To solve the equation \( |x + 6| + 6 = 3x \), follow these steps:

### Step 1: Isolate the absolute value expression.
First, subtract 6 from both sides of the equation:
\[
|x + 6| + 6 - 6 = 3x - 6
\]
\[
|x + 6| = 3x - 6
\]

### Step 2: Consider two cases for the absolute value.
The definition of absolute value gives two possible cases:

#### Case 1: \( x + 6 = 3x - 6 \)
Solve for \(x\):
\[
x + 6 = 3x - 6
\]
Subtract \(x\) from both sides:
\[
6 = 2x - 6
\]
Add 6 to both sides:
\[
12 = 2x
\]
Divide by 2:
\[
x = 6
\]

#### Case 2: \( -(x + 6) = 3x - 6 \)
This simplifies to:
\[
-x - 6 = 3x - 6
\]
Add \(x\) to both sides:
\[
-6 = 4x - 6
\]
Add 6 to both sides:
\[
0 = 4x
\]
Divide by 4:
\[
x = 0
\]

### Step 3: Check for extraneous solutions.
Now, substitute the possible solutions \(x = 6\) and \(x = 0\) into the original equation to verify them.

- For \(x = 6\):
\[
|6 + 6| + 6 = 3(6)
\]
\[
|12| + 6 = 18 \quad \text{which is true.}
\]

- For \(x = 0\):
\[
|0 + 6| + 6 = 3(0)
\]
\[
|6| + 6 = 0 \quad \text{which is false.}
\]

Thus, \(x = 0\) is not a valid solution.

### Final Answer:
The solution is \(x = 6\).

Would you like more details or have any questions?

Here are some related questions:
1. How do you handle absolute value equations with quadratic terms?
2. What are some strategies for solving absolute value inequalities?
3. Can there be extraneous solutions in other types of equations?
4. How do we solve absolute value equations graphically?
5. What happens if there's an absolute value on both sides of the equation?

**Tip:** Always check your solutions in the original equation, especially when dealing with absolute values, to avoid extraneous solutions!

Learn how to solve the absolute value equation ∣x + 6∣ + 6 = 3x with a detailed step-by-step solution. This algebra problem covers the properties of absolute values, how to handle different cases, and checking for extraneous solutions. Suitable for students in Grades 8-10.

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