Question 7224
right, i shall try again. I wrote a lot before then closed the window like a damn fool!!


Before I answer the question, just a little primer on cubics that you should LEARN.


The basic cubic is {{{y=x^3}}}, which looks like {{{graph(200,200,-2,2,-10,10, x^3)}}}. Learn this shape. It starts from way down on bottom left and goes way up to top right. A {{{y=-x^3}}} would be flipped upside down.


Now a cubic equation will typically have 3 roots/solutions (these are where the curve crosses the x-axis). The {{{y=x^3}}} version is a simplified version having just 1 root, at x=0...this really means that all 3 roots have been contorted into the one.


For a general cubic, which does have 3 roots (3 crossing of the x-axis) this means that the curve HAS to have 2 humps in it, so it can do the crossing of the x-axis... this shape is gotten by adding x^2 and x terms to the cubic equation.


As these are added, so the basic x^3 curve stretches and contorts to give these 2 humps, ey {{{y=x^3+6x^2+11x+6}}}, which looks like {{{graph(200,200,-4,1, -4, 10, x^3+6x^2+11x+6)}}}


Now how about {{{y=x^3 + 4x^2+4x}}}, this has 2 roots. So, how can a curve "cross" the x-axis just twice and still start from way down on bottom left and finish way up on top right? It looks like this: {{{graph(200,200,-4,1, -4, 10, x^3 + 4x^2+4x)}}}


As you can see, one of the bumps has moved up/down so that it just touches the x-axis...in effect 2 roots have merged to become a "duplicate"


Right, now to your question. Is is an inequality which says where is {{{x^3-x}}} less than zero. In other words, if we plotted it, where is the curve BELOW the x-axis (where is y negative). To know this, we need to know where the curve crosses the x-axis (where y=0) and then be able to sketch it, so we can then answer the actual question.


so, lets answer {{{x^3-x=0}}}


{{{x(x^2-1) = 0}}}
so either x=0 or {{{x^2-1 = 0}}}
x=0 or {{{x^2 = 1}}}
x=0 or {{{x = sqrt(1)}}} or {{{x = - sqrt(1)}}}
x=0 or x=1 or x=-1


so we have 3 solutions, we also now the rough shape, so we can now SKETCH it (not plot it) --> i have to plot it here, because i cannot "sketch" on here


{{{graph(200,200,-2,2, -5, 5, x^3-x)}}}


so answer is where x<-1 and also 0 < x < 1


Hope this helps? :-)


jon.