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# If 0<2x+3y<10 and -10<3x+2y<0, then which of the following must be tru

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Math Revolution GMAT Instructor
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If 0<2x+3y<10 and -10<3x+2y<0, then which of the following must be tru  [#permalink]

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Updated on: 21 Jan 2018, 06:05
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65% (hard)

Question Stats:

67% (02:52) correct 33% (02:12) wrong based on 69 sessions

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[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

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"Only $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Originally posted by MathRevolution on 18 Jan 2018, 01:44. Last edited by chetan2u on 21 Jan 2018, 06:05, edited 1 time in total. corrected the OA Math Expert Joined: 02 Aug 2009 Posts: 7953 Re: If 0<2x+3y<10 and -10<3x+2y<0, then which of the following must be tru [#permalink] ### Show Tags 18 Jan 2018, 06:53 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 A. I only B. II only C. I & II D. I & III E. I, II, &III MathRevolution, pl correct the OA.. \(0<2x+3y<10$$ and $$-10<3x+2y<0$$............$$3x+2y<0<2x+3y..............(2x+2y)+x<0<(2x+2y)+y$$ from above you can make out that when x is added to 2x+2y it becomes negative and when to same value 2x+2y, y is added, it becomes positive.. so x<0 and y>0. Also xI and III are correct .. D.. Just looking at CHOICES you can eliminate all but 2.. $$0<2x+3y<10$$ and $$-10<3x+2y<0$$ If both x and y are positive, 3x+2y can never be NEGATIVE..... so $$-10<3x+2y<0$$ is not possible If both x and y are negative, 2x+3y can never be POSITIVE..... so $$0<2x+3y<10$$ is not possible therefore both x and y are of opposite sign.. 1) BOTH I and II cannot be true.... eliminate C and E 2) if y<0, x cannot be so either I and III are true or ONLY II is true in choices only B and D are left _________________ Retired Moderator Joined: 25 Feb 2013 Posts: 1179 Location: India GPA: 3.82 Re: If 0<2x+3y<10 and -10<3x+2y<0, then which of the following must be tru [#permalink] ### Show Tags 20 Jan 2018, 05:07 MathRevolution wrote: [GMAT math practice question] If $$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$$ A. I only B. II only C. I & II D. I & III E. I, II, &III $$0<2x+3y<10$$-----------(1) $$-10<3x+2y<0$$----------(2), Multiply this inequality by $$-1$$ $$=>0<-3x-2y<10$$, add this inequality with inequality (1), to get $$0<y-x<20 => y-x>0$$ or $$y>x$$. Hence Statement III must be true Also from inequality (2) we know that $$3x+2y<0$$. This implies both $$x$$ & $$y$$ are of opposite signs. As we have established that III must be true so $$y$$ cannot be negative. Hence $$x<0$$. Therefore statement I must be true Answer: option D Retired Moderator Joined: 25 Feb 2013 Posts: 1179 Location: India GPA: 3.82 Re: If 0<2x+3y<10 and -10<3x+2y<0, then which of the following must be tru [#permalink] ### Show Tags 20 Jan 2018, 05:08 MathRevolution wrote: [GMAT math practice question] If $$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 A. I only B. II only C. I & II D. I & III E. I, II, &III Hi MathRevolution, agree with chetan2u. the OA is wrong, kindly correct it. Target Test Prep Representative Status: Founder & CEO Affiliations: Target Test Prep Joined: 14 Oct 2015 Posts: 8043 Location: United States (CA) Re: If 0<2x+3y<10 and -10<3x+2y<0, then which of the following must be tru [#permalink] ### Show Tags 20 Jan 2018, 07:30 1 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 _________________ # Scott Woodbury-Stewart Founder and CEO Scott@TargetTestPrep.com 122 Reviews 5-star rated online GMAT quant self study course See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews If you find one of my posts helpful, please take a moment to click on the "Kudos" button. Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 8001 GMAT 1: 760 Q51 V42 GPA: 3.82 If 0<2x+3y<10 and -10<3x+2y<0, then which of the following must be tru [#permalink] ### Show Tags 21 Jan 2018, 18:40 => Label the inequalities as follows: 0<2x+3y<10 --- (1) -10<3x+2y<0 --- (2) We consider each statement individually. Statement I: Multiplying (1) by -2 yields -20 < -4x – 6y < 0, and multiplying (2) by 3 yields -30 < 9x + 6y < 0. Adding these inequalities gives -50 < 5x < 0 or -10 < x < 0. This statement is true. Statement II: Multiplying (1) by 3 yields 0 < 6x + 9y < 30, and multiplying (2) by -3 yields 0 < -6x – 4y < 20. Adding these inequalities gives 0 < 5y < 50 or 0 < y < 10. This statement is false. Statement III: Multiplying (1) by – 1 yields -10 < -2x – 3y < 0. Adding this to inequality (2) yields -20 < x – y < 0. This implies that x < y, and statement III is true. Therefore, the answer is D. Answer: D _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 1 month Online Course"
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If 0<2x+3y<10 and -10<3x+2y<0, then which of the following must be tru   [#permalink] 21 Jan 2018, 18:40
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