Solve the equation $|x| + 2(x - 3) = 6$ for $x$.

1. Problem Description

Solve the equation $|x| + 2(x - 3) = 6$ for $x$.

2. Solution Steps

We have the equation $|x| + 2(x - 3) = 6$.

First, we simplify the equation:

$|x| + 2x - 6 = 6$

$|x| + 2x = 12$

We need to consider two cases:

Case 1: $x \ge 0$. Then $|x| = x$, so the equation becomes

$x + 2x = 12$

$3x = 12$

$x = 4$

Since $4 \ge 0$, $x=4$ is a valid solution.

Case 2: $x < 0$. Then $|x| = -x$, so the equation becomes

$-x + 2x = 12$

$x = 12$

However, we assumed $x < 0$, so $x=12$ is not a valid solution.

Therefore, the only solution is $x = 4$.

3. Final Answer

$x = 4$