Last visit was: 23 Apr 2026, 12:06 It is currently 23 Apr 2026, 12:06
Home
Messages
Tests
Schools
Events
Close
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel
Back to Forum
Create Topic
805+ (Hard)|   Inequalities|      
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 23 Apr 2026
Posts: 109,783
Own Kudos:
810,840
 [45]
Given Kudos: 105,853
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 109,783
Kudos: 810,840
 [45]
a, b, c, d are positive integers such that exactly one of 15 Nov 2009, 18:15
17
Kudos
Add Kudos
27
Bookmarks
Bookmark this Post
00:00
Start Timer
Pause Timer
Resume Timer
Show Answer
Hide
Show
History
B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

95% (hard)

Question Stats:

27% (02:23) correct 73% (01:47) wrong based on 940 sessions

History

Date
Time
Result
Not Attempted Yet
a, b, c, d are positive integers such that exactly one of the following inequalities is false. Which inequality is false?

(A) a<b

(B) c<d

(C) a+c<b+c

(D) a+c<b+d

(E) a<b+c+d

I'm not sure whether we can see such problem on GMAT, but I found this one quite interesting so posting for discussion.
ShowHide Answer
Official Answer
B
Most Helpful Reply
User avatar
AKProdigy87
User avatar
Joined: 11 Sep 2009
Last visit: 11 Mar 2015
Posts: 80
Own Kudos:
1,145
 [26]
Given Kudos: 6
Posts: 80
Kudos: 1,145
 [26]
a, b, c, d are positive integers such that exactly one of 15 Nov 2009, 21:00
12
Kudos
Add Kudos
13
Bookmarks
Bookmark this Post
Essentially, we need to find the statement that could be false if all the other statements are true.

The first thing that jumps out to me is statements A and C:

(A) a < b
(C) a+c < b+c

Regardless of the value of c, these statement are either both false or both true. Since only statement CAN be false, we can eliminate these two.

(E) a < b+c+d

Since c and d are both positive integers, and we have determined that statement A is true, statement E must be true as well. This leaves us with:

(B) c < d
(D) a+c < b+d

Since we know statement A (a < b) to be true, if statement B is true, statement D MUST be true. However, if statement D is true, as long as c-d < b-a, statement B does NOT have to be true.

Therefore, the answer is B: c < d . Good question, +1.
User avatar
Zarrolou
User avatar
Joined: 02 Sep 2012
Last visit: 11 Dec 2013
Posts: 842
Own Kudos:
5,187
 [6]
Given Kudos: 219
Status:Far, far away!
Location: Italy
Concentration: Finance, Entrepreneurship
GPA: 3.8
Posts: 842
Kudos: 5,187
 [6]
a, b, c, d are positive integers such that exactly one of 17 Jun 2013, 10:48
6
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
a, b, c, d are positive integers such that exactly one of the following inequalities is false. Which inequality is false?

(A) a<b

(B) c<d

(C) a+c<b+c

(D) a+c<b+d

(E) a<b+c+d

I'm not sure whether we can see such problem on GMAT, but I found this one quite interesting so posting for discussion.
Show more

An algebrical approach:

first of all (A) a<b and (C) a+c<b+c are the same: both tell us that \(a<b\).
Since exactly one inequality is false, both A and C must be true. With \(a<b\) enstablished we can focus on

(B) c<d

(D) a+c<b+d

(E) a<b+c+d

E must be true since \(a<b\) also \(a<b+(+veNumber)+(+veNumber)\) is true as well.

