BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

Shortcut to mixed inequality and absolute DS question

This topic has expert replies
Post new topic Post Reply
Previous Topic Next Topic
Source: — Data Sufficiency |

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 3008
Joined: Mon Aug 22, 2016 6:19 am
Location: Grand Central / New York
Thanked: 470 times
Followed by:34 members

by Jay@ManhattanReview » Tue Mar 14, 2017 5:02 am
Mo2men wrote:Is |3m−n|+|m−2n|>|4m−3n|?

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

OA: C
Hi Mo2men,

We have to see whether |3m−n|+|m−2n| > |4m−3n|?

S1: m > 0

We do not have any information about the value of n.

If m = 1 and n = 0,

|3m−n|+|m−2n| > |4m−3n| => 3 + 1 > 4. The answer is NO.

However, if m = n = 1,

|3m−n|+|m−2n| > |4m−3n| => |3−1|+|1−2| > |4−3| => 2 + 1 > 1. The answer is YES. No unique answer.

S2: 2n < m

=> If m = -2 and n= -2,

|3m−n|+|m−2n| > |4m−3n| => |-6+2|+|-2+4| > |-8+6| => 4 + 2 > 2 => 6 > 2. The answer is YES.

However, if m = 1, and n = 0,

|3m−n|+|m−2n| > |4m−3n| => 3 + 1 > 4 => 4 > 4. The answer is NO. No unique answer.

S1 and S2:

We have m > 0 and 2n < m

2n < m => m - 2n > 0

=> 3m−n > 0, 4m−3n > 0

This implies that |3m−n|+|m−2n| > |4m−3n| => 3m−n + m−2n > 4m−3n => 4m - 3n > 4m - 3n. The answer is NO. Unique answer. Sufficient.

The correct answer: C

Hope this helps!

Relevant book: Manhattan Review GMAT Data Sufficiency Guide

-Jay
_________________
Manhattan Review GMAT Prep

Locations: New York | Bangalore | Guangzhou | Buenos Aires | and many more...

Schedule your free consultation with an experienced GMAT Prep Advisor! Click here.
Top
Legendary Member
Posts: 712
Joined: Fri Sep 25, 2015 4:39 am
Thanked: 14 times
Followed by:5 members

by Mo2men » Tue Mar 14, 2017 6:26 am
Jay@ManhattanReview wrote: S1 and S2:

We have m > 0 and 2n < m

2n < m => m - 2n > 0

=> 3m−n > 0, 4m−3n > 0

This implies that |3m−n|+|m−2n| > |4m−3n| => 3m−n + m−2n > 4m−3n => 4m - 3n > 4m - 3n. The answer is NO. Unique answer. Sufficient.
Hi Jay,

How can you deduce the inequality? can you elaborate please?

Thanks
Top
User avatar
GMAT Instructor
Posts: 15539
Joined: Tue May 25, 2010 12:04 pm
Location: New York, NY
Thanked: 13060 times
Followed by:1906 members
GMAT Score:790

by GMATGuruNY » Tue Mar 14, 2017 6:49 am
Mo2men wrote:Is |3m−n|+|m−2n|>|4m−3n|
(1) m>0
(2) 2n<m
Let x = 3m-n and y = m-2n, with the result that x+y = 4m-3n.
Substituting x = 3m-n, y = m-2n, and x+y = 4m-3n into |3m−n|+|m−2n|>|4m−3n|, we get:
Is |x| + |y| > |x+y|?

Since an absolute value cannot be negative, both sides of the inequality in blue are NONNEGATIVE.
As a result, we can SQUARE THE INEQUALITY:
(|x| + |y|)² > (|x+y|)²
x² + y² + 2|x||y| > x² + y² + 2xy
2|x||y| > 2xy
|x||y| > xy.

The inequality in red will hold true if x and y have DIFFERENT SIGNS.
Since x = 3m-n and y = m-2n, question stem rephrased:
Do 3m-n and m-2n have different signs?

Statement 1: m>0
No way to determine the signs of 3m-n and m-2n.
INSUFFICIENT.

Statement 2: 2n<m
0 < m-2n
m-2n > 0.
No way to determine the sign of 3m-n.
INSUFFICIENT.

Statements combined:
Inequalities can be ADDED TOGETHER.
Adding together m>0 and m-2n > 0, we get:
m + (m-2n) > 0 + 0
2m - 2n > 0
m - n > 0.

Adding together m>0, m>0 and m-n>0, we get:
m + m + (m-n) > 0 + 0 + 0
3m-n > 0.

Since 3m-n > 0 and m-2n > 0, they have the SAME SIGN.
Thus, the answer to the question stem is NO.
SUFFICIENT.

The correct answer is C.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
Top

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 3008
Joined: Mon Aug 22, 2016 6:19 am
Location: Grand Central / New York
Thanked: 470 times
Followed by:34 members

by Jay@ManhattanReview » Wed Mar 15, 2017 12:41 am
Mo2men wrote:
Jay@ManhattanReview wrote: S1 and S2:

We have m > 0 and 2n < m

2n < m => m - 2n > 0

=> 3m−n > 0, 4m−3n > 0

This implies that |3m−n|+|m−2n| > |4m−3n| => 3m−n + m−2n > 4m−3n => 4m - 3n > 4m - 3n. The answer is NO. Unique answer. Sufficient.
Hi Jay,

How can you deduce the inequality? can you elaborate please?

Thanks
Hi Mo2men,

(3m−n) can be written as (6m − 2n)/2 = (5m + m − 2n)/2 = = [5m + (m − 2n)]/2 = [+ive number + +ive number]/2 = +ive number .

Similarly, (4m−3n) can be written as 3(8m/3−2n)/2 = (3/2)*[5m/3 + (m−2n)] = (+ive number)*(+ive number + +ive number) = +ive number

Hope this is clear.

Relevant book: Manhattan Review GMAT Math Essentials Guide

-Jay
_________________
Manhattan Review GMAT Prep

Locations: New York | New Delhi | Seoul | Cairo | and many more...

Schedule your free consultation with an experienced GMAT Prep Advisor! Click here.
Top
Post new topic Post Reply
• Page 1 of 1