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Let's break down both parts of this problem step by step.

### **Part A**:
We are given the equation:

\[
- |x - 2| + 16 = 9
\]

1. **Isolate the absolute value term**:  
   Subtract 16 from both sides to eliminate the constant term.

   \[
   - |x - 2| = 9 - 16
   \]

   Simplifying:

   \[
   - |x - 2| = -7
   \]

2. **Remove the negative sign**:  
   Divide both sides by -1 to get rid of the negative sign.

   \[
   |x - 2| = 7
   \]

3. **Solve the absolute value equation**:  
   An absolute value equation of the form \( |A| = B \) means that \( A = B \) or \( A = -B \). So:

   \[
   x - 2 = 7 \quad \text{or} \quad x - 2 = -7
   \]

4. **Solve each case**:

   - For \( x - 2 = 7 \):
     \[
     x = 7 + 2 = 9
     \]

   - For \( x - 2 = -7 \):
     \[
     x = -7 + 2 = -5
     \]

So the solutions for Part A are:
\[
x = 9 \quad \text{or} \quad x = -5
\]

### **Part B**:
If the leading negative sign is removed, the equation becomes:

\[
|x - 2| + 16 = 9
\]

1. **Isolate the absolute value term**:  
   Subtract 16 from both sides:

   \[
   |x - 2| = 9 - 16
   \]

   Simplifying:

   \[
   |x - 2| = -7
   \]

2. **Analyze the result**:  
   Since the absolute value of an expression cannot be negative, the equation \( |x - 2| = -7 \) has no solution.

### **Conclusion**:
- **Part A**: The solutions are \( x = 9 \) and \( x = -5 \).
- **Part B**: The modified equation has no solution because the absolute value cannot equal a negative number.

Let me know if you'd like further clarification or details!

### Related Questions:
1. What are the general rules for solving absolute value equations?
2. How would the solutions change if the constant term were different (e.g., 16 was replaced with another number)?
3. Can absolute value equations have no solution? When does this happen?
4. What would happen if the absolute value expression involved a more complex term, like \( |2x - 3| \)?
5. How would you handle absolute value inequalities?

### Tip:
Always check for contradictions when solving absolute value equations—if the absolute value is set equal to a negative number, the equation has no solution!

Solve the equation −|x − 2| + 16 = 9. How does removing the leading negative sign affect the solution?

Learn how to solve the absolute value equation −|x − 2| + 16 = 9, with a step-by-step breakdown. Discover how removing the negative sign changes the solution and explore key concepts in algebra and absolute value equations. Suitable for students in Grades 8-10.

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