solve : `|3x-2|le1`

Explanation

To solve the inequality \( |3x - 2| \leq 1 \), we can follow these steps:

### Step 1: Understand the Absolute Value Inequality
The absolute value inequality \( |A| \leq B \) implies that \( -B \leq A \leq B \). In our case, \( A = 3x - 2 \) and \( B = 1 \).

### Step 2: Set Up the Compound Inequality
From the absolute value inequality, we can write:
\[ -1 \leq 3x - 2 \leq 1 \]

### Step 3: Solve the Left Side of the Inequality
Starting with the left side:
\[ -1 \leq 3x - 2 \]
Add 2 to both sides:
\[ -1 + 2 \leq 3x \]
\[ 1 \leq 3x \]
Now, divide by 3:
\[ \frac{1}{3} \leq x \]
This can be rewritten as:
\[ x \geq \frac{1}{3} \]

### Step 4: Solve the Right Side of the Inequality
Now, we solve the right side:
\[ 3x - 2 \leq 1 \]
Add 2 to both sides:
\[ 3x \leq 1 + 2 \]
\[ 3x \leq 3 \]
Now, divide by 3:
\[ x \leq 1 \]

### Step 5: Combine the Results
From the two parts, we have:
\[ \frac{1}{3} \leq x \leq 1 \]
This can be expressed in interval notation as:
\[ x \in \left[\frac{1}{3}, 1\right] \]

### Final Answer
Thus, the solution to the inequality \( |3x - 2| \leq 1 \) is:
\[ x \in \left[\frac{1}{3}, 1\right] \]

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