Question

If [x] denotes the least integer greater than or equal to x, is [x] = 0?
(1) -1 < x < 1
(2) x < 0

• A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.
• E. Statements (1) and (2) together are not sufficient.

Answer 1

Answer: (C)

Both statements together are sufficient, but neither statement alone is sufficient.

Determine if , the least integer greater than or equal to x, is equal to 0, which is the same as determining if x satisfies .

1. Given that −1 < x < 1, then it is possible that (for example, if x = 0, then ) and it is possible that (for example, if , then ); NOT sufficient.
2. Given that x < 0, then it is possible that (for example, if , then ) and it is possible that (for example, if x = −5, then ); NOT sufficient.

• Taking (1) and (2) together gives −1 < x < 0, which implies that .
• The correct answer is C; both statements together are sufficient.