Find the solution set of the inequation `|x-1|lt2`.

Explanation

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

### Step 1: Understand the definition of absolute value
The expression \( |a| < b \) implies that \( -b < a < b \).

### Step 2: Apply the definition to our inequation
For our specific case, we have:
\[ |x - 1| < 2 \]
This means:
\[ -2 < x - 1 < 2 \]

### Step 3: Break it into two inequalities
We can break this compound inequality into two separate inequalities:
1. \( x - 1 > -2 \)
2. \( x - 1 < 2 \)

### Step 4: Solve the first inequality
Starting with the first inequality:
\[ x - 1 > -2 \]
Add 1 to both sides:
\[ x > -2 + 1 \]
\[ x > -1 \]

### Step 5: Solve the second inequality
Now, for the second inequality:
\[ x - 1 < 2 \]
Add 1 to both sides:
\[ x < 2 + 1 \]
\[ x < 3 \]

### Step 6: Combine the results
Now we combine the two results from Step 4 and Step 5:
\[ -1 < x < 3 \]

### Step 7: Write the solution set
The solution set can be expressed in interval notation as:
\[ (-1, 3) \]

### Final Answer:
The solution set of the inequation \( |x - 1| < 2 \) is \( (-1, 3) \).

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