Question

If \( y - x>x + y \), where \( x \) and \( y \) are integers, which of the following must be true?

• A. \( x<0 \)
• B. \( y>0 \)
• C. \( x<y \)
• D. \( x<0 \) and \( y>0 \)
• E. \( x>0 \) and \( y>0 \)

Hint:

When dealing with inequalities, isolate the variable and carefully check the signs and conditions for each variable involved.

Answer 1

The Correct Option is D

Step 1: Analyze the inequality.
We are given \( y - x>x + y \). Let's simplify the inequality: \[ y - x>x + y \] Subtract \( y \) from both sides: \[ -x>x \] Multiply both sides by -1: \[ x<0 \] This shows that \( x \) must be less than 0.
Step 2: Analyze further.
The inequality does not provide any additional restrictions on \( y \), so \( y \) can be either positive or negative. However, we must choose the option that best matches the condition \( x<0 \). Thus, the best answer is (D) \( x<0 \) and \( y>0 \).