Which of the following properties are not correct for three real numbers a, b and c?
A. If a > b and b > c, then a < c
B. If a > b, and c < 0, then ac < bc
C. If a > b, and c < 0, then a ÷ c < b ÷ c
D. If a > b and c > 0, then a ÷ c < b ÷ c
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → A and D only
The question asks for the properties that are not correct.
By the transitive property, a > b and b > c give a > c. Multiplying or dividing by a negative number reverses an inequality, while a positive number keeps it.
So a > c in (A), the sign reverses in (B) and (C) where c < 0, and the sign stays in (D) where c > 0.
Checking the statements
(A) If a > b and b > c, then a < c — Incorrect. a > c; for example 3 > 2 > 1 gives 3 > 1.
(B) If a > b, and c < 0, then ac < bc — Correct. 3 > 2 and c = −1 give −3 < −2.
(C) If a > b, and c < 0, then a ÷ c < b ÷ c — Correct. 3 ÷ (−1) = −3 < −2 = 2 ÷ (−1).
(D) If a > b and c > 0, then a ÷ c < b ÷ c — Incorrect. a ÷ c > b ÷ c; for example 6 ÷ 2 = 3 > 2 = 4 ÷ 2.
So the properties that are not correct are A and D.