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Are x and y both positive?

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Are x and y both positive?

by Bens4vcobra » Mon Jun 20, 2011 4:10 pm
Are x and y both positive?

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

I know 2) is telling me x>y, and I factor 1) 2(x-y) = 1 but that's as far as I get.
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by Ashley@VeritasPrep » Mon Jun 20, 2011 6:47 pm
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.
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by goalevan » Tue Jun 21, 2011 7:57 pm
Statement 1) 2x-2y = 1, 2(x-y) = 1, x-y = 1/2, x = y + 1/2 or y = x - 1/2
Here's an alternative way to think about this statement if you have trouble determining where to go from here. Think about how the graph of this equation looks, y as a function of x. It's a line with a slope of 1 and a y-intercept of -1/2. Thus the plot will cross through quadrants I, III, and IV and you know x and y could both be positive, but also that y could be negative while x is positive or both could be negative. Insufficient.

Statement 2) As was pointed out, avoid multiplying or dividing by variables when the sign is unknown. In this case, x and y could both be positive or both be negative.

Whenever I see x/y > 0 or xy > 0, I think of this as "x and y are the same sign." I recommend memorizing this wording as it seems that this concept is heavily tested on the GMAT.

Statements 1 & 2) Combining and substituting does the trick here: with x = y + 1/2 we can substitute into the second inequality, (y + 1/2)/y > 1, (y + 1/2)/y - 1 > 0, (y + 1/2 - y)/y > 0, (1/2)/y > 0, y > 0. As I mentioned above, statement 2 says "x and y are the same sign," so x must be positive as well. The first statement also confirms this: if y is positive the graph will always be in the first quadrant and x will be positive also.
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by Brent@GMATPrepNow » Tue Sep 17, 2019 8:16 am
Bens4vcobra wrote: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
Brent Hanneson - Creator of GMATPrepNow.com
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