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We are tasked with solving the equation:

\[
3 |2x| + 4 = 12
\]

### Step 1: Isolate the absolute value term
First, subtract 4 from both sides of the equation:

\[
3 |2x| = 12 - 4
\]
\[
3 |2x| = 8
\]

### Step 2: Divide by 3
Next, divide both sides of the equation by 3:

\[
|2x| = \frac{8}{3}
\]

### Step 3: Remove the absolute value
The absolute value expression \( |2x| \) can be rewritten as two equations:

1. \( 2x = \frac{8}{3} \)
2. \( 2x = -\frac{8}{3} \)

### Step 4: Solve for \( x \)
For the first equation \( 2x = \frac{8}{3} \), divide both sides by 2:

\[
x = \frac{8}{3} \div 2 = \frac{8}{3} \times \frac{1}{2} = \frac{4}{3}
\]

For the second equation \( 2x = -\frac{8}{3} \), divide both sides by 2:

\[
x = -\frac{8}{3} \div 2 = -\frac{8}{3} \times \frac{1}{2} = -\frac{4}{3}
\]

### Step 5: Conclusion
The solutions to the equation are:

\[
x = \frac{4}{3} \quad \text{or} \quad x = -\frac{4}{3}
\]

Thus, the correct answer is the option \( \pm \frac{4}{3} \).

---

Do you need more details on any step? Here are five related questions to explore this concept:

1. How do you solve an absolute value equation?
2. What happens if the equation had more complex terms inside the absolute value?
3. How do you check the solution of an absolute value equation?
4. Can absolute value equations have no solutions?
5. How do you solve absolute value inequalities?

**Tip:** When solving absolute value equations, always consider both the positive and negative cases for the expression inside the absolute value!

Learn how to solve the absolute value equation 3|2x| + 4 = 12 with a detailed step-by-step solution. This problem involves isolating the absolute value term, dividing, and solving both the positive and negative cases. Suitable for students in Grades 9-11.

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