Question

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 \(a \div c < b \div c\)
D. If \(a > b\) and \(c > 0\), then \(a \div c < b \div c\)

Choose the correct answer from the options given below:

• A. A and D only
• B. B, C and D only
• C. A and B only
• D. B and C only

Hint:

Multiply or divide by a negative number reverses the inequality.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We need the statements that are wrong. We use the rules of inequalities: order is transitive, and multiplying or dividing by a negative number reverses the inequality.

Step 2: Check statement (A).
If \(a > b\) and \(b > c\), transitivity gives \(a > c\). The statement says \(a < c\). So (A) is NOT correct.

Step 3: Check statement (B).
Multiplying \(a > b\) by a negative \(c\) reverses the sign, so \(ac < bc\). For example \(3 > 2\) and \(c = -1\) give \(-3 < -2\). So (B) is correct.

Step 4: Check statement (C).
Dividing by a negative \(c\) also reverses the sign, so \(a/c < b/c\). For example \(3/(-1) = -3 < -2 = 2/(-1)\). So (C) is correct.

Step 5: Check statement (D).
Dividing by a positive \(c\) keeps the sign, so \(a/c > b/c\). The statement says \(a/c < b/c\). So (D) is NOT correct.

Step 6: Pick the option.
The incorrect statements are A and D, which is option 1.

Final Answer:
The incorrect properties are A and D only. \[ \boxed{\text{A and D only}} \]