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# Is |4m - 3n| > |3m - n| + |m - 2n| ?

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Math Expert
Joined: 02 Sep 2009
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Is |4m - 3n| > |3m - n| + |m - 2n| ?  [#permalink]

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06 Jul 2014, 09:11
10
46
00:00

Difficulty:

95% (hard)

Question Stats:

37% (02:19) correct 63% (02:30) wrong based on 414 sessions

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Is $$|3m - n| + |m - 2n| > |4m - 3n|$$?

(1) $$m > 0$$
(2) $$2n < m$$

Kudos for a correct solution.

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Re: Is |4m - 3n| > |3m - n| + |m - 2n| ?  [#permalink]

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06 Jul 2014, 13:25
15
16
SOLUTION

Is $$|3m - n| + |m - 2n| > |4m - 3n|$$?

One of the properties of absolute values says that $$|x|+|y|\geq|x+y|$$. Note that "=" sign holds for $$xy\geq{0}$$ (or simply when $$x$$ and $$y$$ have the same sign). So, the strict inequality (>) holds when $$xy<0$$. ()

Notice that if we denote $$x=3m - n$$ and $$y=m - 2n$$, then $$x+y=4m-3n$$. So, the question becomes: is $$|x|+|y|>|x+y|$$? Thus, the qeustion basically asks whether $$x$$ and $$y$$, or which is the same $$3m - n$$ and $$m - 2n$$, have the opposite signs.

(1) $$m > 0$$. Clearly insufficient as no info about $$n$$. Not sufficient.

(2) $$2n < m$$. This implies that $$m-2n>0$$. If $$m=3$$ and $$n=1$$, then $$3m - n>0$$ (so in this case $$3m - n$$ and $$m - 2n$$ will have the same sign) but if $$m=-4$$ and $$n=-3$$, then $$3m - n<0$$ (so in this case $$3m - n$$ and $$m - 2n$$ will have different signs sign). Not sufficient.

(1)+(2) We have that $$m > 0$$, or which is the same $$5m>0$$ and $$m>2n$$. Add them: $$6m>2n$$. Reduce by 2 and re-arrange: $$3m-n>0$$. Thus, both $$m-2n$$ and $$3m-n$$ are positive, so we have a NO answer to the question. Sufficient.

Try NEW Absolute Value PS question.
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Re: Is |4m - 3n| > |3m - n| + |m - 2n| ?  [#permalink]

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Updated on: 10 Jul 2014, 08:56
1
Bunuel wrote:

Is $$|3m - n| + |m - 2n| > |4m - 3n|$$?

(1) $$m > 0$$
(2) $$2n < m$$

Kudos for a correct solution.

My take is C.

1) m>0
(m,n) = (+,+);(+,-);(+,0)
in (+,-) LHS=RHS ; so NO LHS is not > RHS
in (+,+) LHS>RHS ; so YES LHS is > RHS
hence, A is inconclusive.

2) 2n<m
based on the aforementioned approach we again get LHS=RHS and LHS>RHS.
hence, B is inconclusive.

combining both shall give us a definite answer. thus C.
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Illegitimi non carborundum.

Originally posted by thefibonacci on 08 Jul 2014, 09:41.
Last edited by thefibonacci on 10 Jul 2014, 08:56, edited 1 time in total.
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Re: Is |4m - 3n| > |3m - n| + |m - 2n| ?  [#permalink]

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08 Jul 2014, 22:17
I am pretty sure , I posted one reply yesterday but dont see it now !!
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Re: Is |4m - 3n| > |3m - n| + |m - 2n| ?  [#permalink]

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08 Jul 2014, 22:30
2
Sol

Putting in values to check for 1 to be sufficient

m = 10 , n =0 makes Is 30+10 > 40? No
m = 10 , n = 20 makes Is 10+30 > 20 ? Yes
m= 10 , n = -10 makes Is 40 + 30 > 70 ? No

So InSufficient

Case 2
Putting in values to check for 2 to be sufficient

m>2n

m = 20 , n = 5 makes Is 55 + 10 > 65 ? No
m= -5 , n = -3 makes Is 12 + 1 > 11 ? Yes
m= 10 , n = -10 makes Is 40 + 30 > 70 ? No

So B alone is insufficient

Taking both of them together we can see , that the contradicting scenarios are getting eliminated. Thus Both are sufficient.

