Online Course Discussion Forum
MC-II A Algebra Lesson 3, example question 3.5
Hi Nicholas, good question! We need to be careful with some of the logic here.
Consider the two statements: (1) "X1 and X2 are both > 2" and (2) "X1+X2 > 4 and X1*X2 > 4".
Pretty clearly (1) implies (2), as if X1 and X2 are both > 2, then there sum and product are both > 4.
However, does (2) imply (1)? Here the answer is no. For example, if X1 = 1.5 and X2 = 4 then statement (2) is true but (1) is not.
This is what goes wrong with your method, as situations like the above X1 = 1.5 and X2 = 4 are not excluded.
What you really want to rely on here is positive and negative numbers, that is pos * pos = pos, pos * neg = neg, etc. This is where the idea of (X1-2) and (X2-2) comes from. If we look at a third statement (3) "(X1-2) + (X1-2) > 0 and (X1-2)(X2-2) > 0" we now do have that statements (1) and (3) are logically equivalent.
Hope this helps!