Let's solve the equation step by step:

\[
\frac{1}{2} \left[ 2x - 3(x - 4) \right] = \frac{2}{3}(x - 12)
\]

### Step 1: Simplify both sides of the equation.

First, simplify the expression inside the brackets on the left-hand side:

\[
2x - 3(x - 4) = 2x - 3x + 12 = -x + 12
\]

Now, substitute this simplified expression back into the equation:

\[
\frac{1}{2}(-x + 12) = \frac{2}{3}(x - 12)
\]

### Step 2: Eliminate the fractions by multiplying both sides by the least common denominator (LCD).

The LCD of 2 and 3 is 6. Multiply the entire equation by 6:

\[
6 \times \frac{1}{2}(-x + 12) = 6 \times \frac{2}{3}(x - 12)
\]

This simplifies to:

\[
3(-x + 12) = 4(x - 12)
\]

### Step 3: Distribute on both sides.

On the left-hand side:

\[
3(-x + 12) = -3x + 36
\]

On the right-hand side:

\[
4(x - 12) = 4x - 48
\]

Now, the equation is:

\[
-3x + 36 = 4x - 48
\]

### Step 4: Solve for \(x\).

First, get all terms with \(x\) on one side and constants on the other:

\[
-3x - 4x = -48 - 36
\]

\[
-7x = -84
\]

### Step 5: Divide both sides by \(-7\):

\[
x = \frac{-84}{-7} = 12
\]

### Solution:

\[
x = 12
\]

Let me know if you'd like further explanation or additional details.

### Here are 5 related questions to expand on this:
1. How do you solve equations with variables on both sides?
2. What is the importance of using the least common denominator (LCD) to eliminate fractions?
3. How does distributing over brackets help in simplifying algebraic expressions?
4. What are the steps to check if the solution \(x = 12\) is correct by substitution?
5. Can you explain how to handle equations with nested parentheses?

### Tip: Always check your final answer by substituting it back into the original equation to ensure it holds true.

This detailed solution walks through solving the linear equation 1/2[2x-3(x-4)] = 2/3(x-12) by simplifying, distributing, and eliminating fractions. Learn key concepts like the distributive property and fraction elimination, ideal for Grades 8-10 students.

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