To solve the equation \( 12 \mid 5x - 15 \mid = 10 \), we can follow these steps:

### Step 1: Isolate the absolute value term
First, divide both sides of the equation by 12:
\[
\mid 5x - 15 \mid = \frac{10}{12} = \frac{5}{6}
\]

### Step 2: Solve for the cases of the absolute value

An absolute value equation \( \mid A \mid = B \) has two cases:
1. \( A = B \)
2. \( A = -B \)

For this equation, that gives us two cases to solve:

#### Case 1: 
\[
5x - 15 = \frac{5}{6}
\]
Add 15 to both sides:
\[
5x = \frac{5}{6} + 15 = \frac{5}{6} + \frac{90}{6} = \frac{95}{6}
\]
Now, solve for \( x \):
\[
x = \frac{95}{6 \times 5} = \frac{95}{30} = \frac{19}{6}
\]

#### Case 2:
\[
5x - 15 = -\frac{5}{6}
\]
Add 15 to both sides:
\[
5x = -\frac{5}{6} + 15 = -\frac{5}{6} + \frac{90}{6} = \frac{85}{6}
\]
Now, solve for \( x \):
\[
x = \frac{85}{6 \times 5} = \frac{85}{30} = \frac{17}{6}
\]

### Final Answer:
The solutions to the equation are:
\[
x = \frac{19}{6}, \quad x = \frac{17}{6}
\]

So the solution provided, \( \{2, 4\} \), is incorrect.

---

Would you like more details or have any questions? 

Here are some related questions to expand on this concept:
1. How do you solve absolute value equations?
2. What happens if there's no solution to an absolute value equation?
3. Can absolute value equations have more than two solutions?
4. How would the answer change if the coefficients were fractions instead of integers?
5. What are some applications of absolute value in real-world problems?

**Tip**: Always check both the positive and negative cases when solving absolute value equations.

Explore how to solve the absolute value equation 12|5x - 15| = 10 step by step. Learn about isolating the absolute value term, solving for both cases, and understanding key algebraic concepts. Ideal for students in Grades 9-11.

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