Verify that `x sgnx=|x|`; `|x|sgnx=x`; `x(sgnx)(sgnx)=x`.

Explanation

To verify the statements \( x \cdot \text{sgn}(x) = |x| \), \( |x| \cdot \text{sgn}(x) = x \), and \( x \cdot \text{sgn}(x) \cdot \text{sgn}(x) = x \), we will analyze each case based on the value of \( x \) (whether \( x < 0 \), \( x = 0 \), or \( x > 0 \)).

### Step 1: Verify \( x \cdot \text{sgn}(x) = |x| \)

**Case 1: \( x < 0 \)**
- Here, \( \text{sgn}(x) = -1 \).
- Thus, \( x \cdot \text{sgn}(x) = x \cdot (-1) = -x \).
- Since \( |x| = -x \) when \( x < 0 \), we have \( x \cdot \text{sgn}(x) = |x| \).
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