Question

Is \(x>y\)?
(1) \(x = y + 2\)
(2) \( \frac{x}{2} = y - 1 \)

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Hint:

For inequality questions like "Is \(x>y\)?", a useful strategy is to rephrase the question as "Is \(x - y>0\)?". Then, use the information in the statements to find the value or the sign of the expression \(x - y\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The question asks whether \(x\) is greater than \(y\). This is equivalent to asking if the expression \(x - y\) is positive. So, the rephrased question is: Is \(x - y>0\)? This is a "Yes/No" Data Sufficiency question.
Step 2: Detailed Explanation:
Analyze Statement (1): \(x = y + 2\).
We can rearrange this equation to find the value of \(x - y\). Subtract \(y\) from both sides: \[ x - y = 2 \] Now we evaluate the rephrased question: Is \(x - y>0\)? Is \(2>0\)? Yes. Since the answer is always "Yes", regardless of the specific values of \(x\) and \(y\), this statement is sufficient.
Analyze Statement (2): \( \frac{x}{2} = y - 1 \).
Let's rearrange this equation to find an expression for \(x - y\). First, multiply both sides by 2 to clear the fraction: \[ x = 2(y - 1) \] \[ x = 2y - 2 \] Now, subtract \(y\) from both sides to get an expression for \(x-y\): \[ x - y = 2y - 2 - y \] \[ x - y = y - 2 \] Now we evaluate the rephrased question: Is \(x - y>0\)? This is equivalent to asking: Is \(y - 2>0\)? Or, Is \(y>2\)? The answer to this question depends on the value of \(y\).

If \(y = 3\), then \(x - y = 3 - 2 = 1\), which is greater than 0. The answer is "Yes".
If \(y = 1\), then \(x - y = 1 - 2 = -1\), which is not greater than 0. The answer is "No".
Since we can get both "Yes" and "No" answers, statement (2) is not sufficient.
Step 3: Final Answer:
Statement (1) alone is sufficient to answer the question, but statement (2) alone is not.