Explanation

The notation of this question is a little distracting, because the vertical bars look like the symbols to indicate absolute value, but in fact they indicate something else. No matter; we will keep the definition in mind. We want to know whether , according to the definition. And applying the definition, that means that we want to know whether zero is the largest integer less than or equal to n. If zero is less than n, then n positive. And if zero is the largest one less than or equal to n, that means, for example, that 1 is not less than or equal to n. So we want to know whether n is greater than 0 and less than 1. In other words, is ? Let's turn to the statements, separately first.

Statement (1) tells us that , when we multiply through by 2. Reordering, it means that . We can analyze by cases. Case I: . That's allowed by this statement. In that case, the answer to our question is "yes," it's true that . Case II: . That's allowed by this statement, but in Case II, the answer to our question is "no." Therefore, we have insufficient information to answer the question definitively. Statement (1) is insufficient.

Statement (2) tells us that n is negative. What does that mean? We can consider a case. n could be -0.5, as we just discussed for Statement (1). That case yields a "no." It could be a number such as -10, which also yields a "no." We notice that in any case allowed by Statement (2), the answer to the question will be "no." Therefore, we have sufficient information to answer the question definitively. Statement (2) is sufficient.

The correct answer is (B).