Solution for the inequality \(|3x-7|\le 2\) is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{5}{3}\le x\le 3\)
For a > 0, \(|t| \le a\) means \(-a \le t \le a\). So
\(-2 \le 3x-7 \le 2\)
Adding 7 throughout:
\(5 \le 3x \le 9\)
Dividing by 3 (positive, so the signs stay):
\( \displaystyle \frac{5}{3} \le x \le 3 \)
Why the other options are incorrect
Option (1) \(x\le 3\) — Incorrect. It drops the lower bound: x = 0 gives |−7| = 7 > 2.
Option (2) \(x\ge\frac{5}{3}\) — Incorrect. It drops the upper bound: x = 4 gives |5| = 5 > 2.
Option (4) \(x\le\frac{5}{3}\) or \(x\ge 3\) — Incorrect. This is the solution of \(|3x-7| \ge 2\), the reverse inequality.