Solve: `0 lt |x-1|lt3`

Answer

To solve the inequality \(0 < |x - 1| < 3\), we can break it down into two parts based on the properties of absolute values.

### Step 1: Understand the Absolute Value Inequality
The expression \( |x - 1| < 3 \) means that the distance between \(x\) and \(1\) is less than \(3\). This can be expressed as:
\[ -3 < x - 1 < 3 \]

### Step 2: Solve the Compound Inequality
We can now solve the compound inequality:
1. Add \(1\) to all parts of the inequality:
\[ -3 + 1 < x - 1 + 1 < 3 + 1 \]
This simplifies to:
\[ -2 < x < 4 \]

### Step 3: Solve the Other Part of the Inequality
Next, we need to consider the other part of the original inequality \(0 < |x - 1|\). This means that \(x - 1\) cannot be equal to \(0\), so:
\[ x - 1 > 0 \quad \text{or} \quad x - 1 < 0 \]
This gives us two cases:
1. \(x - 1 > 0\) implies \(x > 1\)
2. \(x - 1 < 0\) implies \(x < 1\)

### Step 4: Combine the Results
Now we combine the results from the two parts:
1. From \( -2 < x < 4 \), we have the range of \(x\).
2. From \(x > 1\) or \(x < 1\), we can split the solution into two intervals:
- For \(x < 1\), we have \(-2 < x < 1\).
- For \(x > 1\), we have \(1 < x < 4\).

### Final Solution
Thus, the solution to the inequality \(0 < |x - 1| < 3\) is:
\[ (-2, 1) \cup (1, 4) \]