MGMAT Advanced Quant Chapter 4, Question 4, Pg 123

[budoshi]

Data sufficiency: Isthe expression a+a^(-1) >2
1) a>0
2) a<1

I broke down a+a^(-1)>2 into:
a^2 - 2a +1 >0 or a^2 - 2a + 1 <0 (flipped inequalities since we do not know a is positive).

this results in (a-1)^2<0 (this is invalid) or (a-1)^2>0.
so the rephrased question: Is a>1 ?

Statement 1 - Insufficient since a>0
Statement 2 is sufficient since it directly answers the question that a<1.

Why is this WRONG?

[RonPurewal]

Statement 2 is sufficient since it directly answers the question that a<1.

Why is this WRONG?

well, you can quickly and incontrovertibly prove that statement 2 is insufficient by finding specific cases that give different answers to the question.
if a = 1/2, then it satisfies statement 2 (because 1/2 < 1). this value gives YES to the original question, since 1/2 + 2 > 2.
if a = any negative number at all, then it satisfies statement 2 (because all negative numbers are less than 1). any negative number will give NO to the original question, since both terms a and 1/a are negative.

--

the problem with your argument is that you are not thinking about when each of the two inequalities you've generated is valid.

this results in (a-1)^2<0 (this is invalid)

this is the result if a is negative.
it shouldn't surprise anyone that you get something with no solutions, because, as established above, no negative number gives YES to the question.

or (a-1)^2>0.
so the rephrased question: Is a>1 ?

there are actually two things wrong here.

first, you haven't solved the problem correctly; (a - 1)^2 > 0 is actually true for ALL values of a except a = 1. (try plugging some of them in!)

second, you are neglecting the fact that this inequality only applies to positive values of a.

so, combining these observations, the correct interpretation is "YES if a is a positive number ≠ 1".
this is only settled if you have both statements.