If [x] denotes the greatest integer less than or equal to x, then the solutions of the equation 2x-2[x]=1 are

Options

A. `x=n+(1)/(2),n in N`
B. `x=n-(1)/(2),n in N`
C. `x=n+(1)/(2),n in Z`
D. `n lt x lt n+1,n in Z`

Explanation

To solve the equation \( 2x - 2[x] = 1 \), where \([x]\) denotes the greatest integer less than or equal to \(x\), we can follow these steps:

### Step-by-Step Solution:

1. **Rewrite the Equation**:
Start with the given equation:
\[ 2x - 2[x] = 1 \]
We can factor out the 2:
\[ 2(x - [x]) = 1 \]

2. **Isolate the Fractional Part**:
Divide both sides by 2:
\[ x - [x] = \frac{1}{2} \]
Here, \(x - [x]\) represents the fractional part of \(x\), denoted as \(\{x\}\). Thus, we can rewrite the equation as:
\[ \{x\} = \frac{1}{2} \]

3. **Express \(x\) in Terms of the Greatest Integer**:
The fractional part of \(x\) can be expressed as:
\[ x = [x] + \{x\} \]
Substituting \(\{x\} = \frac{1}{2}\) into this equation gives:
\[ x = [x] + \frac{1}{2} \]

4. **Let \([x] = n\)**:
Let \([x] = n\), where \(n\) is an integer. Then we can write:
\[ x = n + \frac{1}{2} \]

5. **Determine the Range of \(x\)**:
Since \(n\) is an integer, \(x\) can take values of the form:
\[ x = n + \frac{1}{2} \quad \text{for } n \in \mathbb{Z} \]
This means \(x\) can be any half-integer value (e.g., \(-1.5, -0.5, 0.5, 1.5, 2.5, \ldots\)).

### Final Solution:
The solutions of the equation \(2x - 2[x] = 1\) are:
\[ x = n + \frac{1}{2} \quad \text{where } n \in \mathbb{Z} \]