`|log_(3) x| lt 2`

Answer

To solve the inequality \( | \log_3 x | < 2 \), we will break it down into steps.

### Step 1: Understand the Absolute Value Inequality
The inequality \( | \log_3 x | < 2 \) means that the expression \( \log_3 x \) must lie between -2 and 2. Thus, we can rewrite the inequality as:
\[ -2 < \log_3 x < 2 \]

### Step 2: Convert the Logarithmic Inequalities to Exponential Form
We will convert both parts of the inequality from logarithmic form to exponential form.

1. For the left part:
\[ -2 < \log_3 x \implies x > 3^{-2} = \frac{1}{9} \]

2. For the right part:
\[ \log_3 x < 2 \implies x < 3^2 = 9 \]

### Step 3: Combine the Results
Now, we combine the results from both inequalities:
\[ \frac{1}{9} < x < 9 \]

### Step 4: Write the Final Solution
The solution set for \( x \) is:
\[ x \in \left( \frac{1}{9}, 9 \right) \]