If x does not = -y, is (x-y)/(x+y) > 1?

[GMAT 5/18]

Source: Gmat Prep, mba.com, Test II

Can someone please explain a good way to solve this problem? Thank you!

If x does not = -y, is (x-y)/(x+y) > 1?

(1) x > 0
(2) y < 0

I will post the answer a little later (for those of you wishing to try it without seeing the answer first).

[StaceyKoprince]

You can simplify the given inequality but, in doing so, you have to split the problem into two parts b/c, when multiplying or dividing an inequality by a negative number, you have to flip the signs. Since we don't know whether the quantity x+y is positive or negative, we have to do the problem both ways.

(x-y)/(x+y) > 1
means either:
if (x+y) is positive, then x-y > x+y or -y > y Note that in this circumstance, y must be negative for this to be true
if (x+y) is negative, then x-y < x+y or -y < y Note that here y must be positive for this to be true

Think about why this is true of y. Also note that, in either case, it doesn't matter what x is (and think about why this is true).

Statement 1 tells us about x and we've already decided that it doesn't matter what x is. Insufficient.
Statement 2 tells us y is negative, which fits the first of our two scenarios above. Sufficient. B.

[Guest]

Stacey -

I believe the answer here is E. Given that (1) x>0 and (2) y<0, consider the two cases below:

x=2, y=-3

(x-y)/(x+y) = 5/-1 = -5 <1
------------------------------------
x=6, y= -3

(x-y)/(x+y) = 9/3 = 3> 1


So (1) and (2) taken together are insufficient to answer the question of whether (x-y)/(x+y) >1. Did I miss something here?

Cheers,
Jeff

[GMAT 5/18]

Jeff,

You are correct; the answer is E.

Stacey,

I think I may have found the flaw in your reasoning. You wrote: Statement 2 tells us y is negative, which fits the first of our two scenarios above. Sufficient. B. We do know y is negative, but we do not know what x+y is positive, which is what we needed for B to be sufficient.

[christiancryan]

You're exactly right, GMAT 5/18, about the flaw in Stacey's reasoning. Her solution structure is a good one, though:

(x-y)/(x+y) > 1?
means either:

a) if (x+y) is positive, then the question becomes
x-y > x+y? (direction of inequality sign stays the same as you cross-multiply)
-y > y? (subtract x -- no change to sign)
0 > 2y? (add y -- no change to sign)
0 > y? (divide by 2 -- no change to sign)
=> is y negative?

b) if (x+y) is negative, then the question becomes
x-y < x+y? (direction of inequality sign flips as you cross-multiply)
same chain of algebra
0 < y?
=> is y positive?

Even with the two conditions together, we never know for sure the sign of x+y, so we can't determine which case we're in and therefore which question is being asked.

Good question!