SOLUTION: Please help and explain: x^3-4x greater than or equal to 0

Question 771758: Please help and explain:
x^3-4x greater than or equal to 0
Found 3 solutions by josgarithmetic, DrBeeee, tommyt3rd:
Answer by josgarithmetic(39629)   (Show Source): You can put this solution on YOUR website!

Factor the left hand side and analyze for signs of factors.

Critical points are at -2, 0, and 2. Check for what the signs do for the inequality at the critical points and in the intervals around them.
Answer by DrBeeee(684)   (Show Source): You can put this solution on YOUR website!
You said the magic word, so I'm your man!
Given:
(1) x^3 - 4x >= 0
This factors into 3 factors
(2) x*(x^2 - 4) >= 0 or
(3) x*(x - 2)*(x + 2) >= 0
The product of three terms is positive (greater or equal to zero) if all three of the them are positive or one is positive and the other two are negative.
Applying this condition to (3) we get
(4) all positive when x >= +2 and
(5) the first two are negative when -2 < x < 0
Answer: The inequality holds in two sections of the real line, when -2 < x < 0 and x > +2. You can graph this answer, but I can't do it here. The "empty" sections are x <= -2 and 0 <= x < +2).

Answer by tommyt3rd(5050)   (Show Source): You can put this solution on YOUR website!
x^3-4x greater than or equal to 0

has 4 intervals to be checked.
Because we are comparing the value to 0 we only need to know what its sign will be (why?)
on [-oo,-2] the signs are (-)(-)(-) -> (-) so false
on [-2,0] the signs are (-)(+)(-) -> (+) so true
on [0,2] the signs are (+)(+)(-) -> (-) so false
on [2, oo] the signs are (+)(+)(+) -> (+) so true
solution:
[-2,0]U[2,oo]

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