Question

If x>2 and y>-1, then which of the following statements is necessarily true?

• A. -x<2y
• B. xy>-2
• C. xy<-2
• D. -x>2y

Hint:

Compare each statement using the conditions on x and y; note that x and y are not stated to be integers.

Answer 1

The Correct Option is A

Step 1: Convert the conditions:
\( x > 2 \) means \( -x < -2 \). \( y > -1 \) means \( 2y > -2 \).

Step 2: Compare -x and 2y:
\( -x < -2 < 2y \), so \( -x < 2y \) is always true. This is option (1). Consequently option (4), \( -x > 2y \), is always false.

Step 3: Show options (2) and (3) are not necessarily true:
Take \( x = 100 \) and \( y = -0.5 \) (both allowed). Then \( xy = -50 \), so \( xy > -2 \) (option 2) is false. Take \( x = 3 \) and \( y = 5 \). Then \( xy = 15 \), so \( xy < -2 \) (option 3) is false.

Final Answer:
The correct option is (1): -x < 2y