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Consider condition 1:
When x>0 , y can be negative or positive
Let y = -5 x =2, Then 2 < 7 + 5 Does not satisfy
Let y = -2, Then 5 < does not satisfy

When y is positive:
Let y = 5 ; x = 2
2 < 8 does not satisfy
Let y=2; x = 5
5 = 5 satisfies In this case y<x

Hence y<x. Condition 1 Sufficient.

Consider Condition 2 only.
y>0; x can be either pos / neg
Let x = -2, y =5
2 < 12 does not satisf
5 < 7 does not satisf
Let x = 2, y =5
2 < 8 does not satisf
Let x=5 y =2
5 = 5 satisfies In this case y<x

Hence y<x. Condition 2 Sufficient.
Either of these are sufficient by themselves, Hence D

In your analysis everything looks good except the first x,y pair for st. 2
(5,2)
|5| >= |5-2| + |2|
5 >= 3 + 2, satisfy the equation, x=5, which is greater than y, our answer to is y>x? is YES

here 2 > 5 is a NO

So, you can consistently answer the question with a NO for St.2.

Hardworker,
The constraints stipulate that either X or Y is always positive and if that is the case we can answer the question with a definite NO for y > x for both st1 & st2, independently.

So, I think the answer is D.

To all,
Is there a quicker way to solve these number properties problems without exhaustively picking numbers?. How can one be smart about picking numbers, if that is the only choice, so that we can definitively answer these kinds of questions.