`|x-1|lt 2 `

Answer

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

### Step 1: Understand the Absolute Value Inequality
The expression \( |x - 1| < 2 \) means that the distance between \( x \) and \( 1 \) is less than \( 2 \). This can be interpreted as:
\[ -2 < x - 1 < 2 \]

### Step 2: Split the Inequality
We can split the compound inequality into two parts:
1. \( x - 1 > -2 \)
2. \( x - 1 < 2 \)

### Step 3: Solve the First Part
For the first part \( x - 1 > -2 \):
\[ x > -2 + 1 \]
\[ x > -1 \]

### Step 4: Solve the Second Part
For the second part \( x - 1 < 2 \):
\[ x < 2 + 1 \]
\[ x < 3 \]

### Step 5: Combine the Results
Now, we combine the results from both parts:
\[ -1 < x < 3 \]

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

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