Algebraically :-

Case 1 m>0 does not indicate anything about the sign of each term as the value of n is uncertain. So in other words |3m-n| could be either 3m-n or n-3m based on the sign of the expression. So insufficient

Case 2 m >2n again does not indicate anything about the sign of any term. So |3m-n| could be 3m-n or n-3m based on whether m and n are both positive or both negative.

Taking together - m >0 and m > 2n

will ensure that 3m-n > 0
m-2n > 0
4m-3n > 0

so the expression after removing mod symbol becomes

Is 3m-n + m-2n > 4m-3n ?

PS - Not satisfied with this dirty approach and the time implications in actual GMAT test , God forbid if I get it
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Re: Is |4m - 3n| > |3m - n| + |m - 2n| ?  [#permalink]

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12 Jul 2014, 05:01
Why not just 'B,? We just need the second statement to know the sign of Modulus expressions.
IMO it should be just "B"
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Re: Is |4m - 3n| > |3m - n| + |m - 2n| ?  [#permalink]

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12 Jul 2014, 11:00
I have done by putting in values. It is not answering conclusively
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Re: Is |4m - 3n| > |3m - n| + |m - 2n| ?  [#permalink]

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12 Jul 2014, 23:03
Not sure - I got E, as it was still inconclusive even after trying values....and the silly timer doesnt help too...
god forbid i get it too....
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Re: Is |4m - 3n| > |3m - n| + |m - 2n| ?  [#permalink]

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13 Jul 2014, 05:23
1
Opening the mod we get if 4m>3n?
Statement 1 says m>0 but we know nothing about n
Eg: if m=3 and n= -1 it is true but if m=1 and n=2 the answer is no

Statement 2 says m>2n but we know nothing about m and n in terms of whether they are positive or negative.
Eg: if m =5 and n =0.5 the answer is yes but if m=-2 and if n=-1.5 the answer is no

Combining both the statements we get that m>0 and m>2n we get m>n and therefore 4m>3n.
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Re: Is |4m - 3n| > |3m - n| + |m - 2n| ?  [#permalink]

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13 Jul 2014, 20:51
Waiting for the Bunuel post with some new twist and dollops of learning.
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Re: Is |4m - 3n| > |3m - n| + |m - 2n| ?  [#permalink]

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13 Jul 2014, 23:40
Bunuel wrote:

Is $$|3m - n| + |m - 2n| > |4m - 3n|$$?

(1) $$m > 0$$
(2) $$2n < m$$

Kudos for a correct solution.

I am getting A as answer, but took too long to solve by plugging numbers.
For statement A, any combination of values of m and n is getting the same answer.
Statement 2 is not sufficient. We get two different answers if we take values as m=10,n=4 and m=-3 and n=-2. Hence insufficient.

Experts please suggest a faster approach to solve such problem.
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Posts: 64101
Re: Is |4m - 3n| > |3m - n| + |m - 2n| ?  [#permalink]

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15 Jul 2014, 10:33
2
2
SOLUTION

Is $$|3m - n| + |m - 2n| > |4m - 3n|$$?

One of the properties of absolute values says that $$|x|+|y|\geq|x+y|$$. Note that "=" sign holds for $$xy\geq{0}$$ (or simply when $$x$$ and $$y$$ have the same sign). So, the strict inequality (>) holds when $$xy<0$$. ()

Notice that if we denote $$x=3m - n$$ and $$y=m - 2n$$, then $$x+y=4m-3n$$. So, the question becomes: is $$|x|+|y|>|x+y|$$? Thus, the qeustion basically asks whether $$x$$ and $$y$$, or which is the same $$3m - n$$ and $$m - 2n$$, have the opposite signs.

(1) $$m > 0$$. Clearly insufficient as no info about $$n$$. Not sufficient.

