Find the number of solutions of equation (where sgn represent signum function) `sgn (x)=|x|` .

Answer

To find the number of solutions for the equation \( \text{sgn}(x) = |x| \), we will analyze both sides of the equation step by step.

### Step 1: Understand the Signum Function
The signum function, denoted as \( \text{sgn}(x) \), is defined as follows:
- \( \text{sgn}(x) = 1 \) if \( x > 0 \)
- \( \text{sgn}(x) = 0 \) if \( x = 0 \)
- \( \text{sgn}(x) = -1 \) if \( x < 0 \)

### Step 2: Understand the Absolute Value Function
The absolute value function, denoted as \( |x| \), is defined as:
- \( |x| = x \) if \( x \geq 0 \)
- \( |x| = -x \) if \( x < 0 \)

### Step 3: Set Up Cases Based on the Value of \( x \)
We will analyze the equation \( \text{sgn}(x) = |x| \) by considering different cases based on the value of \( x \).

#### Case 1: \( x > 0 \)
- Here, \( \text{sgn}(x) = 1 \) and \( |x| = x \).
- The equation becomes \( 1 = x \).
- This gives us the solution \( x = 1 \).

#### Case 2: \( x = 0 \)
- Here, \( \text{sgn}(x) = 0 \) and \( |x| = 0 \).
- The equation becomes \( 0 = 0 \).
- This is true, so \( x = 0 \) is also a solution.

#### Case 3: \( x < 0 \)
- Here, \( \text{sgn}(x) = -1 \) and \( |x| = -x \).
- The equation becomes \( -1 = -x \) or \( x = 1 \).
- However, since \( x < 0 \), there are no solutions from this case.

### Step 4: Count the Solutions
From the analysis above, we found:
- One solution from Case 1: \( x = 1 \)
- One solution from Case 2: \( x = 0 \)
- No solutions from Case 3.

Thus, the total number of solutions is \( 2 \).

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

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