MathRevolution wrote:
[GMAT math practice question]
If \(0<2x+3y<10\) and \(-10<3x+2y<0\), then which of the following must be true?
\(I. x<0\)
\(II. y<0\)
\(III. x<y\)
A. I only
B. II only
C. I & II
D. I & III
E. I, II, &III
We see that x and y can’t be both positive or both negative. If x and y are both positive, then the second inequality will not hold. Similarly, if x and y are both negative, then the first inequality will not hold. Therefore, we must consider two separate cases: (1) x is negative and y is positive and (2) x is positive and y is negative.
Case 1. Let’s assume that x is negative and y is positive.
For example, If x = -5 and y = 5, we see that we do have 0 < 2x + 3y < 10 and -10 < 3x + 2y < 0.
Case 2. Now let’s assume that x is positive and y is negative. We see that the absolute value of y must be greater than the absolute value of x in order for the second inequality to hold.
For example, if x = 2, y has to be less than -3 in order to have -10 < 3x + 2y < 0. However, in that case, the first inequality will never hold since 2x + 3y will be negative. Thus we can’t have x as positive and y as negative.
Thus it must be true that x is negative and y is positive and in that case we also have x < y. Thus, Statements I and III must be true.
Answer: D
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