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The given problem is:

\[
\frac{14}{9} \div \left( \frac{1}{3} - \frac{1}{6} \right)
\]

### Step-by-step solution:

1. **Simplify the denominator:**
   \[
   \frac{1}{3} - \frac{1}{6}
   \]
   To subtract these fractions, find the least common denominator (LCD). The LCD of 3 and 6 is 6. Rewrite \(\frac{1}{3}\) as \(\frac{2}{6}\):
   \[
   \frac{2}{6} - \frac{1}{6} = \frac{1}{6}
   \]

2. **Rewrite the division:**
   Now the expression becomes:
   \[
   \frac{14}{9} \div \frac{1}{6}
   \]
   Dividing by a fraction is the same as multiplying by its reciprocal:
   \[
   \frac{14}{9} \times \frac{6}{1}
   \]

3. **Multiply the fractions:**
   Multiply the numerators and the denominators:
   \[
   \frac{14 \times 6}{9 \times 1} = \frac{84}{9}
   \]

4. **Simplify the fraction:**
   \(\frac{84}{9}\) can be simplified by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 3:
   \[
   \frac{84 \div 3}{9 \div 3} = \frac{28}{3}
   \]

Thus, the final answer is:
\[
\boxed{\frac{28}{3}}
\]

Would you like more details on any of these steps? Here are some related questions for further practice:

1. How do you multiply fractions and simplify the result?
2. What is the rule for dividing fractions?
3. How do you find the least common denominator for subtraction of fractions?
4. How can you determine the greatest common divisor (GCD) of two numbers?
5. Can you simplify the fraction \(\frac{28}{3}\) further?

**Tip:** Always check if fractions can be simplified by dividing both numerator and denominator by their GCD!

Evaluate the expression: 14/9 ÷ (1/3 - 1/6)

Learn how to evaluate the expression 14/9 ÷ (1/3 - 1/6) with a detailed step-by-step solution. This problem involves fraction subtraction, division, and simplification. Suitable for middle school students in grades 6-8.

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