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

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New post Updated on: 21 Jan 2018, 06:05
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A
B
C
D
E

Difficulty:

  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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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
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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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New post 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
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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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New post 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
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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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New post 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.
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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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New post 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
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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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New post 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
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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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