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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8029
GMAT 1: 760 Q51 V42 GPA: 3.82
If 0<2x+3y<50 and -50<3x+2y<0, then which of the following must be tr  [#permalink]

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Question Stats: 53% (02:16) correct 47% (02:01) wrong based on 92 sessions

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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

_________________

Originally posted by MathRevolution on 21 Nov 2017, 18:20.
Last edited by MathRevolution on 23 Nov 2017, 13:05, 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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1
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

Originally posted by niks18 on 22 Nov 2017, 09:50.
Last edited by niks18 on 23 Nov 2017, 21:28, edited 2 times in total.
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8029
GMAT 1: 760 Q51 V42 GPA: 3.82
If 0<2x+3y<50 and -50<3x+2y<0, then which of the following must be tr  [#permalink]

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=>

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.

_________________

Originally posted by MathRevolution on 22 Nov 2017, 19:53.
Last edited by MathRevolution on 23 Nov 2017, 13:09, 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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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.

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 $$0<2x+3y<50$$
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8029
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: If 0<2x+3y<50 and -50<3x+2y<0, then which of the following must be tr  [#permalink]

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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.

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

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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.

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 Intern  B
Joined: 27 Apr 2015
Posts: 39
GMAT 1: 370 Q29 V13 Re: If 0<2x+3y<50 and -50<3x+2y<0, then which of the following must be tr  [#permalink]

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4
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

Given $$0<2x+3y<50$$ and $$-50<3x+2y<0$$
Since
=> $$0<2x+3y$$ and
=> $$3x+2y<0$$
Therefore
=> $$3x+2y<2x+3y$$
=> OR $$3x-2x<3y-2y$$
=> OR $$x<y$$ so satisfy III

Now x,y
=>both CANNOT be +ve since given $$3x+2y<0$$ and
=>both CANNOT be -ve since given $$2x+3y>0$$
Therefore
=>both are OPPOSITE sign
=>AND Since $$x<y$$ THEREFORE $$x<0$$ AND $$y>0$$ so Satisfy II

Option E

Regards
Dinesh
Non-Human User Joined: 09 Sep 2013
Posts: 13410
Re: If 0<2x+3y<50 and -50<3x+2y<0, then which of the following must be tr  [#permalink]

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_________________ Re: If 0<2x+3y<50 and -50<3x+2y<0, then which of the following must be tr   [#permalink] 24 Jul 2019, 07:12
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