Is x > 0?

[BTGModeratorVI]

Is x > 0?

(1) (x + y)^2 < (x - y)^2
(2) x + y < x - y

Answer: C
Source: Magoosh

[Brent@GMATPrepNow]

Is x > 0?

(1) (x + y)^2 < (x - y)^2
(2) x + y < x - y

Answer: C
Source: Magoosh

Target question: Is x POSITIVE?

Statement 1: (x + y )² > (x - y)²
Take: (x + y )² > (x - y)²
Expand and simplify both sides to get: x² + 2xy + y ² > x² - 2xy + y ²
Subtract x² from both sides: 2xy + y ² > -2xy + y ²
Subtract y² from both sides: 2xy > -2xy
Add 2xy both sides: 4xy > 0
Divide both sides by 4 to get: xy > 0
If the product xy> 0, then there are two possibilities:
Case a: x is positive and y is positive. In this case, the answer to the target question is YES, x is positive
Case b: x is negative and y is negative. In this case, the answer to the target question is NO, x is not positive
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x + y > x - y
Take: x + y > x - y
Add y to both sides: x + 2y > x
Subtract x from both sides: 2y > 0
Divide both sides by 2 to get: y > 0
So, we know that y is positive, but we have no information about x
So, we cannot answer the target question with certainty.
Statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that there are two possible cases (case a: x and y are both positive OR case b: x and y are both negative)
Statement 2 tells us that y is positive
So, statement 2 ELIMINATES case b, which means x and y are both positive
The answer to the target question is YES, x is positive
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

[beatGMAT1221]

Is x > 0?

(1) (x + y)^2 < (x - y)^2
(2) x + y < x - y

Answer: C
Source: Magoosh

Target question: Is x POSITIVE?

Statement 1: (x + y )² > (x - y)²
Take: (x + y )² > (x - y)²
Expand and simplify both sides to get: x² + 2xy + y ² > x² - 2xy + y ²
Subtract x² from both sides: 2xy + y ² > -2xy + y ²
Subtract y² from both sides: 2xy > -2xy
Add 2xy both sides: 4xy > 0
Divide both sides by 4 to get: xy > 0
If the product xy> 0, then there are two possibilities:
Case a: x is positive and y is positive. In this case, the answer to the target question is YES, x is positive
Case b: x is negative and y is negative. In this case, the answer to the target question is NO, x is not positive
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x + y > x - y
Take: x + y > x - y
Add y to both sides: x + 2y > x
Subtract x from both sides: 2y > 0
Divide both sides by 2 to get: y > 0
So, we know that y is positive, but we have no information about x
So, we cannot answer the target question with certainty.
Statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that there are two possible cases (case a: x and y are both positive OR case b: x and y are both negative)
Statement 2 tells us that y is positive
So, statement 2 ELIMINATES case b, which means x and y are both positive
The answer to the target question is YES, x is positive
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

In the question, first statement is
1) (x + y)^2 < (x - y)^2
But in the solution, you have taken
(x + y )² > (x - y)²

Similarly, for the second statement,
in the question -> x + y < x - y
in the answer -> x + y > x - y

Please make the correction.