Question 7224: Please help me solve this problem.
Solve the inequality. State and graph the solution set.
x3 < x
Can you explain the graph in english language, because I do not have graphing software.
Answer by longjonsilver(2297)   (Show Source):
You can put this solution on YOUR website! 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  , which looks like  . Learn this shape. It starts from way down on bottom left and goes way up to top right. A  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 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  , which looks like 
Now how about  , 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: 
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  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 
so either x=0 or 
x=0 or 
x=0 or  or 
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
so answer is where x<-1 and also 0 < x < 1
Hope this helps? :-)
jon.
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