Is x^2>y^2? GMAT Data Sufficiency

Question: Is x^2>y^2?

1.  x < y
2. -y > x

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are not sufficient.

Correct Answer: C

Solution and Explanation:
Approach Solution 1:
The problem statement asks to find out if  x^2 is greater than y^2.
It should be noted that x^2 and y^2 are always positive numbers despite their sign.
Statement one alone: x < y 
If x < y and both x and y are both positive, then if we square both sides then x^2 < y^2.
So in that case y^2 will be larger.
Hence, statement 1 alone is not sufficient.

Statement two alone: -y > x 
But if x is negative and y is positive but the integer value of x is larger than y. Then x^2 will be larger.
For example- x = -7 and y = 6
Then x^2> y^2, However, if both x and y are negative and -y>x then, y^2 is greater.
Hence, statement 2 alone is also not sufficient.

Now taking both statements into account we get:
x< y and x < -y
X < y < -x
So y lies between -x and x, therefore the integer value of y will always be less than x.
So after squaring both sides x^2 will always be greater than y^2.
Therefore both statements are sufficient to find the answer.

Approach Solution 2:
Given to us that there are two numbers x and y. It is not mentioned whether both the numbers are positive or not. So x and y, both can be negative or positive.
The problem statement has asked to find out if x^2 > y^2
It should be noted that x^2 and y^2 are always positive numbers despite their sign.
It has asked if x^2 > y^2
It can be written as x^2 - y^2 > 0
(x-y)(x+y) > 0
It has two cases:

Both (x+y) and (x-y) have to be positive in order to be greater than 0.
Both (x+y) and (x-y) have to be negative in order to be greater than 0.

Statement 1: x< y
This implies that x-y < 0.. i.e the value of x-y is negative. We cannot surely say whether (x+y) is also negative or positive to comment that the product is greater than 0.
Hence statement 1 alone is insufficient.

Statement 2: -y > x
This implies that -y -x > 0
Or, -(x+y) > 0
Or, (x+y) < 0 (Please note that when we multiply both sides of an  inequality by -ve value, both the sign changes )
Hence, we have (x+y) < 0

However, We cannot surely say whether (x-y) is also negative or positive to comment that the product is greater than 0.

Hence statement 2 alone is insufficient.

Combining both statements together, we get:
Both (x+y) < 0 and x-y < 0
The product of two negative values is a positive value. Hence (x-y) * (x+y) > 0.
Hence x^2 – y^2 is greater than 0, in other terms we can say, x^2>y^2. 
Therefore both statements together are sufficient to find the answer.

Approach Solution 3:
The problem statement asks to find out if  x^2 is greater than y^2.
It should be noted that x^2 and y^2 are always positive numbers despite their sign.
Let’s consider that x and y are real numbers.

To solve the question, we should know if x^2 is greater than y^2 or if |x| >|y|.
Therefore, we need:
The exact value of x and y.
Any relation between x and y
Any characteristics of x and y that can give the range of x and y.

Statement 1 indicates that x < y.
We will test two values of (x,y).
Case 1: (x,y) is (1,2). For this result, |x| >|y| does not hold true.
Case 2: (x,y) is (-2,1). For this result, |x| >|y| holds true.

Therefore, we cannot determine if x^2 > y^2. Hence statement 1 alone is INSUFFICIENT. 

Statement 2 indicates that x <-y
Again we take 2 values
Case 1: (x,y) = (2,-3). For this result, |x| >|y| does not hold true.
Case 2: (x,y) = (-4, -3). For this result, |x| >|y| hold true. 

Therefore, we cannot determine if x^2 > y^2. Hence statement 2 alone is INSUFFICIENT.

Combining Statement 1 & Statement 2:
We need to solve this by assuming some specific cases.
Consider the value of y to be 4. For this, the value of x could be in the range of (-infinity. 4).
From statement 2, the final domain comes as (-infinity. -4).
All the values in this range have greater modulus values as compared to y.
Hence both statements together are sufficient to find the answer.

“Is x^2>y^2”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This question has been taken from the book “GMAT Official Guide 2021”. The GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency contains 15 questions which are two-fifths of the total 31 GMAT quant questions.

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