Let's solve the equation step by step:

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

### Step 1: Distribute \(\frac{2}{3}\) over \((x + 2)\)
\[
\frac{2}{3}(x + 2) = \frac{2}{3}x + \frac{4}{3}
\]
Now, the equation becomes:
\[
\frac{2}{3}x + \frac{4}{3} - (x + 4) = 3x + 2
\]

### Step 2: Distribute the negative sign through \((x + 4)\)
\[
\frac{2}{3}x + \frac{4}{3} - x - 4 = 3x + 2
\]

### Step 3: Combine like terms on the left-hand side
\[
\left(\frac{2}{3}x - x\right) + \left(\frac{4}{3} - 4\right) = 3x + 2
\]

To simplify \(\frac{2}{3}x - x\), rewrite \(x\) as \(\frac{3}{3}x\):
\[
\frac{2}{3}x - \frac{3}{3}x = -\frac{1}{3}x
\]
Now simplify the constants:
\[
\frac{4}{3} - 4 = \frac{4}{3} - \frac{12}{3} = -\frac{8}{3}
\]
The equation is now:
\[
-\frac{1}{3}x - \frac{8}{3} = 3x + 2
\]

### Step 4: Eliminate fractions by multiplying the entire equation by 3
\[
3\left(-\frac{1}{3}x - \frac{8}{3}\right) = 3(3x + 2)
\]
This gives:
\[
-x - 8 = 9x + 6
\]

### Step 5: Move all terms involving \(x\) to one side and constants to the other side
\[
-x - 9x = 6 + 8
\]
Simplify:
\[
-10x = 14
\]

### Step 6: Solve for \(x\)
\[
x = \frac{14}{-10} = -\frac{7}{5}
\]

So, the solution is:
\[
x = -\frac{7}{5}
\]

Would you like further details or explanations on any step?

Here are 5 related questions for practice:
1. Solve \( \frac{4}{5}(x + 3) - (x + 5) = 2x + 7 \).
2. Solve \( \frac{3}{4}(x + 1) - 2(x + 3) = 4x + 1 \).
3. Solve \( 2(x + 3) - 3(x + 2) = 4x + 7 \).
4. Solve \( \frac{5}{6}(x + 4) - (x + 5) = 4x + 3 \).
5. Solve \( \frac{7}{8}(x + 2) - (x + 7) = 5x + 9 \).

**Tip:** To clear fractions in equations, multiply the entire equation by the least common denominator (LCD).

This step-by-step guide teaches you how to solve the linear equation 2/3(x + 2) - (x + 4) = 3x + 2. The solution uses key algebraic concepts, such as distributing fractions, combining like terms, and eliminating fractions. Ideal for students in Grades 8-10.

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