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).