solve `|x-1|le2`.

Answer

To solve the inequality \( |x - 1| \leq 2 \), we will follow these steps:

### Step 1: Understand the Absolute Value Inequality
The expression \( |x - 1| \leq 2 \) means that the distance between \( x \) and \( 1 \) on the number line is at most \( 2 \). This can be translated into a compound inequality.

### Step 2: Set Up the Compound Inequality
Using the property of absolute values, we can rewrite the inequality as:
\[ -2 \leq x - 1 \leq 2 \]

### Step 3: Solve the Compound Inequality
Now, we will solve the compound inequality by isolating \( x \).

1. **Left Side**:
\[ -2 \leq x - 1 \]
Add \( 1 \) to both sides:
\[ -2 + 1 \leq x \implies -1 \leq x \]

2. **Right Side**:
\[ x - 1 \leq 2 \]
Add \( 1 \) to both sides:
\[ x \leq 2 + 1 \implies x \leq 3 \]

### Step 4: Combine the Results
From the two parts, we have:
\[ -1 \leq x \leq 3 \]

### Final Answer
The solution to the inequality \( |x - 1| \leq 2 \) is:
\[ x \in [-1, 3] \]