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

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

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Is \(|3m - n| + |m - 2n| > |4m - 3n|\)?

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

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

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New post 06 Jul 2014, 14:25
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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 \(xy<0\). (Check here: tips-and-hints-for-specific-quant-topics-with-examples-172096.html#p1381430)

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.

Answer: C.

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

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

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New post 08 Jul 2014, 23:17
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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New post 08 Jul 2014, 23:30
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.

Answer is C.


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 ?
Answer is No So sufficient.

Answer is C



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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New post 12 Jul 2014, 06:01
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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New post 12 Jul 2014, 12:00
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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New post 13 Jul 2014, 00:03
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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New post 13 Jul 2014, 06:23
1
Answer must be C.
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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New post 13 Jul 2014, 21:51
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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New post 14 Jul 2014, 00:40
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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Re: Is |4m - 3n| > |3m - n| + |m - 2n| ?  [#permalink]

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New post 15 Jul 2014, 11:33
1
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 \(xy<0\). (Check here: tips-and-hints-for-specific-quant-topics-with-examples-172096.html#p1381430)

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.

Answer: C.

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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New post 15 Jul 2014, 20:32
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]


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

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New post 15 Jul 2014, 20:34
Hi bunuel,
Is my solution incorrect?
Quote:
Answer must be C.
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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New post 16 Jul 2014, 02:22
peacewarriors wrote:
Hi bunuel,
Is my solution incorrect?
Quote:
Answer must be C.
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.


I don't understand your solution...
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Re: Is |4m - 3n| > |3m - n| + |m - 2n| ?  [#permalink]

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New post 16 Jul 2014, 02:30
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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New post 16 Jul 2014, 02:36
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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PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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

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New post 16 Jul 2014, 02:54
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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New post 17 Jul 2014, 11:01
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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New post 17 Jul 2014, 11:03
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Re: Is |4m - 3n| > |3m - n| + |m - 2n| ? &nbs [#permalink] 17 Jul 2014, 11:03

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