gmat800live wrote:
Hi all! Thanks for your help in advance. This basic question is driving me crazy! Imagine that I have two inequalities,let's call them A and B.
A is (2 < x)
B is (y + 3< x).
My understanding is that if I merge these two inequalities, I am going to get a new inequality C that will yield valid solutions for itself and also the previous two inequalities. My understanding is that to merge A and B, I can only add them up when the signs face the same direction. I did that and got a new inequality C which is
C is (y + 5 < 2x).
Now, if x is 2.5 (which is greater than 2 hence meets inequality A), then substituting 2.5 for x on C gives me y<0 HOWEVER substituting in B gives me y<-0.5. Why? Shouldn't both give me the same?
Thanks a lot.
If you add together two inequalities that are both true, and the signs are pointing the same way, you'll get a third inequality that's also true.
For example, if x < 10, and y < 3, you can be completely certain that x + y < 13.
The problem you're running into is that
it doesn't go the other way. If x + y < 13, that doesn't mean that x < 10 and y < 3. (You can see this by testing numbers, just like you did - x could be 11 and y could be 1, in which case the first of those inequalities is false!)
This is what you're doing in your example, because you're starting with the 'new' inequality you got after doing the addition, and then trying to apply the results you get from it to the 'old' inequalities you added together. That doesn't necessarily work.
In short:
- Adding two correct inequalities together will give you a new, correct inequality
- However, 'splitting up' one correct inequality
won't always give you two correct inequalities
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