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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.

Learn how to solve equations involving fractions step-by-step. This problem focuses on simplifying equations with fractions and applying fraction operations to find the solution. Suitable for high school students.

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