Expert: Sherry Wallin - 4/2/2011

Question
Dear,

Actually I have a simple question on maths notation.

Informally speaking, I have a variable A that can assume any real value in the interval Ai <= A <= Aj, depending on the value of variable B. How to declare that relationship through a math notation?

Thanks a lot in advance.

Answer
Fabricio~

You need to give me more explicit information. How is the variable B related to the variable A? For instance, suppose you have a rational expression like A = 12/(B-12). Then A can be anything except for when B = 12. Now further suppose you want only values of A >=1, i.e., you want 12/(B-12)>=1. => 12/(B-12) -1 >= 0. =>
(12 - B + 12)/(B - 12) >= 0. => 24 - B >= 0 and B - !2 >= 0. => B <= 24 and B >= 12. Now break up the number line with 12 and 24 marked off. This means you are going to check the interval from (-oo, 12), (12,24), (24, oo) and see where the rational expression (24 - B)/(B-12) >= 0. Choose 0 in (-oo,12) and you will find (24 - B)/(B-12) < 0, that is negative. Now check a value in the interval (12,24) use 15 and you will find that (24 - B)/(B-12)> 0, and finally check a value in the interval (24, oo), say 25, and you will find that (24 - B)/(B-12) < 0. Recall that
12/(B-12)>=1 is equivalent to (24 - B)/(B-12) >= 0 so the values of B that make this true is B in (12,24). And now finally recall that A = 12/(B-12)thus clearly A depends on B in this example.


Math Prof

Note: in this particular example you could have easily seen that B/= 12, that B needs to be bigger than 12 and when B > 24 then A would be smaller than 1. Above I gave you the tools to algebraically solve any rational expression.