Last visit was: 02 Sep 2026, 15:39 It is currently 02 Sep 2026, 15:39
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.
Close
Request Expert Reply
Confirm Cancel
705-805 (Hard)|   Inequalities|   Must or Could be True|                              
User avatar
chetan2u
User avatar
GMAT Expert
Joined: 02 Aug 2009
Last visit: 02 Sep 2026
Posts: 11,288
Own Kudos:
46,066
 [3]
Given Kudos: 339
Status:Math and DI Expert
Location: India
Concentration: Human Resources, General Management
GMAT Focus 1: 735 Q90 V89 DI81
Products:
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
GMAT Focus 1: 735 Q90 V89 DI81
Posts: 11,288
Kudos: 46,066
 [3]
1
Kudos
Add Kudos
2
Bookmarks
Bookmark this Post
User avatar
dante11
Joined: 06 Feb 2021
Last visit: 30 Aug 2023
Posts: 10
Own Kudos:
3
 [1]
Given Kudos: 412
Location: India
Schools: Haas '24
Schools: Haas '24
Posts: 10
Kudos: 3
 [1]
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
User avatar
egmat
User avatar
e-GMAT Representative
Joined: 02 Nov 2011
Last visit: 02 Sep 2026
Posts: 6,287
Own Kudos:
33,892
 [5]
Given Kudos: 715
GMAT Date: 08-19-2020
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 6,287
Kudos: 33,892
 [5]
4
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
User avatar
RenB
Joined: 13 Jul 2022
Last visit: 06 Jul 2026
Posts: 388
Own Kudos:
Given Kudos: 302
Location: India
Concentration: Finance, Nonprofit
GMAT Focus 1: 715 Q90 V84 DI82
GPA: 3.74
WE:Corporate Finance (Consulting)
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
If \(4<\frac{7-x}{3}\), which of the following must be true?

I. \(5<x\)
II. \(|x+3|>2\)
III. \(-(x+5)\) is positive

(A) II only
(B) III only
(C) I and II only
(D) II and III only
(E) I, II and III
­Writing down how I solved it-

The qs gives me x<-5

Now what I have to do is identify which of the following is true given that x<-5

1. x>5- I know x<-5 as per the qs. So this is not possible.

2. |x+3|>2
This will be true in 2 cases
a. x+3>2: x>-1 
b. -(x+3)>2
i.e -x-3>2: -5>x or x<-5

I know that the range of x<-5. Thus the qs satisfies case 2 of option b. Since it satisfies, I can say that the given question prompt can be moulded into |x+3|>2. And hence it must be true that |x+3|>2 given 4<(7−x)/3

3.  -(x+5) is positive
Or i can say that (x+5) is negative. i.e x+5<0
Thus x<-5.
Hence this must be true.

Thus D
User avatar
egmat
User avatar
e-GMAT Representative
Joined: 02 Nov 2011
Last visit: 02 Sep 2026
Posts: 6,287
Own Kudos:
33,892
 [1]
Given Kudos: 715
GMAT Date: 08-19-2020
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 6,287
Kudos: 33,892
 [1]
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
This is a classic GMAT trap that tests your ability to work systematically with inequalities and Roman numeral statements. Many students rush into evaluating the statements without first establishing the constraint on x.

Strategic Framework:

Step 1: Solve the Core Constraint
Given: (\frac{4-7x}{3} > 3)
Multiply both sides by 3: (4-7x > 9)
Subtract 4: (-7x > 5)
Divide by -7 (flip the inequality): (x < -\frac{5}{7})
Wait - this means (x < -5), not (x > -5). Critical insight: The constraint severely limits our x values.

Step 2: Systematic Statement Evaluation
Now that we know (x < -5), let's check what MUST always be true:
Statement I: (x > 5)
Since (x < -5), this can never be true. FALSE

Statement II: (|x + 3| > 2)
If (x < -5), then (x + 3 < -2)
Since (x + 3) is negative and less than -2: (|x + 3| = -(x + 3) > 2) ✓ TRUE

Statement III: (-(x + 5)) is positive
Since (x < -5), we have (x + 5 < 0)
Therefore (-(x + 5) > 0) ✓ TRUE

Answer: D) II and III only

The key insight here is recognizing that "must be true" problems require you to find what's always true given the constraint, not what's sometimes true. This pattern appears frequently in GMAT inequalities.

