Is |4m - 3n| > |3m - n| + |m - 2n| ? : Data Sufficiency (DS)

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

08 Jul 2014, 23:30

4

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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]

13 Jul 2014, 06:23

2

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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]

23 Nov 2016, 15:39

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Statement 2 says 2n<m which translates to m-2n>0. I thought, by filling in numbers, this would mean that 3m-n certainly is bigger than 0. However, in the explanation above m=-4 and n=-3 are given as an example to fill in. But if m =-4 and n=-3 then 2n is not smaller than m. This would mean that these numbers could not be used with this premise right? m has to be bigger than 2n so m has to be >0. I'd say statement 2 is sufficient to answer the question. Could someone help me please?

L

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

24 Nov 2016, 00:47

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esmeeaugustijn wrote:

Statement 2 says 2n<m which translates to m-2n>0. I thought, by filling in numbers, this would mean that 3m-n certainly is bigger than 0. However, in the explanation above m=-4 and n=-3 are given as an example to fill in. But if m =-4 and n=-3 then 2n is not smaller than m. This would mean that these numbers could not be used with this premise right? m has to be bigger than 2n so m has to be >0. I'd say statement 2 is sufficient to answer the question. Could someone help me please?

If m =-4 and n=-3, then \((2n = -6) < (m = -4)\)

L

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

12 Mar 2017, 20:41

Bunuel wrote:

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: https://gmatclub.com/forum/tips-and-hint ... l#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.

Could someone please help to explain \(x=3m - n\) 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?

I thought it needs to be the same sign.

L

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

13 Mar 2017, 00:21

Expert Reply

ziyuen wrote:

Bunuel wrote:

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: https://gmatclub.com/forum/tips-and-hint ... l#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.

Could someone please help to explain \(x=3m - n\) 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?

I thought it needs to be the same sign.

Plug the numbers and check. Try x=1 and y=2 AND x=1 and y=-2.

L

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

11 Mar 2018, 05:04

Bunuel wrote:

SOLUTION

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

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.

Dear Bunuel

Can you prove that statement 2 is insufficient through algebraic approach? I failed to do so

Thanks

L

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

11 Mar 2018, 05:16

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Mo2men wrote:

Dear Bunuel

Can you prove that statement 2 is insufficient through algebraic approach? I failed to do so

Thanks

Hey Mo2men ,

Here I go:

We are given 2n < m

This means m - 2n > 0 and

RHS could be 4m - 3n or 3n - 4m

Now, I will take the 2nd and 3rd modulus positive while I am not sure of the 1st modulus, so I will evaluate both + and - scenarios for it.

We know that our RHS = 4m - 3n because 4m > 3n.

When 3m - n > 0

We have LHS = 3m - n + m - 2n = 4m - 3n , which is also equal to RHS if it is 4m - 3n. But not otherwise.

When 3m - n < 0

We have LHS = - 3m + n + m - 2n = -2m - n , which is not equal to RHS.

Hence, 2nd statement is giving us both scenarios. Hence, insufficient.

Does that make sense?

L

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

11 Mar 2018, 05:25

abhimahna wrote:

Mo2men wrote:

Dear Bunuel

Can you prove that statement 2 is insufficient through algebraic approach? I failed to do so

Thanks

Hey Mo2men ,

Here I go:

We are given 2n < m

This means m - 2n > 0 and

if m - 2n > 0 => m > 2n or 2m > 4n or 4m > 8n. => 4m > 3n as well

Now, I will take the 2nd and 3rd modulus positive while I am not sure of the 1st modulus, so I will evaluate both + and - scenarios for it.

We know that our RHS = 4m - 3n because 4m > 3n.

When 3m - n > 0

We have LHS = 3m - n + m - 2n = 4m - 3n , which is also equal to RHS

When 3m - n < 0

We have LHS = - 3m + n + m - 2n = -2m - n , which is not equal to RHS.

Hence, 2nd statement is giving us both scenarios. Hence, insufficient.

Does that make sense?

Hi abhimahna

The highlighted dedcution is not always true

Take Bunuel example above

m = -4 & n =-3

Statement 2 says : m > 2n ......... -4 > -6...true

In your conclusion that 4m > 3n, you can consider same same example above

-16 > -9..........which is not true.

L

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

11 Mar 2018, 05:41

Expert Reply

Mo2men wrote:

Hi abhimahna

The highlighted dedcution is not always true

Take Bunuel example above

m = -4 & n =-3

Statement 2 says : m > 2n ......... -4 > -6...true

In your conclusion that 4m > 3n, you can consider same same example above

-16 > -9..........which is not true.

Hey Mo2men ,

I agree to your point. I missed the nature of variables point.

Anyways, to conclude we can still have two different RHS values and some of the scenarios would work while others won't. Hence, B is insufficient.

L

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

11 Mar 2018, 16:30

abhimahna wrote:

Hey Mo2men ,

I agree to your point. I missed the nature of variables point.

