The number of solution of the equation `sgn({x})=|1-x|` is/are (where `{*}` represent fractional part function and sgn represent signum function)

Options

A. `4`
B. `3`
C. `2`
D. `1`

Answer

To solve the equation \( \text{sgn}(\{x\}) = |1 - x| \), we will analyze both sides of the equation step by step.

### Step 1: Understanding the Functions
The left-hand side, \( \text{sgn}(\{x\}) \), is the signum function of the fractional part of \( x \). The fractional part \( \{x\} = x - \lfloor x \rfloor \) lies in the interval \([0, 1)\):
- \( \text{sgn}(\{x\}) = 1 \) if \( \{x\} > 0 \) (i.e., \( x \) is not an integer),
- \( \text{sgn}(\{x\}) = 0 \) if \( \{x\} = 0 \) (i.e., \( x \) is an integer).

### Step 2: Analyzing the Right-Hand Side
The right-hand side is \( |1 - x| \):
- \( |1 - x| = 1 - x \) when \( x \leq 1 \),
- \( |1 - x| = x - 1 \) when \( x > 1 \).

### Step 3: Case Analysis
We will analyze the equation in two cases based on the value of \( x \).

#### Case 1: \( x < 1 \)
In this case, \( |1 - x| = 1 - x \).
- If \( x \) is an integer (i.e., \( x = 0 \)), then \( \text{sgn}(\{x\}) = 0 \) and \( |1 - 0| = 1 \). This does not satisfy the equation.
- If \( x \) is not an integer (i.e., \( 0 < x < 1 \)), then \( \text{sgn}(\{x\}) = 1 \) and the equation becomes \( 1 = 1 - x \). Solving this gives:
\[ x = 0 \]
However, \( x = 0 \) is not in the interval \( (0, 1) \), so there are no solutions in this case.

#### Case 2: \( x \geq 1 \)
In this case, \( |1 - x| = x - 1 \).
- If \( x \) is an integer (i.e., \( x = n \) where \( n \) is an integer), then \( \text{sgn}(\{x\}) = 0 \) and the equation becomes \( 0 = n - 1 \). This gives \( n = 1 \), which is a solution.
- If \( x \) is not an integer (i.e., \( x = n + d \) where \( 0 < d < 1 \)), then \( \text{sgn}(\{x\}) = 1 \) and the equation becomes \( 1 = x - 1 \). Solving this gives:
\[ x = 2 \]
This is valid since \( x = 2 \) is not an integer.

### Step 4: Summary of Solutions
From our analysis:
- The only integer solution is \( x = 1 \).
- The non-integer solution is \( x = 2 \).

Thus, the total number of solutions is **2**.

### Final Answer
The number of solutions of the equation \( \text{sgn}(\{x\}) = |1 - x| \) is **2**.