Let's solve the equation:

\[
-4|x - 2| - 9 = -37
\]

### Step 1: Isolate the absolute value expression
Add 9 to both sides of the equation:

\[
-4|x - 2| = -37 + 9
\]

\[
-4|x - 2| = -28
\]

### Step 2: Divide both sides by -4
To get rid of the coefficient of the absolute value, divide both sides by -4:

\[
|x - 2| = \frac{-28}{-4}
\]

\[
|x - 2| = 7
\]

### Step 3: Solve for \(x\)
Now we need to solve the absolute value equation \( |x - 2| = 7 \). This gives us two possible equations:

1. \( x - 2 = 7 \)
2. \( x - 2 = -7 \)

#### Case 1:
\[
x - 2 = 7
\]
Add 2 to both sides:
\[
x = 7 + 2 = 9
\]

#### Case 2:
\[
x - 2 = -7
\]
Add 2 to both sides:
\[
x = -7 + 2 = -5
\]

### Step 4: Solutions
The two solutions are:

\[
x = 9 \quad \text{or} \quad x = -5
\]

Would you like more details or have any questions?

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Here are five related questions:

1. How do you solve absolute value equations in general?
2. What if the absolute value term was negative? How would that affect the solution?
3. Can an absolute value equation ever have no solution?
4. How would this change if the absolute value expression was part of an inequality?
5. How can we check the solutions of an absolute value equation?

**Tip**: Always check your solutions by substituting them back into the original equation!

Learn how to solve the absolute value equation -4|x − 2| − 9 = −37. This problem demonstrates key algebraic concepts such as isolating absolute values, solving two linear equations, and checking solutions. Ideal for students in Grades 8-10.

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