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We are tasked with determining whether the equation has no solution, one solution, or infinitely many solutions. The equation provided is:

\[
-2(11 - 12x) = -4(1 - 6x)
\]

### Step-by-step Solution:

1. **Distribute the constants** on both sides of the equation:
   - On the left-hand side: \(-2 \cdot 11 = -22\) and \(-2 \cdot (-12x) = 24x\).
     So, the left-hand side becomes:
     \[
     -22 + 24x
     \]
   - On the right-hand side: \(-4 \cdot 1 = -4\) and \(-4 \cdot (-6x) = 24x\).
     So, the right-hand side becomes:
     \[
     -4 + 24x
     \]

   After distribution, the equation is:
   \[
   -22 + 24x = -4 + 24x
   \]

2. **Simplify the equation** by subtracting \(24x\) from both sides:
   \[
   -22 = -4
   \]

3. **Analyze the result**:
   - The equation simplifies to \(-22 = -4\), which is a false statement.
   - Since this is a contradiction, it means that the original equation has **no solution**.

### Conclusion:
The equation has no solution because simplifying it leads to a contradiction. The left-hand side and right-hand side cannot be equal, regardless of the value of \(x\).

---

Would you like further clarification on any step?

Here are 5 related questions for further exploration:
1. What does it mean when an equation has infinitely many solutions?
2. How can you tell when an equation has exactly one solution?
3. Can you provide an example of an equation with infinitely many solutions?
4. How do we approach solving a system of equations with multiple variables?
5. What are some other types of contradictions in equations?

**Tip**: Always simplify both sides of an equation as much as possible before drawing conclusions about the number of solutions.

Determine whether the equation has no solution, one solution, or infinitely many solutions. -2(11 - 12x) = -4(1 - 6x)

This linear equation explores whether the given expression has no solution, one solution, or infinitely many solutions. The problem involves using the distributive property and simplifying both sides. Suitable for students in Grades 8-10.

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