(B) \(0<d-c\)
(D) \(a-b<d-c\) but since a-b is negative => \(-ve<d-c\)

(B) \(d-c>0(>-ve)\) number
(D) \(d-c>-ve\) number

if B is true, also D is true for sure (\(d-c=7\) \(7>-ve\) and \(7>0\)) this goes against the text of the question
if D is true, B could be false (\(d-c=-1\) \(-1>-ve(-2)\)(for example) and \(-1>0\) FALSE)
Hence B must be the false inequality

Simpler approach: given \(a<b\) if B is true \(c<d\) also D is true
\(a+c<b+d\), once more this is against the question
General Discussion
User avatar
lagomez
User avatar
Joined: 05 Mar 2008
Last visit: 18 Sep 2011
Posts: 1,224
Own Kudos:
563
 [1]
Given Kudos: 31
Posts: 1,224
Kudos: 563
 [1]
a, b, c, d are positive integers such that exactly one of 15 Nov 2009, 18:27
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
Bunuel
a, b, c, d are positive integers such that exactly one of the following inequalities is false. Which inequality is false?

(A) a<b

(B) c<d

(C) a+c<b+c

(D) a+c<b+d

(E) a<b+c+d

I'm not sure whether we can see such problem on GMAT, but I found this one quite interesting so posting for discussion.
Show more

I'm going to guess B

A). a < b to prove this false let's say a = 4 and b = 3 this would make C) false as well and only one can be false

we are left with B, D, E

c<d

let's say c = 4 and d = 3 to prove this false

and let's say a = 1 and b = 5

d) 1 + 4 < 5 + 3 is still true
e)1 < 5 + 3 + 4 still true

so my guess is B if i'm reading the question correctly
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 23 Apr 2026
Posts: 109,783
Own Kudos:
810,840
Given Kudos: 105,853
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 109,783
Kudos: 810,840
a, b, c, d are positive integers such that exactly one of 16 Nov 2009, 12:38
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Yes, OA is B.

Not hard but good question.
User avatar
lonewolf
User avatar
Joined: 22 Sep 2009
Last visit: 16 Feb 2011
Posts: 151
Own Kudos:
30
Given Kudos: 8
Location: Tokyo, Japan
Posts: 151
Kudos: 30
a, b, c, d are positive integers such that exactly one of 16 Nov 2009, 21:10
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Definitely an interesting question... I have to work on my inequalities
User avatar
lagomez
User avatar
Joined: 05 Mar 2008
Last visit: 18 Sep 2011
Posts: 1,224
Own Kudos:
563
Given Kudos: 31
Posts: 1,224
Kudos: 563
a, b, c, d are positive integers such that exactly one of 16 Nov 2009, 21:13
Kudos
Add Kudos
Bookmarks
Bookmark this Post
do you have a set of ds geometry?

thanks for the inequality set
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 23 Apr 2026
Posts: 109,783
Own Kudos:
810,840
 [3]
Given Kudos: 105,853
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 109,783
Kudos: 810,840
 [3]
a, b, c, d are positive integers such that exactly one of 17 Jun 2013, 05:51
Kudos
Add Kudos
3
Bookmarks
Bookmark this Post
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

Theory on Inequalities:
x2-4x-94661.html#p731476
inequalities-trick-91482.html
data-suff-inequalities-109078.html
range-for-variable-x-in-a-given-inequality-109468.html
everything-is-less-than-zero-108884.html
graphic-approach-to-problems-with-inequalities-68037.html

All DS Inequalities Problems to practice: search.php?search_id=tag&tag_id=184
All PS Inequalities Problems to practice: search.php?search_id=tag&tag_id=189

