For the inequality \(7 - 2x \ge x - 8\), which of the following statements are correct?
(A) When \(x\) is a natural number, the solution set is \(\{1, 2, 3, 4, 5\}\).
(B) When \(x\) is an integer, the greatest value of \(x\) satisfying it is 5.
(C) \(x = 5.5\) satisfies the inequality.
(D) When \(x\) is a real number, the solution set is \((-\infty, 5)\).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (B) only
\(7 - 2x \ge x - 8\)
\(7 + 8 \ge x + 2x\)
\(15 \ge 3x\)
\(x \le 5\)
So the inequality holds exactly when \(x \le 5\), with 5 itself included.
Checking the statements
(A) When \(x\) is a natural number, the solution set is \(\{1, 2, 3, 4, 5\}\). — Correct. The natural numbers \(\le 5\) are 1 to 5.
(B) When \(x\) is an integer, the greatest value of \(x\) satisfying it is 5. — Correct. 5 satisfies \(x \le 5\) and 6 does not.
(C) \(x = 5.5\) satisfies the inequality. — Incorrect. \(7 - 11 = -4\) and \(5.5 - 8 = -2.5\), and \(-4 \lt -2.5\).
(D) When \(x\) is a real number, the solution set is \((-\infty, 5)\). — Incorrect. \(x = 5\) gives \(-3 \ge -3\), so the set is \((-\infty, 5]\).