (x-1)( x+1)( x -1)?
Michele_Laino
  • Michele_Laino
not exactly, we have this: \[\Large x\left( {x - 1} \right)\left( {{x^2} + x + 1} \right) \leqslant 0\]
Michele_Laino
  • Michele_Laino
since: \[\Large {x^3} - 1 = \left( {x - 1} \right)\left( {{x^2} + x + 1} \right)\]
anonymous
  • anonymous
I see
Michele_Laino
  • Michele_Laino
now, we can note that \[\Large {{x^2} + x + 1}\] is always positive
Michele_Laino
  • Michele_Laino
so we have to study the sign of x and x-1 only
Michele_Laino
  • Michele_Laino
for example, please solve this inequality: \[\Large x - 1 \geqslant 0\]
anonymous
  • anonymous
\[x \ge 1\]
Michele_Laino
  • Michele_Laino
correct! So we have this drawing: |dw:1440745389311:dw|
Michele_Laino
  • Michele_Laino
a continuous line stands for positivity, whereas a dashed line stands for negativity
Michele_Laino
  • Michele_Laino
so we have the subsequent drawing: |dw:1440745570772:dw|
Michele_Laino
  • Michele_Laino
whereas the signs +, -, + indicate the sign of x^4-x
anonymous
  • anonymous
Ok i get that
Michele_Laino
  • Michele_Laino
I have used the usual rule: \[\Large \begin{gathered} \left( - \right) \cdot \left( - \right) \cdot \left( + \right) = + \hfill \\ \left( + \right) \cdot \left( - \right) \cdot \left( + \right) = - \hfill \\ \left( + \right) \cdot \left( + \right) \cdot \left( + \right) = + \hfill \\ \end{gathered} \]
Michele_Laino
  • Michele_Laino
so, what is your solution?
anonymous
  • anonymous
I would like to say Everything less than 1 including 1
Michele_Laino
  • Michele_Laino
you have to search for minus sign
anonymous
  • anonymous
So the solution would be [0,1]?
Michele_Laino
  • Michele_Laino
correct!!
anonymous
  • anonymous
I can see it in the lines so thank you for drawing those out
Michele_Laino
  • Michele_Laino
:)
anonymous
  • anonymous
Ill go over the thread a couple times to study, thank you!
Michele_Laino
  • Michele_Laino
:)

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