Let's solve the equation \( -4 |3 + x| - 7 = -15 \).

### Step 1: Isolate the absolute value term
Start by adding 7 to both sides of the equation:

\[
-4 |3 + x| - 7 + 7 = -15 + 7
\]
\[
-4 |3 + x| = -8
\]

### Step 2: Divide by -4 to further isolate the absolute value

\[
|3 + x| = \frac{-8}{-4}
\]
\[
|3 + x| = 2
\]

### Step 3: Solve the absolute value equation

An absolute value equation \( |A| = B \) can be split into two cases:

1. \( 3 + x = 2 \)
2. \( 3 + x = -2 \)

#### Case 1: \( 3 + x = 2 \)
\[
x = 2 - 3
\]
\[
x = -1
\]

#### Case 2: \( 3 + x = -2 \)
\[
x = -2 - 3
\]
\[
x = -5
\]

### Step 4: Final solution
The solutions to the equation are:
\[
x = -1 \quad \text{or} \quad x = -5
\]

Would you like further details or have any questions?

Here are 5 related questions to expand on this concept:
1. How do you solve an absolute value equation like \( |x + 4| = 7 \)?
2. What happens if the absolute value equation has no solutions?
3. Can you solve \( |2x - 1| = 5 \)?
4. How do you solve an absolute value inequality like \( |x - 2| \leq 4 \)?
5. How can you graph \( |x + 1| = 3 \) on a coordinate plane?

**Tip:** When solving absolute value equations, always remember to split them into two separate cases!

Learn how to solve the absolute value equation -4 |3 + x| - 7 = -15 with a step-by-step solution. This problem covers key algebraic concepts and the application of the absolute value equation. Suitable for students in Grades 8-10.

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