700+ Inequalities problems: inequality-and-absolute-value-questions-from-my-collection-86939.html
avatar
rskanumuri
avatar
Joined: 01 Feb 2013
Last visit: 14 Jan 2021
Posts: 32
Own Kudos:
32
Given Kudos: 45
Location: India
Concentration: Technology, Leadership
Schools: Tepper '17 (A) Johnson '17 (D) Stanford '17 (D) Haas '17 (D) Tuck '17 (D) Yale '17 (D)
GMAT 1: 750 Q50 V41
GPA: 3.49
WE:Engineering (Computer Software)
Products:
My Reviews
Schools: Tepper '17 (A) Johnson '17 (D) Stanford '17 (D) Haas '17 (D) Tuck '17 (D) Yale '17 (D)
GMAT 1: 750 Q50 V41
Posts: 32
Kudos: 32
a, b, c, d are positive integers such that exactly one of 09 Sep 2013, 06:32
Kudos
Add Kudos
Bookmarks
Bookmark this Post
C, D & E depend on the true/false of the equation A (a < b)
C ( a+c<b+c ) will be true for any a < b
D ( a+c<b+d ) can be true for any a < b . It can be true when c > d, if c-d < b-a
E is true always if a < b.
Since C D E depend on A for the major part, the one inequality that can be wrong is B
User avatar
b2bt
User avatar
Joined: 25 Sep 2012
Last visit: 14 Apr 2024
Posts: 192
Own Kudos:
651
Given Kudos: 242
Location: India
Concentration: Strategy, Marketing
Schools: Weatherhead '18 (A$) Northeastern '18 (WL)
GMAT 1: 660 Q49 V31
GMAT 2: 680 Q48 V34
Products:
My Reviews
Schools: Weatherhead '18 (A$) Northeastern '18 (WL)
GMAT 2: 680 Q48 V34
Posts: 192
Kudos: 651
a, b, c, d are positive integers such that exactly one of 10 Sep 2013, 01:24
Kudos
Add Kudos
Bookmarks
Bookmark this Post
I substituted a,b,c,d as 2,1,3,4 and 1,2,4,3 (as only A or B is true ).. plugged in the options.. got the right anser in 2 mins..
User avatar
mathivanan
User avatar
Joined: 10 Jun 2015
Last visit: 20 Sep 2015
Posts: 83
Own Kudos:
79
Posts: 83
Kudos: 79
a, b, c, d are positive integers such that exactly one of 14 Jun 2015, 05:14
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
a, b, c, d are positive integers such that exactly one of the following inequalities is false. Which inequality is false?

(A) a<b

(B) c<d

(C) a+c<b+c

(D) a+c<b+d

(E) a<b+c+d

I'm not sure whether we can see such problem on GMAT, but I found this one quite interesting so posting for discussion.
Show more
If (A) is true then (C) and (E) also true. If (A) is wrong, then (C) is also wrong. That means either (B) or (D) is wrong. For the value 2, 5, 3, and 1, (D) is true but (B) is wrong.
User avatar
iPen
User avatar
Joined: 08 Jun 2015
Last visit: 20 Dec 2020
Posts: 80
Own Kudos:
109
Given Kudos: 40
Posts: 80
Kudos: 109
a, b, c, d are positive integers such that exactly one of 27 Jul 2015, 01:24
Kudos
Add Kudos
Bookmarks
Bookmark this Post
No plugging in is necessary.


(A) a < b
(C) a+c < b+c

These two statements are equal. Subtract c from both sides in (C). Since we must find one of the answer choices that does not belong, then we know that (A) and (C) must be true.

(E) a<b+c+d

This must be true because adding two positive integers to b, which is greater than a, will always be greater than a.

(D) a−b<d−c

Rearranging (E):

(E) \(a<b+c+d\)
\(a-b-d<c\)
\(c>a-b-d\)

Rearranging (D):
(D) \(a−b<d−c\)
\(a-b-d<-c\)
\(-c>a-b-d\)
\(c<-a+b+d\)

This does not mean that D and E are necessarily contradictory.

Just E: a-b-d<c; -c<-a+b+d (Must be true)
Just D: a-b-d<-c; c<-a+b+d

Combining D and E we get:
\(a-b-d<c<-a+b+d\\
a-b-d<-c<-a+b+d\)

Possible scenarios (think of it as numbers added/subtracted in order of smallest to biggest):
\(a-b-d<-c<c<-a+b+d\)
OR
\(-c<a-b-d<c<-a+b+d\)
OR
\(-c<a-b-d<-a+b+d<c\)

The values in question, c and d, are never contradictory. They're always positive on the right side and negative on the left side. For instance, you don't see a positive d value on the left with a negative d value on the right. All of the choices are consistent/possible. c can be larger than d, d can be larger than c; all we know for sure is that b is larger than a.