Hi abhimahna

It seems you forgot math and focus on other school issues now

I was challenged by statement 2 and reached algebraic proof and I hope Bunuel could feedback. The xlation might be long for details.

Question: |3m−n|+|m−2n|>|4m−3n|

Quote Bunuel solution:

One of the properties of absolute values says that |x|+|y|≥|x+y|. Note that "=" sign holds for xy≥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 question basically asks whether x and y, or which is the same 3m−n and m−2n, have the opposite signs.

Statement 2:

\(2n < m\)

Case 1:

m > 0 & n>= 0

m > 2n & 2n >n....hence m>n and consequently 4m > 3n & 3m > n

First term: 3m−n = +++

second term: m−2n = +++

x & y have same sign.............Answer to question is NO

Case 2:

m > 0 & n < 0

Same as above excalty

x & y have same sign.............Answer to question is NO

The trick starts here when 'm' is negative and 'n' is negative because m could be larger or less than n. (let m=-4 & n =-2) or ( m=-2 & n =-3). Both examples hold statement 2 true. let's see the algebraic approach

By plugging numbers, I proved that '3m-n' is negative and hence x & y have different sign and answer is Yes.

But I could not reach it algebraically. I hope to get some help.

Thanks

L

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

12 Mar 2018, 00:43

Expert Reply

Mo2men wrote:

Hi abhimahna

It seems you forgot math and focus on other school issues now

I was challenged by statement 2 and reached algebraic proof and I hope Bunuel could feedback. The xlation might be long for details.

Question: |3m−n|+|m−2n|>|4m−3n|

Quote Bunuel solution:

One of the properties of absolute values says that |x|+|y|≥|x+y|. Note that "=" sign holds for xy≥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 question basically asks whether x and y, or which is the same 3m−n and m−2n, have the opposite signs.

Statement 2:

\(2n < m\)

Case 1:

m > 0 & n>= 0

m > 2n & 2n >n....hence m>n and consequently 4m > 3n & 3m > n

First term: 3m−n = +++

second term: m−2n = +++

x & y have same sign.............Answer to question is NO

Case 2:

m > 0 & n < 0

Same as above excalty

x & y have same sign.............Answer to question is NO

The trick starts here when 'm' is negative and 'n' is negative because m could be larger or less than n. (let m=-4 & n =-2) or ( m=-2 & n =-3). Both examples hold statement 2 true. let's see the algebraic approach

By plugging numbers, I proved that '3m-n' is negative and hence x & y have different sign and answer is Yes.

But I could not reach it algebraically. I hope to get some help.

Thanks

Hey Mo2men ,

Lol, I have to politely deny your statement "It seems you forgot math and focus on other school issues now". I am pretty much sure that if I give the exam at this moment also, I will end up getting Q50.

Anyways, you need to understand the fact that there are n number of ways to solve any question. The way I approached above can also be used to break the 2nd statement and call it insufficient. And the way I solved is also an algebraic way to solve any modulus questions.

You need to understand that if we have any complex modulus, we can always end up making multiple possibilities and see which case will actually work. That's what I did. Hence. I got 2nd statement insufficient.

I hope that makes sense.

D

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

26 Mar 2020, 11:25

Bunuel wrote:

New project from GMAT Club: Topic-wise questions with tips and hints!

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

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

Kudos for a correct solution.

1) Lets take m= 1 and n = 0. In that case the inequality holds true. Now if we take m =1 and n =-2, | 3 +2| +|1+4| > |4+6|. 9 < 10. What if m =1, n = 4, |3-4| +|1-8| > |4-12|. In that case the two sides become equal. Insufficient.

2) Lets take n = -2, m =-3. |-9 + 2|+|-3+4| > |-12+6| .8 > 6. Now taking n = 1, m =3. |9-1|+|3-2|> |12-3|. Both sides become equal . Not sufficient.

Together,m =1, n =0, |3-0| +|1-0| > |4 -0|.Both sides equal. when we have taken m =1 and n =-2, the left side was smaller than the right side. Again when , we take m =3, n =1, both sides become equal. So, the inequality will not hold true. Sufficient.

C is the answer.

B

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

05 Jun 2022, 06:00

My 2 cents on this:

Whenever it comes to modulus, I'm personally more inclined to choose numbers and make distinct cases. In this question, it can get tricky and long, but I find it better than opening the modulus sign which I find more time consuming.

S1: I used the following values for m and n given the m>0 condition and plugged it into original equation

m=5 n = 20 (Original equation not satisfied)
m = 10 n = -5 (Original equation satisfied)

Insufficient

S1: m/2 > n

m = 10, n = 4 ( Original equation not satisfied)
m = -10 n = -15 ( Original equation satisifed)

Insufficient.

1 and 2 together,

m = 10, n = 4
m = 6 n = 1

We will get a Definitely No answer. Hence Sufficient. Answer C

On a side note, Bunuel - I know the above approach does not include fractions and even the question doesn't specify anything around it. What do you suggest in this case? Any input is helpful

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

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

16 Jul 2023, 11:05

GMAT Data Sufficiency (DS) Questions

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