GMAT Prep inequality

[rommysingh]

are x and y both positive?

1. 2x-2y=1
2. x/y >1

[Brent@GMATPrepNow]

Are x and y both positive ?

1) 2x - 2y = 1
2) x/y > 1

Target question: Are x and y both positive?

Statement 1: 2x - 2y = 1
There are several pairs of numbers that satisfy this condition. Here are two:
Case a: x = 1 and y = 0.5, in which case x and y are both positive
Case b: x = -0.5 and y = -1, in which case x and y are not both positive
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x/y > 1
This tells us that x/y is positive. This means that either x and y are both positive or x and y are both negative. Here are two possible cases:
Case a: x = 4 and y = 2, in which case x and y are both positive
Case b: x = -4 and y = -2, in which case x and y are not both positive
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2
Statement 1 tells us that 2x - 2y = 1.
Divide both sides by 2 to get: x - y = 1/2
Solve for x to get x = y + 1/2

Now take the statement 2 inequality (x/y > 1) and replace x with y + 1/2 to get:
(y + 1/2)/y > 1
Rewrite as: y/y + (1/2)/y > 1
Simplify: 1 + 1/(2y) > 1
Subtract 1 from both sides: 1/(2y) > 0
If 1/(2y) is positive, then y must be positive.

Statement 2 tells us that either x and y are both positive or x and y are both negative.
Now that we know that y is positive, it must be the case that x and y are both positive
Since we can now answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C

Cheers,
Brent

[Matt@VeritasPrep]

Here's a quick approach:

S1 tells us that x = y + 1/2. So x > y, but we don't know if they're positive, negative, whatever. NOT SUFFICIENT

S2 tells us one of two things:

If y > 0, then x > y

If y < 0, then x < y

This is close, but NOT SUFFICIENT either.

Together, from S1 we know that x > y, since x = y + 1/2. Now using S2, that means we must have y > 0, since that is the only case in S2 where x > y.

So y > 0 and x > y, which gives x > y > 0, and both terms are positive; SUFFICIENT!