If `5{x}=x +[X] and [X] -{x}=(1)/(2)`, where {x} and [X] are fractional and integral part of x then x= ….

Answer

To solve the given problem, we need to analyze the equations involving the integral part and the fractional part of \( x \).

### Step-by-Step Solution

1. **Understanding the Notation**:
- Let \( [x] \) denote the integral part of \( x \) (the greatest integer less than or equal to \( x \)).
- Let \( \{x\} \) denote the fractional part of \( x \) (which is \( x - [x] \)).
- Therefore, we can express \( x \) as:
\[ x = [x] + \{x\} \]

2. **Setting Up the Equations**:
- From the problem, we have two equations:
1. \( 5\{x\} = x + [x] \)
2. \( [x] - \{x\} = \frac{1}{2} \)

3. **Substituting \( x \)**:
- Using the expression for \( x \):
\[ 5\{x\} = ([x] + \{x\}) + [x] \]
- Simplifying this gives:
\[ 5\{x\} = 2[x] + \{x\} \]
- Rearranging the equation:
\[ 5\{x\} - \{x\} = 2[x] \]
\[ 4\{x\} = 2[x] \]
\[ \{x\} = \frac{1}{2}[x] \]

4. **Using the Second Equation**:
- Now, substitute \( \{x\} = \frac{1}{2}[x] \) into the second equation:
\[ [x] - \frac{1}{2}[x] = \frac{1}{2} \]
- This simplifies to:
\[ \frac{1}{2}[x] = \frac{1}{2} \]
- Multiplying both sides by 2:
\[ [x] = 1 \]

5. **Finding the Fractional Part**:
- Now substitute \( [x] = 1 \) back into the equation for \( \{x\} \):
\[ \{x\} = \frac{1}{2}[x] = \frac{1}{2}(1) = \frac{1}{2} \]

6. **Finding \( x \)**:
- Now we can find \( x \):
\[ x = [x] + \{x\} = 1 + \frac{1}{2} = \frac{3}{2} \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{\frac{3}{2}} \]