SOLUTION: If a^3b^5c^7 < 0 and b > 0, which of the following must be true? I. ab < 0 II. ac < 0 III. a^2c^2 > 0

Algebra ->  Finance -> SOLUTION: If a^3b^5c^7 < 0 and b > 0, which of the following must be true? I. ab < 0 II. ac < 0 III. a^2c^2 > 0       Log On


   

Question 880568: If a^3b^5c^7 < 0 and b > 0, which of the following must be true?
I. ab < 0
II. ac < 0
III. a^2c^2 > 0
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Since a%5E3b%5E5c%5E7%3C0 , we know that a%5E3b%5E5c%5E7%3C%3E0 , so system%28a%3C%3E0%2Cb%3C%3E0%2C%22and%22%2Cc%3C%3E0%29 .
That is good to know, in case we want to divide by those numbers.
For all a%3C%3E0 and c%3C%3E0 , a%5E2%3E0 and c%5E6=%28c%5E3%29%5E2%3E0 , because they are squares.
If b%3E0 , then b%5E5%3E0 .
So a%5E2 , b%5E5 and c%5E6 are positive, and so is their product, a%5E2b%5E5c%5E6 .
Then, starting with
a%5E3b%5E5c%5E7%3C0 ,
we can divide both sides of the inequality by that positive product:
a%5E3b%5E5c%5E7%2Fa%5E2b%5E5c%5E6%22%3C%220%2Fa%5E2b%5E5c%5E6
That simplifies to highlight%28ac%3C0%29

NOTE:
Of course, a%5E2c%5E2%3E0 also must be true if system%28a%3C%3E0%2C%22and%22%2Cc%3C%3E0%29 , which is a consequence of a%5E3b%5E5c%5E7%3C0 .
So for a%5E3b%5E5c%5E7%3C0 , a%5E2c%5E2%3E0 is true if b%3E0 , and it is also true if b%3C0 ,
but III is not the answer they want,
and the answer that will help your grade is highlight%28II%29 .