Question

If \[ |3x - 2| \leq \frac{1}{2}, \] then \( x \in \)

• A. \( \left[ \frac{1}{2}, \frac{5}{6} \right] \)
• B. \( \left( \frac{1}{2}, \frac{5}{6} \right) \)
• C. \( \left( \frac{1}{2}, \frac{5}{6} \right] \)
• D. \( \left[ \frac{1}{2}, \frac{5}{6} \right) \)

Hint:

When solving absolute value inequalities, split the inequality into two parts, then solve for the variable.

Answer 1

The Correct Option is A

Step 1: Solve the inequality.
We are given the inequality: \[ |3x - 2| \leq \frac{1}{2}. \] This implies: \[ -\frac{1}{2} \leq 3x - 2 \leq \frac{1}{2}. \] Now, solve for \( x \) by adding 2 to all parts of the inequality: \[ \frac{3}{2} \leq 3x \leq \frac{5}{2}. \] Next, divide by 3: \[ \frac{1}{2} \leq x \leq \frac{5}{6}. \]
Step 2: Conclusion.
Thus, \( x \) lies in the interval \( \left[ \frac{1}{2}, \frac{5}{6} \right] \), which corresponds to option (A).