Data Sufficiency

Is 1 / (a-b) < (b-a)? (1) a < b (2) 1 < |a-b|

Answer: A

Approach:

Rephrase: Is 1<(a-b)(b-a)?

1) a < b
If a, b are both +ve, Both -ve, or -ve & +ve
1 > (a-b) (b-a)
Statement (1) is sufficient.

2) 1 < |a-b|

case 1: a-b>1 if a-b>0 or a>b
case 2: b-a>1 if a-b<0 or a<b

For case 2 we know that the statement 1 > (a-b) (b-a) is always true

Even for case 1: a>b, (a,b) = (9,7) or (-7,-10) or (7,-2) we get: 1 > (a-b) (b-a) is always true

and therefore statement 2 is also sufficient.

My answer is D but the correct answer is A. Can anyone please help me understand why statement 2 is not sufficient and where is the problem with my approach.

Thanks in advance