(2) $$2n < m$$. This implies that $$m-2n>0$$. If $$m=3$$ and $$n=1$$, then $$3m - n>0$$ (so in this case $$3m - n$$ and $$m - 2n$$ will have the same sign) but if $$m=-4$$ and $$n=-3$$, then $$3m - n<0$$ (so in this case $$3m - n$$ and $$m - 2n$$ will have different signs sign). Not sufficient.

(1)+(2) We have that $$m > 0$$, or which is the same $$5m>0$$ and $$m>2n$$. Add them: $$6m>2n$$. Reduce by 2 and re-arrange: $$3m-n>0$$. Thus, both $$m-2n$$ and $$3m-n$$ are positive, so we have a NO answer to the question. Sufficient.

Kudos points given to correct solutions above.

Try NEW Absolute Value PS question.
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Re: Is |4m - 3n| > |3m - n| + |m - 2n| ?  [#permalink]

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15 Jul 2014, 19:32
Wow. A kudos from Bunuel. GMAC should consider Bunuel's kudos in its exam process [SMILING FACE WITH SMILING EYES][SMILING FACE WITH SMILING EYES]

Sent from my iPhone using Tapatalk
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Re: Is |4m - 3n| > |3m - n| + |m - 2n| ?  [#permalink]

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15 Jul 2014, 19:34
Hi bunuel,
Is my solution incorrect?
Quote:
Opening the mod we get if 4m>3n?
Statement 1 says m>0 but we know nothing about n
Eg: if m=3 and n= -1 it is true but if m=1 and n=2 the answer is no

Statement 2 says m>2n but we know nothing about m and n in terms of whether they are positive or negative.
Eg: if m =5 and n =0.5 the answer is yes but if m=-2 and if n=-1.5 the answer is no

Combining both the statements we get that m>0 and m>2n we get m>n and therefore 4m>3n.
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Posts: 64101
Re: Is |4m - 3n| > |3m - n| + |m - 2n| ?  [#permalink]

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16 Jul 2014, 01:22
peacewarriors wrote:
Hi bunuel,
Is my solution incorrect?
Quote:
Opening the mod we get if 4m>3n?
Statement 1 says m>0 but we know nothing about n
Eg: if m=3 and n= -1 it is true but if m=1 and n=2 the answer is no

Statement 2 says m>2n but we know nothing about m and n in terms of whether they are positive or negative.
Eg: if m =5 and n =0.5 the answer is yes but if m=-2 and if n=-1.5 the answer is no

Combining both the statements we get that m>0 and m>2n we get m>n and therefore 4m>3n.

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Re: Is |4m - 3n| > |3m - n| + |m - 2n| ?  [#permalink]

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16 Jul 2014, 01:30
HI,
Opening the modulus I mean 4m - 3n| > |3m - n| + |m - 2n|: 4m-3n>-3m+n - m+2n= Consolidating the similar terms,we get 8m>6n? dividing by 2 we get 4m>3n
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Re: Is |4m - 3n| > |3m - n| + |m - 2n| ?  [#permalink]

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16 Jul 2014, 01:36
peacewarriors wrote:
HI,
Opening the modulus I mean 4m - 3n| > |3m - n| + |m - 2n|: 4m-3n>-3m+n - m+2n= Consolidating the similar terms,we get 8m>6n? dividing by 2 we get 4m>3n

How do you know that |4m - 3n| = 4m-3n, |3m - n| = -3m+n and |m - 2n| = - m+2n???
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Re: Is |4m - 3n| > |3m - n| + |m - 2n| ?  [#permalink]

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16 Jul 2014, 01:54
Got my mistake.
I assumed that lhs = rhs and followed the approach |lhs|=|rhs| or -|lhs|=|rhs| or vice versa.
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Re: Is |4m - 3n| > |3m - n| + |m - 2n| ?  [#permalink]

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17 Jul 2014, 10:01
this is not og questions and should not be studied
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Re: Is |4m - 3n| > |3m - n| + |m - 2n| ?  [#permalink]

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17 Jul 2014, 10:03
vietmoi999 wrote:
this is not og questions and should not be studied

This is GMAT Club's question.
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Re: Is |4m - 3n| > |3m - n| + |m - 2n| ?   [#permalink] 17 Jul 2014, 10:03

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