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massyeah
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Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1 11 Aug 2022, 04:13
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Bunuel
Are x and y both positive?


(1) \(2x-2y=1\). Well this one is clearly insufficient. You can do it with number plugging OR consider the following: both x and y are positive means that point (x,y) is in the I quadrant. \(2x-2y=1\) --> \(y=x-\frac{1}{2}\). We know it's an equation of a line and basically the question asks whether this line (all (x,y) points of this line) is only in I quadrant. This line for sure passes I quadrant (for example, x = 1.5 and y = 1) but it cannot entirely be only in I quadrant, so there must be some (x, y) points whose coordinates are not both positive. Not sufficient.


(2) \(\frac{x}{y}>1\) --> x and y have the same sign. But we don't know whether they are both positive or both negative. Not sufficient.


(1)+(2) Again it can be done with different approaches. You should just find the one which is the less time-consuming and comfortable for you personally. One of the approaches:

From (1): \(2x-2y=1\), so \(x=y+\frac{1}{2}\)

From (2): \(\frac{x}{y}>1\);

Substitute x into (2): \(\frac{y + \frac{1}{2}}{y}>1\);

\(1 + \frac{1}{2y}>1\);

\(\frac{1}{2y}>0\);

\(y > 0\). \(y\) is positive. Thus, \(x=y+\frac{1}{2}=positive+positive=positive\). So, \(x\) is positive too. Sufficient.


Answer: C.


You can also check GRAPHIC APPROACH below.
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What if x=-1/2 and y=-3/4
Both equations are satisfied.

It should be E
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Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1 11 Aug 2022, 04:18
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BrentGMATPrepNow

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
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What if x=-1/2 and y=-3/4
Both equations are satisfied.
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Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1 17 Jun 2023, 08:42
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Manbehindthecurtain
Are x and y both positive?


(1) \(2x - 2y = 1\)

(2) \(\frac{x}{y} > 1\)
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I have a doubt in this one:
Here using 2 we can say \(\frac{x}{y}\) >1
hence x > y
if I take x=-0.5 and y=-1
it satisfy x>y
also putting it in eq 1 :
2(-0.5)-2(-1) = -1+2 = 1

hence -0.5 and -1 satisfy both 1 and 2 still x and y both are negative so the answer should be e?
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Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1 12 Nov 2025, 09:57
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from statement 2, we know that x and y have the same sign. so if we plug in values of x and y with same sign in stmnt 1.

For x=-1/2 and y=-1, we get 2x-2y=1

For x= 1 and y= 1/2 we get 2x-2y =1.

From this it seems that both negative and positive values of x and y can satisfy statement 1. Please correct me if I am wrong but I think the answer should be E.
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Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1 12 Nov 2025, 10:03
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eaminima
from statement 2, we know that x and y have the same sign. so if we plug in values of x and y with same sign in stmnt 1.

For x=-1/2 and y=-1, we get 2x-2y=1

For x= 1 and y= 1/2 we get 2x-2y =1.

From this it seems that both negative and positive values of x and y can satisfy statement 1. Please correct me if I am wrong but I think the answer should be E.
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If x = -1/2 and y = -1, x/y = 1/2, which is not greater than 1 as (2) says.

Please study the previous 4 pages of discussion for more.

P.S. Pure algebraic questions are no longer a part of the DS syllabus of the GMAT.

DS questions in GMAT Focus encompass various types of word problems, such as:

  • Word Problems
  • Work Problems
  • Distance Problems
  • Mixture Problems
  • Percent and Interest Problems
  • Overlapping Sets Problems
  • Statistics Problems
  • Combination and Probability Problems

While these questions may involve or necessitate knowledge of algebra, arithmetic, inequalities, etc., they will always be presented in the form of word problems. You won’t encounter pure "algebra" questions like, "Is x > y?" or "A positive integer n has two prime factors..."

Check GMAT Syllabus for Focus Edition

You can also visit the Data Sufficiency forum and filter questions by OG 2024-2025, GMAT Prep (Focus), and Data Insights Review 2024-2025 sources to see the types of questions currently tested on the GMAT.

So, you can ignore this question.

Hope it helps.­
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