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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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18 Jan 2018, 00:44
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58% (02:17) correct 42% (01:24) wrong based on 48 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
[Reveal] Spoiler: OA

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Last edited by chetan2u on 21 Jan 2018, 05:05, edited 1 time in total.
corrected the OA
Math Expert
Joined: 02 Aug 2009
Posts: 5660
Re: If 0<2x+3y<10 and -10<3x+2y<0, then which of the following must be tru [#permalink]

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18 Jan 2018, 05:53
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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
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Re: If 0<2x+3y<10 and -10<3x+2y<0, then which of the following must be tru [#permalink]

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20 Jan 2018, 04:07
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

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Joined: 25 Feb 2013
Posts: 946
Location: India
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Re: If 0<2x+3y<10 and -10<3x+2y<0, then which of the following must be tru [#permalink]

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20 Jan 2018, 04:08
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: 2192 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, 06:30 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.

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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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21 Jan 2018, 17:40
=>

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.

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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, 17:40
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