If xy 0 and x + y

Question

If xy < 0 and x + y < 0 , then which of the following must be true? I. x - y > 0 II. x > 0 or y > 0 III. \frac{x}{y} > -1 or \frac{y}{x} > -1 A. I only B. II only C. III only D. II and III only E. I, II, and III

Bunuel · Answer

Official Solution: If xy < 0 and x + y < 0 , then which of the following must be true? I. x - y > 0 II. x > 0 or y > 0 III. \frac{x}{y} > -1 or \frac{y}{x} > -1 A. I only B. II only C. III only D. II and III only E. I, II, and III xy < 0 implies that x and y have opposite signs. Hence, in conjunction with x + y < 0 , either x > 0 > y and y is further from 0 than x (to make x + y negative, for example x = 1 and y = -2 ), or y > 0 > x and x is further from 0 than y (to make x + y negative, for example x = -2 and y = 1 ). Let's examine the options: I. x - y > 0 The above implies that x > y . This option is not always true because we know that either x > 0 > y or y > 0 > x . Discard. II. x > 0 or y > 0 This option must be true. Since from xy < 0 , it follows that one of them must be positive. III. \frac{x}{y} > -1 or \frac{y}{x} > -1 This one is indeed tricky. If x > 0 > y and y is further from 0 than x , then \frac{x}{y} > -1 must be true. We can derive this by dividing the inequality x + y < 0 by the negative value y , and then flipping the inequality sign to get \frac{x}{y} + 1 > 0 , which leads to \frac{x}{y} > -1 . On the other hand, if y > 0 > x and x is further from 0 than y , then \frac{y}{x} > -1 must be true. We can derive this by dividing the inequality x + y < 0 by the negative value x , and then flipping the inequality sign to get 1 + \frac{y}{x} > 0 , which leads to \frac{y}{x} > -1 . Therefore, in either case, either \frac{x}{y} > -1 or \frac{y}{x} > -1 is always true. Hence, only options II and III are always true. Answer: D

shubhim20 · Answer

Bunuel  can you explain point 3 by giving numbers please

Bunuel · Answer

It's a must be true question, so showing one pair of (x, y) satisfying III does not prove that it must be true. You need to derive it algebraically. However, if you just need a pair (x, y) that satisfies III, you can take x = -2 and y = 1, giving y/x = -1/2 > -1.

dhruva09 · Answer

For point 3, if I consider y= -2 and x=1. It does not satisfy the condition for must be true. Can someone pls guide here.

Bunuel · Answer

III.  \frac{x}{y} > -1   or   \frac{y}{x} > -1

If x = 1 and y = -2, then  \frac{x}{y}= -\frac{1}{2}> -1 .

If xy > 0 and x + y < 0, then which of the following must be true?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions