Explanation:

Let's breakdown the problem:

- We need to find the values of \( x \) that satisfy the inequality \(|3x| \geq |6 - 3x|\).

- First, consider when \(3x \geq 0\) (so, \(x \geq 0\)) and \(6 - 3x \geq 0\) (so, \(x \leq 2\)). In the range \(0 \leq x \leq 2\), \(|3x| = 3x\) and \(|6 - 3x| = 6 - 3x\). Solving \(3x \geq 6 - 3x\), we find \(x \geq 1\). Thus, the solution is \([1, 2]\).

- Second, consider when \(3x \geq 0\) (so, \(x \geq 0\)) and \(6 - 3x < 0\) (so, \(x > 2\)). In this range, \(|3x| = 3x\) and \(|6 - 3x| = - (6 - 3x) = 3x - 6\). Solving \(3x \geq 3x - 6\), we find the inequality always holds. So, \(x > 2\) is part of the solution.

- For negative \(x\), consider when \(3x < 0\), hence \(x < 0\) and \(|3x| = -3x\). Since \(|6 - 3x|\) remains \(6 - 3x\) or \(-(6 - 3x)\), the range doesn't yield any feasible results where \(-3x \geq |6 - 3x|\).

- Combining results, the values of \(x\) that satisfy the inequality are \(x \in [1, 2] \cup (2, \infty)\), which simplifies to \(x \in [1, \infty)\).

Correct conclusions:

- Option B \([1, 4]\) is valid because it fits within the found solution set.

- Option C \((4, \infty)\) is valid since it also fits within the solution set.

- Option 3 (B and C only) is indeed correct.