Combining (E) and (B) we get:
Just E: \(a-b-d<c\); \(-c<-a+b+d\) (Must be true)
Just B: \(d>c\); \(-c>-d\)

Possible scenarios:
\(-c<a-b-d<-a+b+d<c<d\) (Not possible to have \(d> -a +b +d\))
OR
\(-d<-c<a-b-d<-a+b+d<c\) (Not possible to have \(-d< a -b -d\))
OR
\(-c<a-b-d<c<d<-a+b+d\)

(B) does not work with the inequality.
avatar
AMahmoodi
avatar
Joined: 08 Jan 2017
Last visit: 24 Apr 2017
Posts: 21
Own Kudos:
26
Given Kudos: 230
Posts: 21
Kudos: 26
a, b, c, d are positive integers such that exactly one of 30 Jan 2017, 04:12
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Zarrolou
Bunuel
a, b, c, d are positive integers such that exactly one of the following inequalities is false. Which inequality is false?

(A) a<b

(B) c<d

(C) a+c<b+c

(D) a+c<b+d

(E) a<b+c+d

I'm not sure whether we can see such problem on GMAT, but I found this one quite interesting so posting for discussion.

An algebrical approach:

first of all (A) a<b and (C) a+c<b+c are the same: both tell us that \(a<b\).
Since exactly one inequality is false, both A and C must be true. With \(a<b\) enstablished we can focus on

(B) c<d

(D) a+c<b+d

(E) a<b+c+d

E must be true since \(a<b\) also \(a<b+(+veNumber)+(+veNumber)\) is true as well.

(B) \(0<d-c\)
(D) \(a-b<d-c\) but since a-b is negative => \(-ve<d-c\)

(B) \(d-c>0(>-ve)\) number
(D) \(d-c>-ve\) number

if B is true, also D is true for sure (\(d-c=7\) \(7>-ve\) and \(7>0\)) this goes against the text of the question
if D is true, B could be false (\(d-c=-1\) \(-1>-ve(-2)\)(for example) and \(-1>0\) FALSE)
Hence B must be the false inequality

Simpler approach: given \(a<b\) if B is true \(c<d\) also D is true
\(a+c<b+d\), once more this is against the question
Show more

Hi, it could be a little more faster though :-D

An algebrical approach:

first of all (A) a<b and (C) a+c<b+c are the same: both tell us that \(a<b\).
Since exactly one inequality is false, both A and C must be true. With \(a<b\) enstablished we can focus on

(B) c<d

(D) a+c<b+d

(E) a<b+c+d

E must be true since \(a<b\) also \(a<b+(+veNumber)+(+veNumber)\) is true as well.

I agree with all of above but finally we conclude that either B or D must be incorrect. if B is going to be correct we then can add it with A and it will be a+c<b+d which is D so D must be correct also and we don't have a false inequality. therefore we can conclude that B must be incorrect.

anyway thanks for your astute approach.
avatar
Perezstephen91
avatar
Joined: 19 Apr 2018
Last visit: 03 Apr 2020
Posts: 8
Own Kudos:
3
Given Kudos: 58
Posts: 8
Kudos: 3
a, b, c, d are positive integers such that exactly one of 10 Jul 2019, 15:29
Kudos
Add Kudos
Bookmarks
Bookmark this Post
IMO the question would have to specify 1 of the following is false when each of the others are true. This question is way to ambiguous with out correct parameters or specifications there is no way this question could be allowed.
User avatar
bumpbot
User avatar
Non-Human User
Joined: 09 Sep 2013
Last visit: 04 Jan 2021
Posts: 38,964
Own Kudos:
1,117
Posts: 38,964
Kudos: 1,117
a, b, c, d are positive integers such that exactly one of 22 Jul 2023, 19:58
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Automated notice from GMAT Club BumpBot:

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

This post was generated automatically.
Moderators:
Bunuel
Math Expert
109783 posts
Krunaal
Tuck School Moderator
853 posts