Question 251837: Please help me solve the inequality of this equation:
(x-3)(x+4)(x-1)<0.
Whenever I solve this the solution I get is : x^3-13x+12<0
And I don't know what I'll do next.
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! Actually the way to solve this is to leave it factored.
The inequality you have says that the product of three numbers is less than zero. In other words you have a product that is negative. The key to solving this comes from figuring out how the product of three numbers can be negative. With some basic knowledge of multiplication and a little thought we should be able to figure out that a product of three numbers is negative when- All three factors are negative, or
- One factor is negative and the other two are positive.
So we can solve this if we can use inequalities to express the idea that "all three factors are negative or two are positive and one is negative". The straightforward approach would be to write expressions like:
x-3 < 0 and x+4 < 0 and x-1 < 0
for "all three factors are negative" and for "two are positive and one is negative" we could write:
((x-3 < 0 and x+4 > 0 and x-1 > 0) or (x-3 > 0 and x+4 < 0 and x-1 > 0) or (x-3 > 0 and x+4 > 0 and x-1 < 0))
The combined inequalities would be:
(x-3 < 0 and x+4 < 0 and x-1 < 0) or ((x-3 < 0 and x+4 > 0 and x-1 > 0) or (x-3 > 0 and x+4 < 0 and x-1 > 0) or (x-3 > 0 and x+4 > 0 and x-1 < 0))
We could use this fairly long compound inequality. But there is a better way. We can use some logic and come up with a simpler inequality. First we can determine the order of the factors. Which is largest? Which is smallest and which is in between? Even though we do not know what number x might be we can still determine the order. It shouldn't take long to realize that x+4 will always be greater than x-1 and x-3. And x-3 will always be smaller than x-1 and x+4.
Now that we know the order of the factors we can take advantage of it. If the largest of three numbers is negative, doesn't that mean that all three would have to be negative? Think about this until you understand. Think about three numbers on a number line. The largest one will be to the right of the other two. And if that rightmost number is negative (left of zero) won't the other two also be negative (left of zero)? So the short way to say that "all three factors are negative" is to say that the largest one is negative:
x+4 < 0
Using similar logic we can some up with a short way to say "two factors are positive and one is negative". Knowing the order of the factors, we can guarantee two positive factors and one negative one if the smallest one is negative and the other two are positive:
x-3 < 0 and x-1 > 0
Think about this. x-3 is the smallest of the three factors and if only one factor is to be negative that the smallest one has to be the one. But we also have to make sure the other two are positive. This is done with x-1 > 0. x-1 is the in between number of the three factors. If it is positive won't the largest one be positive, too? Again picture a number line. We want three dots (numbers) and we want the zero of the number line to be between the leftmost dot and the in between dot.
So instead of
(x-3 < 0 and x+4 < 0 and x-1 < 0) or ((x-3 < 0 and x+4 > 0 and x-1 > 0) or (x-3 > 0 and x+4 < 0 and x-1 > 0) or (x-3 > 0 and x+4 > 0 and x-1 < 0))
we can use
(x+4 < 0) or (x-3 < 0 and x-1 > 0)
Both will give us the correct answer. Which do you want to solve? I hope you agree that the extra effort to find the shorter expressions for "all three are negative or two are positive and one is negative" is worth it.
Solving
(x+4 < 0) or (x-3 < 0 and x-1 > 0)
we get
(x < -4) or (x < 3 and x > 1)
which says that any number below -4 works or any number between 1 and 3 (not including 1 or 3) works.
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