Question

If \( x \) satisfies \( |3x-2| + |3x-4| \geq |3x-6| \), then

• A. \( 0 \leq x \leq \frac{8}{3} \)
• B. \( x \geq \frac{8}{3} \)
• C. \( x \leq 0 \) or \( x \geq \frac{8}{3} \)
• D. \( x \geq 2 \) only

Hint:

When solving inequalities involving absolute values, consider the different cases where the expressions inside the absolute values change sign.

Answer 1

The Correct Option is C

Step 1: Break down the absolute value expressions.
We will consider the different cases where each of the absolute values changes based on the values of \( x \).
Step 2: Solve for \( x \).
Solving the inequality for \( x \), we find that the solution is \( x \leq 0 \) or \( x \geq \frac{8}{3} \). Final Answer: \[ \boxed{x \leq 0 \text{ or } x \geq \frac{8}{3}} \]