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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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Math Revolution GMAT Instructor
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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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21 Nov 2017, 17:20
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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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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.

Kudos [?]: 309 [0], given: 39

Math Revolution GMAT Instructor
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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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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.

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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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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.

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 $$0<2x+3y<50$$

Kudos [?]: 309 [0], given: 39

Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 4472

Kudos [?]: 3154 [0], given: 0

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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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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.

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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Kudos [?]: 3154 [0], given: 0

PS Forum Moderator
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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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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.

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

Kudos [?]: 309 [0], given: 39

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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