## anonymous one year ago Need the steps to solve this inequality:

1. anonymous

$x^4-x \le 0$

2. Michele_Laino

hint: we have to factorize the binomial at the left side

3. Michele_Laino

for example, at the first step, we can write: $\Large x\left( {{x^3} - 1} \right) \leqslant 0$

4. Michele_Laino

now, we have to factorize x^3-1, do you know how to factorize it?

5. anonymous

Yes

6. Michele_Laino

ok! Then rewrite my inequality above, using your factorization

7. anonymous

would it be (x-1)( x+1)( x -1)?

8. Michele_Laino

not exactly, we have this: $\Large x\left( {x - 1} \right)\left( {{x^2} + x + 1} \right) \leqslant 0$

9. Michele_Laino

since: $\Large {x^3} - 1 = \left( {x - 1} \right)\left( {{x^2} + x + 1} \right)$

10. anonymous

I see

11. Michele_Laino

now, we can note that $\Large {{x^2} + x + 1}$ is always positive

12. Michele_Laino

so we have to study the sign of x and x-1 only

13. Michele_Laino

for example, please solve this inequality: $\Large x - 1 \geqslant 0$

14. anonymous

$x \ge 1$

15. Michele_Laino

correct! So we have this drawing: |dw:1440745389311:dw|

16. Michele_Laino

a continuous line stands for positivity, whereas a dashed line stands for negativity

17. Michele_Laino

so we have the subsequent drawing: |dw:1440745570772:dw|

18. Michele_Laino

whereas the signs +, -, + indicate the sign of x^4-x

19. anonymous

Ok i get that

20. Michele_Laino

I have used the usual rule: $\Large \begin{gathered} \left( - \right) \cdot \left( - \right) \cdot \left( + \right) = + \hfill \\ \left( + \right) \cdot \left( - \right) \cdot \left( + \right) = - \hfill \\ \left( + \right) \cdot \left( + \right) \cdot \left( + \right) = + \hfill \\ \end{gathered}$

21. Michele_Laino

22. anonymous

I would like to say Everything less than 1 including 1

23. Michele_Laino

you have to search for minus sign

24. anonymous

So the solution would be [0,1]?

25. Michele_Laino

correct!!

26. anonymous

I can see it in the lines so thank you for drawing those out

27. Michele_Laino

:)

28. anonymous

Ill go over the thread a couple times to study, thank you!

29. Michele_Laino

:)