If the sum of the greatest integer less than or equal to x and the least integer than or equal to x is 5, then the solution set for x is

Options

A. [2,3]
B. (0,5)
C. [5,6)
D. [2,3)

Answer

To solve the problem, we need to find the solution set for \( x \) given that the sum of the greatest integer less than or equal to \( x \) (denoted as \( \lfloor x \rfloor \)) and the least integer greater than or equal to \( x \) (denoted as \( \lceil x \rceil \)) is equal to 5.

### Step-by-Step Solution:

1. **Understanding the Functions**:
- The greatest integer function \( \lfloor x \rfloor \) gives the largest integer less than or equal to \( x \).
- The least integer function \( \lceil x \rceil \) gives the smallest integer greater than or equal to \( x \).

2. **Setting Up the Equation**:
- We are given the equation:
\[ \lfloor x \rfloor + \lceil x \rceil = 5 \]

3. **Using the Property of the Functions**:
- There is a known property that states:
\[ \lceil x \rceil = \lfloor x \rfloor + 1 \quad \text{(if \( x \) is not an integer)} \]
\[ \lceil x \rceil = \lfloor x \rfloor \quad \text{(if \( x \) is an integer)} \]

4. **Case 1: \( x \) is an Integer**:
- If \( x \) is an integer, then:
\[ \lfloor x \rfloor = \lceil x \rceil = x \]
- Thus, the equation becomes:
\[ x + x = 5 \implies 2x = 5 \implies x = 2.5 \]
- Since \( x \) must be an integer, this case does not yield any valid solutions.

5. **Case 2: \( x \) is Not an Integer**:
- If \( x \) is not an integer, we can use the property:
\[ \lceil x \rceil = \lfloor x \rfloor + 1 \]
- Substituting this into our equation gives:
\[ \lfloor x \rfloor + (\lfloor x \rfloor + 1) = 5 \]
- Simplifying this:
\[ 2\lfloor x \rfloor + 1 = 5 \implies 2\lfloor x \rfloor = 4 \implies \lfloor x \rfloor = 2 \]

6. **Finding \( x \)**:
- Since \( \lfloor x \rfloor = 2 \), we have:
\[ 2 \leq x < 3 \]
- Therefore, the solution set for \( x \) is:
\[ [2, 3) \]

### Final Solution:
The solution set for \( x \) is \( [2, 3) \).