Problem:
The sum of the greatest integer less than or equal to x and the least integer greater than or equal to x is 5. The solution set for x is
Answer Choices:
A. {25}
B. {x∣2≤x≤3}
C. {x∣2≤x<3}
D. {x∣2<x≤3}
E. {x∣2<x<3}
Solution:
Let ⌊x⌋ be the greatest integer less than or equal to x, and let ⌈x⌉ be the least integer greater than or equal to x. If x≤2, then ⌊x⌋+⌈x⌉≤2+2<5, so there are no solutions with x≤2. Similarly, if x≥3, then ⌊x⌋+⌈x⌉≥3+3, so there are no solutions here. Finally, if 2<x<3, then ⌊x⌋+⌈⌉=2+3=5, so every such x is a solution. Thus the solution set is (E).