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Let's solve the given equation step by step.

The equation is:

\[
\frac{2}{3}(x + 2) - 2 = 3 + \frac{1}{3}(x - 5)
\]

### Step 1: Distribute the fractions.

Expand the terms that have fractions:

\[
\frac{2}{3}(x + 2) = \frac{2}{3}x + \frac{4}{3}
\]
\[
\frac{1}{3}(x - 5) = \frac{1}{3}x - \frac{5}{3}
\]

Now the equation becomes:

\[
\left(\frac{2}{3}x + \frac{4}{3}\right) - 2 = 3 + \left(\frac{1}{3}x - \frac{5}{3}\right)
\]

### Step 2: Simplify constants.

First, handle the subtraction of 2 on the left-hand side:

\[
\frac{4}{3} - 2 = \frac{4}{3} - \frac{6}{3} = -\frac{2}{3}
\]

Now the equation is:

\[
\frac{2}{3}x - \frac{2}{3} = 3 + \frac{1}{3}x - \frac{5}{3}
\]

### Step 3: Combine constants on the right-hand side.

Simplify the constants on the right-hand side:

\[
3 - \frac{5}{3} = \frac{9}{3} - \frac{5}{3} = \frac{4}{3}
\]

Now the equation is:

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

### Step 4: Eliminate the fractions by multiplying through by 3.

To simplify further, multiply both sides of the equation by 3 to eliminate the fractions:

\[
3\left(\frac{2}{3}x - \frac{2}{3}\right) = 3\left(\frac{4}{3} + \frac{1}{3}x\right)
\]

This gives:

\[
2x - 2 = 4 + x
\]

### Step 5: Solve for \(x\).

Now subtract \(x\) from both sides:

\[
2x - x - 2 = 4
\]

Simplify:

\[
x - 2 = 4
\]

Add 2 to both sides:

\[
x = 6
\]

Thus, the value of \(x\) is \(6\).

### Final Answer: \(x = 6\)

Would you like more details on any step? Here are some related questions you might want to explore:

1. How do you solve equations involving fractions?
2. How does distributing work with fractional coefficients?
3. What techniques can simplify solving multi-step equations?
4. Can you explain the logic behind multiplying both sides by the same value?
5. How do you check if a solution to an equation is correct?

**Tip:** When working with equations that involve fractions, a good strategy is to eliminate fractions early by multiplying the entire equation by the least common denominator (LCD).

In the equation \(\frac{2}{3}(x + 2) - 2 = 3 + \frac{1}{3}(x - 5)\), what is the value of x?

Learn how to solve the linear equation \(\frac{2}{3}(x + 2) - 2 = 3 + \frac{1}{3}(x - 5)\). This detailed step-by-step guide covers algebraic concepts such as distributing fractions and solving for x. Ideal for students in Grades 8-10.

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