The solution set \( \{x:x\in R,3x-2\lt x+2\le 2x+6\} \) (Where \( R \) is set of real numbers) is :
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → [−4,2)
A double inequality is two inequalities that must hold together.
\( \displaystyle 3x-2\lt x+2 \)
\( \displaystyle 2x\lt 4\ \Rightarrow\ x\lt 2 \)
\( \displaystyle x+2\le 2x+6\ \Rightarrow\ -4\le x \)
Together, \( -4\le x\lt 2 \), so the solution set is [−4, 2).
Why the other options are incorrect
Option (2) [−4,−2) — Incorrect. The right end is 2, not −2; for example x = 0 satisfies both inequalities.
Option (3) [−4,2] — Incorrect. x = 2 is excluded, since \( 3(2)-2=4 \) is not less than \( 2+2=4 \).
Option (4) (−4,2) — Incorrect. x = −4 is included, since \( -4+2=-2\le 2(-4)+6=-2 \) holds with equality.