To solve the equation \( |3x + 9| - 7 = x \), let's follow these steps:

### Step 1: Isolate the absolute value
We begin by isolating the absolute value on one side of the equation:
\[
|3x + 9| - 7 = x
\]
Add 7 to both sides:
\[
|3x + 9| = x + 7
\]

### Step 2: Remove the absolute value
To eliminate the absolute value, we need to consider two cases:

#### Case 1: \( 3x + 9 = x + 7 \)
In this case, the expression inside the absolute value is equal to the right-hand side. Let's solve for \( x \):
\[
3x + 9 = x + 7
\]
Subtract \( x \) from both sides:
\[
2x + 9 = 7
\]
Subtract 9 from both sides:
\[
2x = -2
\]
Divide by 2:
\[
x = -1
\]

#### Case 2: \( 3x + 9 = -(x + 7) \)
In this case, the expression inside the absolute value is equal to the negative of the right-hand side. Let's solve for \( x \):
\[
3x + 9 = -(x + 7)
\]
Distribute the negative sign on the right-hand side:
\[
3x + 9 = -x - 7
\]
Add \( x \) to both sides:
\[
4x + 9 = -7
\]
Subtract 9 from both sides:
\[
4x = -16
\]
Divide by 4:
\[
x = -4
\]

### Step 3: Verify the solutions
We have two potential solutions: \( x = -1 \) and \( x = -4 \). Let's check them by substituting back into the original equation.

- For \( x = -1 \):
\[
|3(-1) + 9| - 7 = -1
\]
\[
| -3 + 9 | - 7 = -1
\]
\[
|6| - 7 = -1 \quad \text{(True)}
\]

- For \( x = -4 \):
\[
|3(-4) + 9| - 7 = -4
\]
\[
| -12 + 9 | - 7 = -4
\]
\[
| -3 | - 7 = -4
\]
\[
3 - 7 = -4 \quad \text{(True)}
\]

### Final Answer:
The solutions are \( x = -1 \) and \( x = -4 \).

Would you like more details on any step? 

Here are some related questions:
1. How do absolute value equations work?
2. What are the steps for solving multi-case absolute value equations?
3. Can you explain how to graph absolute value functions?
4. How do you verify solutions in equations with absolute values?
5. What happens if there's no solution in an absolute value equation?

**Tip:** Always check both cases when solving absolute value equations since the expression inside the absolute value can be both positive and negative.

Learn how to solve the absolute value equation |3x + 9| - 7 = x through a detailed step-by-step solution. This guide covers the isolation of absolute values, considering different cases, and verifying solutions. Suitable for students in Grades 9-10.

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