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Question 261339: x^3 - 3x^2 -x +3 <= 0
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! 
If this was an equation, we would factor the left side and use the Zero Product Property. Solving an inequality like this also starts with factoring.
The left side does not factor easily. It will factor if you use factoring by grouping or factoring by trial and error of the possible rational roots. I will use factoring by grouping. When I factor by grouping I start by rewriting the expression with all additions. This allows me to rearrange and regroup the terms freely (which is sometimes needed by the technique). It also makes the handling of minus signs easier to get right. Your expression with additions would be:
When factoring by grouping you look for pairs of terms which have a common factor. In your expression, the first two terms have a common factor of  . The last two terms do not appear to have a common factor but, as we will soon see, we will find one to use. Factoring out the common factor of the first two terms gives us:
After factoring out the other factor is (x + (-3)). This happens to be the opposite of the last two terms. So if we factor out a -1 from the last two terms we get:
(Note: when factoring by grouping, be prepared to factor out 1's and/or -1's. Sometimes, like in this problem, it is the key to successful factoring.) Now we can factor out the (x +(-3))'s:
The second factor is a difference of squares which we can factor even further. (Always keep factoring until you can factor no further.):
Now that we have it factored, what next? Well the inequality describes a product of three factors which is negative or zero. (As long as we continue to use "or equal to" inequalities, the possibility of a zero product will be handled automatically. So from here on I am only going to discuss how we determine the x values that make the product negative.) So how do we get a negative when we multiply three numbers? Answer: If all three numbers are negative or if two numbers are positive and one is negative. So our solution will come from the values that make all three factors negative or two factors positive and one negative.
To express the idea of all three factors algebraically we could use:
 and  and 
And for two positive and one negative factor we could write:
( and  and  ) or ( and and ) or ( and and )
This is a big mess but it will work. However, we can do better if we are a little clever. Can you see that the x+1 factor will always be bigger than the other two, regardless of what number x is? And can you see that the x + (-3) will always smaller than the other two, no matter what x is? After a little thought I hope these facts are clear to you. We can use these facts to our advantage. For example, if the largest of three numbers is negative, won't the other two have to be negative, too? (Think about it.) So to find the x's that make all three factors negative we only have to find the x's that make x+1 negative. So algebraically we can replace
and and
with the much simpler and easier
And for two positives and one negative we can replace
( and and ) or ( and and ) or ( and and )
with
and
because, if only one factor is negative then it must be the smallest one, x + (-3). And the other two will be positive if the middle factor is positive. So our solution will be the solution to:
or ( and )
Solving these we get:
 or ( and  )
This tells us that any number less than or equal to -1 will work and any number between 1 and 3, inclusive, will work. (Any other numbers will not be solutions to your inequality.)
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