Hi there,
You're right to start off factoring from the first statement as you have: 2(x-y) = 1. From there we can divide both sides of the equation by 2 and get to x-y = 1/2, and then, adding y to both sides, we land at x = y + 1/2. All this tells us is that x is .5 greater than y, but that could be the case if they're both positive, or if they're both negative, or even if x is a just-above-zero positive and y is a just-below-zero negative. So Statement (1) alone is insufficient.
With statement (2), be careful! Remember that when you're dealing with an inequality, if you divide or multiply both sides by a negative number, you must flip the direction of the inequality sign. In this case, if we multiply through by y, we don't KNOW whether we're multiplying through by a positive or a negative number. So, we wind up with two possibilities: if y is positive, the statement becomes x>y, but if y is negative, the statement becomes precisely the opposite -- x<y. This alone isn't enough to tell us whether x and y are both positive -- though it IS enough to tell us that x and y have the same sign (if y is positive, x is "even more" positive; if y is negative, x is "even more" negative. Insufficient alone; now let's combine.
In combination, we know that x and y could both be positive, with x being 1/2 greater than y. Could it equally well be that x and y are both negative? Well, no, because if they're negative, x must be less than y, but we KNOW that x is .5 greater than y. So now the only possibility is indeed that they're both positive. So in combination, the statements are sufficient.
Ashley Newman-Owens
GMAT Instructor
Veritas Prep
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