To solve the equation $|x + 6| + 6 = 3x$, follow these steps:

Step 1: Isolate the absolute value term

Subtract 6 from both sides to isolate the absolute value expression:

$|x + 6| = 3x - 6$

Step 2: Solve for both cases of the absolute value

The absolute value equation $|x + 6| = 3x - 6$ splits into two cases:

Case 1: $x + 6 = 3x - 6$

$x + 6 = 3x - 6$

Subtract $x$ from both sides:

$6 = 2x - 6$

Add 6 to both sides:

$12 = 2x$

Divide by 2:

$x = 6$

Case 2: $-(x + 6) = 3x - 6$

$-x - 6 = 3x - 6$

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

Plug both potential solutions back into the original equation to check if they are valid:

• For $x = 6$:

$|6 + 6| + 6 = 3(6)$ $|12| + 6 = 18 \quad \text{True}$

• For $x = 0$:

$|0 + 6| + 6 = 3(0)$ $6 + 6 = 0 \quad \text{False}$

So, $x = 0$ is an extraneous solution.

Final solution:

$x = 6$

Would you like further details or have any questions?

Here are some related questions to explore:

1. How do you solve equations involving multiple absolute values?
2. Can absolute value equations have no solutions?
3. What are the properties of absolute value that are useful for solving equations?
4. How would the solution change if $|x + 6| - 6 = 3x$?
5. How can absolute value inequalities be solved?

Tip: Always check for extraneous solutions when solving absolute value equations.