Is x > y ? GMAT Problem Solving

Question: Is x > y ?

(1) x^(1/2) > y

(2) x^3 > y

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

Correct Answer: C

Solution and Explanation:
Approach Solution 1:
The given situation assumes, x > y only when two of the following given statements are proven to be sufficient.

The first statement states that \(\sqrt{x}\) > y only if x > y. For instance, if x is equal to 4 and y is equal to 1 then, yes x > y.

However, if x is equal to ¼ and y is equal to ⅓, then no x is not greater than y.

Considering for the same statement the resultant answers are different, it can be evaluated that the statement is not sufficient.

Further, there is another statement that needs to answer to find out whether it is sufficient for proving x > y.

Accordingly, with \(x^3\) > y, where x is 4 and y is 1, then we have permission. However, if x is 2 and y is 3, then the answer would be different hence, it won’t be considered sufficient.

Combining both statements:
From the above evaluations, we have \(x\geq{y}\).

Further, when these two conditions are equated with one another, with \(x\geq{y}\)placing them on the number line would help us find the answer.

Considerably, for \(x\geq{1}\) for \(\sqrt{x}\), x and \(x^3\) and the assumption of\(x\geq{1}\), it can be evaluated that \(1\leq{\sqrt{x}\leq{x}\leq{x^3}}\). y can be considered in the green zone being less than\(y<{\sqrt{x}<{x}<{x^3}}\) and in this case, the answer is sufficient.

For the case of \(0\leq{{x}<{1}}\), the number line would have 0 to \(x^3\) to x to \(\sqrt{x}\) . y is always in the green zone. It can be stated that \(y<{\sqrt{x}<{x}<{x^3}}\) hence, x > y

Thus, both statements together are sufficient.

Approach Solution 2:
The problem statement asks to find whether x is greater than y.

Statement 1: √x>y.
If we assume the value of x = 4 and the value of y = 1, then yes, we can derive the solution to the problem.
If we assume the value of x = 1/4 and the value of y = 1/3, then No, we cannot derive the solution to the problem.
Hence statement (1) is not sufficient.

Statement 2: x^3>y
Let’s assume the value of x = 2 and y = 3, then we cannot derive the solution to the problem.
Let’s assume the value of x = 2 and y = 1, then yes, we can derive the solution to the problem.
Hence statement (2) is not sufficient.

Combining statements 1 and 2 together.
Either the value of y is negative or positive.
If y is negative then x >y always holds true.
If y is positive then x>y^2 and x^3>y => x^6 > y^2

Therefore by dividing both the inequalities, we get:
x^5 >1
=> x >1.

Since the square root of x (> 1) is greater than y => x>y.
Thus, both statements together are sufficient.

“Is x > y ?”- is a topic of the GMAT Quantitative reasoning section of GMAT. 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 are featured with a problem statement that is followed by two factual statements. GMAT data sufficiency constitutes 15 questions which are two-fifths of the total 31 GMAT quant questions.

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