|
Question 356999: Hi I cant seem to figure out this problem and had asked for help before but the steps and solution were confusing and I did not understand. So I am hoping you will be able to help me understand the solution.
I need to solve,
(x+11)(x-9)(x+3)>0
I need to find out the solution set, { x|______}?
Thank you so much. :o)
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! It will simplify matters if we take a moment to look at the factors and determine their order. With a little thought I hope it is obvious that
(x+11) will be the largest factor,
(x-9) will be the smallest factor, and
(x+3) will be the "in-between" factor
no matter what value x has!
Now we are ready to solve the problem.
(x+11)(x-9)(x+3) > 0
What we have is a product of three factors that is greater than zero. In other words, we have a product that is positive. Now think about how multiplication works with three numbers. Under what circumstances do you end up with a positive result? Answer (which I hope is obvious): when all three numbers are positive or when two of the three numbers are negative and the third one is positive. Any other product of three numbers will either be negative or zero.
So how do we express the idea of "all three factors are positive." The straightforward way to say this would be:
(x+11) > 0 and (x-9) > 0 and (x+3) > 0
But we can simplify this. If the smallest factor is positive, won't the other two (which are larger) also be positive? The answer is yes. So we can abbreviate our "all three factors are positive" statement to:
(x-9) > 0
Solving this for x we get:
x > 9
For "two factors are negative and one positive", we could take a long time to write and solve:
((x+11) < 0 and (x-9) < 0 and (x+3) > 0) or ((x+11) < 0 and (x-9) > 0 and (x+3) < 0) or ((x+11) > 0 and (x-9) < 0 and (x+3) < 0)
But we can simplify this if we take a moment to think. If only one factor can be positive, then that factor must be the largest factor, (x+11). And if the other two must be negative then this will happen if the "in-between" factor is negative. (Think about this. I hope the logic is clear. If not, then you will be forced to us the very long compound inequality above.) To say the largest factor must be positive and the "in-between factor must be negative we use:
(x+11) > 0 and (x+3) < 0
Solving these we get:
x > -11 and x < -3
which is equivalent to:
-11 < x < -3
Now we can put these together. Our solution is found using "All three factors are positive or two factors are negative and one is positive":
((x-9) > 0) or ((x+11) > 0 and (x+3) < 0)
which, when solved for "x", give us:
(x > 9) or (-11 < x < -3)
In plain English, the product will be positive for any x that is greater than 9 or for any x that is between -11 and -2.
Note: If the use of logic to determine the order of the factors and to simplify the inequalities we solve is too much for you then you will have to use the much longer and more complicated expressions I have shown above. And if you solve them correctly, you end up with exactly the same solution as we did above with the simpler, logic-based inequalities. It just takes a lot more work.
P.S. In response to the question in your "Thank you" note: x > 9 is only part of the solution. The full solution is:
(x > 9) or (-11 < x < -3)
|
|
|
| |
