Let's solve the given equation step by step.

The equation is:

$\frac{3}{2} \left[ \frac{2}{3} \left( \frac{1}{2} x + 1 \right) + 2 \right] - \frac{2}{3} = \frac{3}{2} x$

First, simplify inside the brackets:

$\frac{2}{3} \left( \frac{1}{2} x + 1 \right)$

Distribute $\frac{2}{3}$:

$\frac{2}{3} \cdot \frac{1}{2} x + \frac{2}{3} \cdot 1 = \frac{2}{6} x + \frac{2}{3} = \frac{1}{3} x + \frac{2}{3}$

Now, add 2 inside the brackets:

$\frac{1}{3} x + \frac{2}{3} + 2$

Convert 2 to a fraction with a common denominator of 3:

$\frac{1}{3} x + \frac{2}{3} + \frac{6}{3} = \frac{1}{3} x + \frac{8}{3}$

Now, multiply by $\frac{3}{2}$:

$\frac{3}{2} \left( \frac{1}{3} x + \frac{8}{3} \right)$

Distribute $\frac{3}{2}$:

$\frac{3}{2} \cdot \frac{1}{3} x + \frac{3}{2} \cdot \frac{8}{3} = \frac{3}{6} x + \frac{24}{6} = \frac{1}{2} x + 4$

So the equation now is:

$\frac{1}{2} x + 4 - \frac{2}{3} = \frac{3}{2} x$

Combine constants on the left side:

$\frac{1}{2} x + 4 - \frac{2}{3}$

Convert 4 to a fraction with a common denominator of 3:

$\frac{1}{2} x + \frac{12}{3} - \frac{2}{3} = \frac{1}{2} x + \frac{10}{3}$

So the equation now is:

$\frac{1}{2} x + \frac{10}{3} = \frac{3}{2} x$

Subtract $\frac{1}{2} x$ from both sides:

$\frac{10}{3} = \frac{3}{2} x - \frac{1}{2} x$

$\frac{10}{3} = \frac{2}{2} x = x$

So, we have:

$x = \frac{10}{3}$

The solution to the equation is $x = \frac{10}{3}$.

Would you like more details or have any questions?

Here are 5 related questions you can ask:

1. How do you solve equations involving fractions?
2. Can you explain how to simplify complex fractions?
3. What are the steps to solve a linear equation?
4. How can you verify the solution to an equation?
5. What are some common mistakes to avoid when solving equations?

Tip: When working with equations involving fractions, always try to find a common denominator to simplify the process.