To solve the equation step by step, let’s analyze it carefully:

The given equation is:

$-3| -6x + 7| = 12$

Step 1: Isolate the absolute value term

Divide both sides of the equation by $-3$ to simplify:

$| -6x + 7| = \frac{12}{-3}$

$| -6x + 7| = -4$

Step 2: Analyze the result

An absolute value (denoted by $|x|$) represents the distance from 0 and is always non-negative. In other words:

$| -6x + 7| \geq 0$

Thus, it cannot equal -4, as absolute values are never negative.

Step 3: Conclude

Since the equation leads to an impossible condition, the equation has no solutions.

Final Answer:

The correct choice is:

$\mathbf{D. \text{There are no solutions.}}$

If you have any follow-up questions or would like me to clarify further, feel free to ask!

Expanding on the Concept:

1. Why can’t absolute values be negative?
2. How would the solution change if the right-hand side were positive instead of negative?
3. Can you graph the function $-3| -6x + 7| + 12 = 0$ to confirm visually?
4. What are other types of equations involving absolute values?
5. How do we solve absolute value inequalities instead of equations?

Tip:

Always check if the absolute value condition makes sense before solving the equation!