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The equation you provided is:

\[
- |x - 4| + 12 = 7
\]

Let's solve this step by step:

### Step 1: Isolate the absolute value term
Subtract 12 from both sides to isolate the absolute value:

\[
- |x - 4| = 7 - 12
\]
\[
- |x - 4| = -5
\]

### Step 2: Eliminate the negative sign
Divide both sides by \(-1\) to get rid of the negative sign:

\[
|x - 4| = 5
\]

### Step 3: Solve the absolute value equation
Now that we have \( |x - 4| = 5 \), this gives us two possible cases:

1. \( x - 4 = 5 \)
2. \( x - 4 = -5 \)

#### Case 1:
\[
x - 4 = 5 \implies x = 9
\]

#### Case 2:
\[
x - 4 = -5 \implies x = -1
\]

### Final Solution:
The solutions are \( x = 9 \) and \( x = -1 \).

Would you like further details on this process or have any additional questions? 

Here are 5 related questions to explore:

1. How would the solution change if the equation were \( |x - 4| - 12 = 7 \)?
2. What if the absolute value term was \( |x + 4| \) instead of \( |x - 4| \)?
3. How can absolute value inequalities be solved, such as \( |x - 4| > 5 \)?
4. How do you graph \( |x - 4| = 5 \) on a number line?
5. What happens when there is no solution to an absolute value equation?

**Tip:** When solving absolute value equations, always consider both the positive and negative scenarios inside the absolute value function.

Step-by-step solution of the equation -|x - 4| + 12 = 7. Learn how to handle absolute value equations and isolate variables to find the solutions. Suitable for Grades 7-9.

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