The solution to the inequality \( -12\lt\displaystyle\frac{2x-1}{3}\lt 12 \) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \left(\displaystyle\frac{-35}{2},\displaystyle\frac{37}{2}\right) \)
The same operation is applied to all three parts of the double inequality.
Multiplying by 3 (positive, so the signs stay):
\( \displaystyle -36\lt 2x-1\lt 36 \)
Adding 1:
\( \displaystyle -35\lt 2x\lt 37 \)
Dividing by 2:
\( \displaystyle -\frac{35}{2}\lt x\lt\frac{37}{2} \)
The inequalities are strict, so the end points are excluded: \( x\in\left(\frac{-35}{2},\frac{37}{2}\right) \).
Why the other options are incorrect
Option (2) \( \left[\displaystyle\frac{-35}{2},\displaystyle\frac{37}{2}\right] \) — Incorrect. The square brackets include the end points, but at \( x=\frac{-35}{2} \) or \( \frac{37}{2} \) the expression equals −12 or 12, which the strict inequality does not allow.
Option (3) \( \left(\displaystyle\frac{-37}{2},\displaystyle\frac{35}{2}\right) \) — Incorrect. This comes from subtracting 1 instead of adding 1 to all three parts.
Option (4) \( \left[\displaystyle\frac{-37}{2},\displaystyle\frac{35}{2}\right] \) — Incorrect. This has both slips: 1 subtracted instead of added, and the end points included.