For the inequality \( 3-2x\ge x-32 \), which of the following are true ?
(A) When \( x\in N \) (set of natural numbers), the solution set is \( \{1,2,3,\ldots,11\} \).
(B) When \( x\in\mathrm{I} \) (set of integers), the solution set is \( \{-\infty,\ldots,-3,-2,-1,0,1,2,3,\ldots,10,11\} \).
(C) When \( x\in W \) (set of whole numbers), the solution set is \( \{0,1,2,3,\ldots,11\} \).
(D) When \( x\in R \) (set of real numbers), the solution set is \( (-\infty,\infty) \).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (B) and (C) only
\( 3-2x\ge x-32 \)
\( 35\ge 3x \)
\( \displaystyle x\le\frac{35}{3}=11\frac{2}{3} \)
So the largest integer solution is 11, and the real solution set is \( \left(-\infty,\frac{35}{3}\right] \).
Checking the statements
(A) When \( x\in N \) (set of natural numbers), the solution set is \( \{1,2,3,\ldots,11\} \). — Correct. The natural numbers up to \( 11\frac{2}{3} \) are 1 to 11.
(B) When \( x\in\mathrm{I} \) (set of integers), the solution set is \( \{-\infty,\ldots,-3,-2,-1,0,1,2,3,\ldots,10,11\} \). — Correct. Every integer up to 11 satisfies \( x\le 11\frac{2}{3} \).
(C) When \( x\in W \) (set of whole numbers), the solution set is \( \{0,1,2,3,\ldots,11\} \). — Correct. The whole numbers up to \( 11\frac{2}{3} \) are 0 to 11.
(D) When \( x\in R \) (set of real numbers), the solution set is \( (-\infty,\infty) \). — Incorrect. The real solution set is \( \left(-\infty,\frac{35}{3}\right] \); for example x = 12 fails.