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: Test with numbers.
Take \(a = 6\), \(b = 4\).

Step 2: Statement A.
Take \(c = 2\). Then \(6 > 4 > 2\), but the statement claims \(6 < 2\). It fails.

Step 3: Statement B and C.
With \(c = -2\): \(ac = -12\) and \(bc = -8\), so \(ac < bc\) holds. Also \(a/c = -3\) and \(b/c = -2\), so \(a/c < b/c\) holds. Both are true.

Step 4: Statement D.
With \(c = 2\): \(a/c = 3\) and \(b/c = 2\). So \(a/c > b/c\), and the claim of "less than" fails.

Step 5: Conclude.
Only A and D are wrong, so option 1.

Final Answer:
Not correct: A and D. \[ \boxed{\text{Option 1}} \]