Ok Bryan88,
I'm unfamiliar with the particular method you are referring to. But heres what I can tell you:
in your question, the range of mn would lie over the entire number line because both can take negative values.
The multiplication that you have done would be valid in two cases:
A>3 , B>4
AB>12
i.e., both LOWER limits are positive.
If even 1 LOWER limit is negative, than the product can not have a lower limit and since it has an unbounded upper limit, the product too would have no upper limit either
A<-3, B<-4
AB > 12
i.e., Both UPPER bounds are negative.
If even one upper limit is positive, than the product can not have a upper limit and since it has an unbounded lower limit, the product too would have no lower limit either
Now for these sums and where A & B have both upper and lower limits, it is always usefull to draw the number line and check the value of the expression(in this case the product), at limits of the variable:
Lets take 1 example,
-1<a<5, b<3
Now since b has no lower limit, we can assume a arbitrarily large negative number to help us say -10,000<b<3
Now, ab at a=-1 would be 10,000 for b = -10,000, and -3 for b = 3
ab at a = 5 would be -10,000 for b = -10,000, and 15 for b = 3
As we can see ab can take arbitrarily large value in both both positive and negative direction so ab has no bound.
Lets take 1 more:
50<a<60, b>-2
so, -2<b<10K
ab at a = 50, would be -100 and some large positve number at b = 10K
ab at a =60, would be -120 and some large positve number at b = 10K
As we can see, the value of ab is restricted from below so:
-120<ab
