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If 0<2x+3y<50 and -50<3x+2y<0, then which of the following must be tr

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If 0<2x+3y<50 and -50<3x+2y<0, then which of the following must be tr [#permalink]

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[GMAT math practice question]
If \(0<2x+3y<50\) and \(-50<3x+2y<0\), then which of the following must be true?

I. \(x>0\)
II. \(y>0\)
III. \(x<y\)


A. I only
B . II only
C. III only
D. I and III
E. II, and III
[Reveal] Spoiler: OA

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Last edited by MathRevolution on 23 Nov 2017, 12:05, edited 1 time in total.

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If 0<2x+3y<50 and -50<3x+2y<0, then which of the following must be tr [#permalink]

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New post 22 Nov 2017, 08:50
MathRevolution wrote:
[GMAT math practice question]
If \(0<2x+3y<50\) and \(-50<3x+2y<0\), then which of the following must be true?

I. \(x>0\)
II. \(y>0\)
III. \(x<y\)


A. I only
B . II only
C. III only
D. I and III
E. II and III


\(0<2x+3y<50 => 2x+3y\) is positive ---------(1)

\(-50<3x+2y<0 =>3x+2y\) is negative -----------(2)

notice that by adding \(x-y\) to equation (1) it becomes equation (2) i.e a negative value.

so \(x-y<0 => x<y\). Statement III must be true

\(x\) can be negative try values \(x=-1\) & \(y=1\). Statement I is not always true

Statement II: As \(0<2x+3y<50\) is positive and we have already derived that \(y>x\), so if \(y\) is negative then \(x\) has to be negative which will mean that \(2x+3y<0\) which is not possible. So we can say that \(y\) must be positive.

Statement II must be true.

Option E

Last edited by niks18 on 23 Nov 2017, 20:28, edited 2 times in total.

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If 0<2x+3y<50 and -50<3x+2y<0, then which of the following must be tr [#permalink]

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New post 22 Nov 2017, 18:53
=>

When we add the two inequalities \(0<2x+3y<50\) and \(-50<3x+2y<0\), we obtain \(-50<5x+5y<50\), or \(-20<-2x-2y< 20\).

Statement I.
Adding the two inequalities \(-50<3x+2y<0\) and \(-20<-2x-2y< 20\) yields \(-70<x<20\).
So x may not be greater than zero.
Statement I may not be true.

Statement II.
By multiplying all sides of \(0<2x+3y<50\) by \(-3\), we have \(-150<-6x-9y< 0\).
By multiplying all sides of \(-50<3x+2y<0\) by \(2\), we have \(-100<6x+4y< 0\).
By adding the above inequalities, we have \(-250<-5y<0\) or \(0<y<50\).

Statement II is true.

Statement III.
Since \(0<2x+3y<50\) is equivalent to \(-50<-2x-3y<0\) and \(-50<3x+2y<0\), adding the two inequalities yields
\(-100<x-y<0\). This implies that \(x < y\).
Statement III must be true.

Therefore, the answer is E.

Answer : E
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Last edited by MathRevolution on 23 Nov 2017, 12:09, edited 1 time in total.

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Re: If 0<2x+3y<50 and -50<3x+2y<0, then which of the following must be tr [#permalink]

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New post 23 Nov 2017, 10:26
MathRevolution wrote:
=>

When we add the two inequalities \(0<2x+3y<50\) and \(-50<3x+2y<0\), we obtain \(-50<5x+5y<50\), or \(-20<-2x-2y< 20\).

Statement I.
Adding the two inequalities \(-50<3x+2y<0\) and \(-20<-2x-2y< 20\) yields \(-70<x<20\).
So x may not be greater than zero.
Statement I may not be true.

Statement II.
Adding the two inequalities \(0<2x+3y<50\) and \(-20<-2x-2y< 20\) yields \(-20<y<70\).
So y may not be greater than zero.
Statement II may not be true, either.

Statement III.
Since \(0<2x+3y<50\) is equivalent to \(-50<-2x-3y<0\) and \(-50<3x+2y<0\), adding the two inequalities yields
\(-100<x-y<0\). This implies that \(x < y\).
Statement III must be true.

Therefore, the answer is C.

Answer : C


Hi MathRevolution,

Need a clarity in Statement II.

if x & y both can be negative then how will they both satisfy \(0<2x+3y<50\) and \(-50<3x+2y<0\) simultaneously? Negative x & negative y will not satisfy the \(0<2x+3y<50\)

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Re: If 0<2x+3y<50 and -50<3x+2y<0, then which of the following must be tr [#permalink]

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New post 23 Nov 2017, 12:10
niks18 wrote:
MathRevolution wrote:
=>

When we add the two inequalities \(0<2x+3y<50\) and \(-50<3x+2y<0\), we obtain \(-50<5x+5y<50\), or \(-20<-2x-2y< 20\).

Statement I.
Adding the two inequalities \(-50<3x+2y<0\) and \(-20<-2x-2y< 20\) yields \(-70<x<20\).
So x may not be greater than zero.
Statement I may not be true.

Statement II.
Adding the two inequalities \(0<2x+3y<50\) and \(-20<-2x-2y< 20\) yields \(-20<y<70\).
So y may not be greater than zero.
Statement II may not be true, either.

Statement III.
Since \(0<2x+3y<50\) is equivalent to \(-50<-2x-3y<0\) and \(-50<3x+2y<0\), adding the two inequalities yields
\(-100<x-y<0\). This implies that \(x < y\).
Statement III must be true.

Therefore, the answer is C.

Answer : C


Hi MathRevolution,

Need a clarity in Statement II.

if x & y both can be negative then how will they both satisfy \(0<2x+3y<50\) and \(-50<3x+2y<0\) simultaneously? Negative x & negative y will not satisfy the \(0<2x+3y<50\)



Yes, you are right.
The solution is fixed. Please look at the above solution again.
_________________

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The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
Find a 10% off coupon code for GMAT Club members.
“Receive 5 Math Questions & Solutions Daily”
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See our Youtube demo

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If 0<2x+3y<50 and -50<3x+2y<0, then which of the following must be tr [#permalink]

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New post 23 Nov 2017, 20:23
MathRevolution wrote:
niks18 wrote:
MathRevolution wrote:
=>

When we add the two inequalities \(0<2x+3y<50\) and \(-50<3x+2y<0\), we obtain \(-50<5x+5y<50\), or \(-20<-2x-2y< 20\).

Statement I.
Adding the two inequalities \(-50<3x+2y<0\) and \(-20<-2x-2y< 20\) yields \(-70<x<20\).
So x may not be greater than zero.
Statement I may not be true.

Statement II.
Adding the two inequalities \(0<2x+3y<50\) and \(-20<-2x-2y< 20\) yields \(-20<y<70\).
So y may not be greater than zero.
Statement II may not be true, either.

Statement III.
Since \(0<2x+3y<50\) is equivalent to \(-50<-2x-3y<0\) and \(-50<3x+2y<0\), adding the two inequalities yields
\(-100<x-y<0\). This implies that \(x < y\).
Statement III must be true.

Therefore, the answer is C.

Answer : C


Hi MathRevolution,

Need a clarity in Statement II.

if x & y both can be negative then how will they both satisfy \(0<2x+3y<50\) and \(-50<3x+2y<0\) simultaneously? Negative x & negative y will not satisfy the \(0<2x+3y<50\)



Yes, you are right.
The solution is fixed. Please look at the above solution again.


Thanks MathRevolution for the reply and clarifying :thumbup:

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If 0<2x+3y<50 and -50<3x+2y<0, then which of the following must be tr   [#permalink] 23 Nov 2017, 20:23
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