Solve : `|x - 3| le 1`

Explanation

To solve the inequality \( |x - 3| \leq 1 \), we will break it down into two cases based on the definition of absolute value.

### Step 1: Set up the cases
The expression \( |x - 3| \leq 1 \) means that the distance between \( x \) and \( 3 \) is at most \( 1 \). This can be expressed as two separate inequalities:

1. \( x - 3 \leq 1 \) (Case 1)
2. \( -(x - 3) \leq 1 \) or \( 3 - x \leq 1 \) (Case 2)

### Step 2: Solve Case 1
From Case 1, we have:
\[ x - 3 \leq 1 \]
Adding \( 3 \) to both sides gives:
\[ x \leq 4 \]

### Step 3: Solve Case 2
From Case 2, we have:
\[ 3 - x \leq 1 \]
Subtracting \( 3 \) from both sides gives:
\[ -x \leq -2 \]
Multiplying both sides by \( -1 \) (remember to flip the inequality sign) gives:
\[ x \geq 2 \]

### Step 4: Combine the results
Now we have two inequalities:
1. \( x \leq 4 \)
2. \( x \geq 2 \)

Combining these, we get:
\[ 2 \leq x \leq 4 \]

### Step 5: Write the final answer
In interval notation, the solution is:
\[ x \in [2, 4] \]