Question

Solve the equation for all values of x.
open vertical bar, x, plus, 6, close vertical bar, plus, 6, equals, 3, x
∣x+6∣+6=
3x

Answer 1

To solve the equation \( |x + 6| + 6 = 3x \), we can break it down into two cases based on the absolute value expression.

### Case 1: \( x + 6 \geq 0 \)
In this case, \( |x + 6| = x + 6 \). Substituting this into the equation gives us:

\[
x + 6 + 6 = 3x
\]

This simplifies to:

\[
x + 12 = 3x
\]

Next, rearranging the equation by subtracting \( x \) from both sides:

\[
12 = 2x
\]

Dividing both sides by 2:

\[
x = 6
\]

Now we need to check if this solution satisfies the condition for Case 1 (i.e., \( x + 6 \geq 0 \)):

\[
6 + 6 = 12 \geq 0 \quad \text{(True)}
\]

So, \( x = 6 \) is a valid solution for this case.

### Case 2: \( x + 6 < 0 \)
In this case, \( |x + 6| = -(x + 6) = -x - 6 \). Substituting this into the equation gives us:

\[
-x - 6 + 6 = 3x
\]

This simplifies to:

\[
-x = 3x
\]

Adding \( x \) to both sides yields:

\[
0 = 4x
\]

Therefore:

\[
x = 0
\]

Now we verify if this solution satisfies the condition for Case 2 (i.e., \( x + 6 < 0 \)):

\[
0 + 6 = 6 \not< 0 \quad \text{(False)}
\]

So, \( x = 0 \) is not a valid solution for this case.

### Conclusion
The only valid solution to the equation \( |x + 6| + 6 = 3x \) is:

\[
\boxed{6}
\]