The given expression is:

$\frac{b}{3} \left( 4a - \frac{3}{2b} \right)$

To simplify this expression, let's follow a step-by-step approach:

1. Distribute $\frac{b}{3}$ to both terms inside the parentheses: $= \frac{b}{3} \times 4a - \frac{b}{3} \times \frac{3}{2b}$

2. Simplify the first term: $= \frac{4ab}{3}$

3. Simplify the second term: $= \frac{b}{3} \times \frac{3}{2b} = \frac{3b}{3 \times 2b} = \frac{1}{2}$

4. So, the simplified expression becomes: $= \frac{4ab}{3} - \frac{1}{2}$

Final Answer:

$\frac{4ab}{3} - \frac{1}{2}$

Would you like further clarification on any step? Let me know if you need additional details!

Here are some related questions you might find useful:

1. How do I combine terms with fractions in algebraic expressions?
2. What are the rules for distributing multiplication over addition or subtraction?
3. How do you simplify complex fractions involving variables?
4. What is the process for factoring expressions like $\frac{4ab}{3} - \frac{1}{2}$?
5. How do I solve an equation that involves fractions and variables?

Tip: When distributing terms, always ensure that each part of the parentheses is multiplied by the outer term. It helps to break down the expression step-by-step for clarity.