For the complete breakdown showing the systematic approach to all Roman numeral inequality problems, plus the 3 most common trap patterns students fall into: https://neuron.e-gmat.com/quant/questions/if-4-7-x-3-which-of-the-following-must-be-true-1617.html
User avatar
rajdeep212003
Joined: 11 Dec 2024
Last visit: 02 Sep 2026
Posts: 156
Own Kudos:
Given Kudos: 17
Location: India
Products:
Posts: 156
Kudos: 78
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Hope it will be helpfull for everyone
Attachments

17570927547737329154115391755426.jpg
17570927547737329154115391755426.jpg [ 3.07 MiB | Viewed 470 times ]

User avatar
rajdeep212003
Joined: 11 Dec 2024
Last visit: 02 Sep 2026
Posts: 156
Own Kudos:
Given Kudos: 17
Location: India
Products:
Posts: 156
Kudos: 78
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Hope it will be best
Attachments

17570975004362735063474686543572.jpg
17570975004362735063474686543572.jpg [ 3.07 MiB | Viewed 460 times ]

User avatar
totaltestprepNick
Joined: 25 Aug 2014
Last visit: 02 Sep 2026
Posts: 469
Own Kudos:
Given Kudos: 2
GMAT 1: 750 Q49 V42
GMAT 1: 750 Q49 V42
Posts: 469
Kudos: 9
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
If \(4<\frac{7-x}{3}\), which of the following must be true?

I. \(5<x\)
II. \(|x+3|>2\)
III. \(-(x+5)\) is positive

(A) II only
(B) III only
(C) I and II only
(D) II and III only
(E) I, II and III





Nick Slavkovich, GMAT/GRE tutor with 20+ years of experience

[email protected]
User avatar
PGTLrowanhand
Joined: 30 Oct 2012
Last visit: 02 Sep 2026
Posts: 159
Own Kudos:
Given Kudos: 3
Status:London UK GMAT Consultant / Tutor
Expert
Expert reply
Posts: 159
Kudos: 189
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Here's my video solution:
User avatar
Shlok02
Joined: 08 Jan 2024
Last visit: 02 Sep 2026
Posts: 65
Own Kudos:
Given Kudos: 6
Products:
Posts: 65
Kudos: 41
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
SOLUTION

If 4<(7-x)/3, which of the following must be true?

I. 5<x
II. |x+3|>2
III. -(x+5) is positive


(A) II only
(B) III only
(C) I and II only
(D) II and III only
(E) I, II and III

Note that we are asked to determine which MUST be true, not could be true.

\(4<\frac{7-x}{3}\)

\(12<7-x\)

\(x<-5\).

So we know that \(x<-5\), it's given as a fact. Now, taking this info we should find out which of the following inequalities will be true OR which of the following inequalities will be true for the range \(x<-5\).

Basically the question asks: if \(x<-5\) which of the following is true?

I. \(5<x\) --> not true as \(x<-5\).

II. \(|x+3|>2\), this inequality holds true for 2 cases, (for 2 ranges): 1. when \(x+3>2\), so when \(x>-1\) or 2. when \(-x-3>2\), so when \(x<-5\). We are given that second range is true (\(x<-5\)), so this inequality holds true.

Or another way: ANY \(x\) from the range \(x<-5\) (-5.1, -6, -7, ...) will make \(|x+3|>2\) true, so as \(x<-5\), then \(|x+3|>2\) is always true.

III. \(-(x+5)>0\) --> \(x<-5\) --> true.

Answer: D.
I had a query regarding Statement 2
We get 2 conditions- x>-1 and x<-5
since the second range satisfies the inequality given, but the first one does not, how can we say that (2) MUST BE TRUE??
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 02 Sep 2026
Posts: 113,061
Own Kudos:
Given Kudos: 111,271
Products:
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 113,061
Kudos: 838,696
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Shlok02

I had a query regarding Statement 2
We get 2 conditions- x>-1 and x<-5
since the second range satisfies the inequality given, but the first one does not, how can we say that (2) MUST BE TRUE??

Statement II is true whenever x < -5 or x > -1. Since the given condition guarantees x < -5, every possible value of x satisfies Statement II. The other range, x > -1, is irrelevant.

To understand the underline concept better practice other Trickiest Inequality Questions Type: Confusing Ranges.
Hope it helps.
User avatar
egmat
User avatar
e-GMAT Representative
Joined: 02 Nov 2011
Last visit: 02 Sep 2026
Posts: 6,287
Own Kudos:
Given Kudos: 715
GMAT Date: 08-19-2020
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 6,287
Kudos: 33,892
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Hi Shlok02,

Your algebra is perfect. You correctly cracked |x+3| > 2 into its two ranges: x > -1 OR x < -5. The only thing that's flipped is the direction you're checking.

Here's the key idea. Statement II is satisfied by a big set of values - everything with x > -1and everything with x < -5. The word "OR" means you only need to land in one of those ranges for |x+3| > 2 to hold.

Now look at what the question actually hands you: x < -5. Every value you could possibly have (-5.1, -6, -100, ...) sits inside one of statement II's ranges - the x < -5 piece. So for every allowed value of x, statement II is true. That's exactly what "must be true" means.

The x > -1 range isn't a problem. It just means II happens to be true for some extra values too - values our x never takes. Extra coverage never hurts you. You don't need our range to match both of II's pieces; you only need our range to fall within II's true set. And it does.

So the test is one-directional: Is (given range) a subset of (statement's true range)? Here, x < -5 is fully inside x < -5 OR x > -1. Yes - must be true.

A cleaner-to-feel version:

Given: x < -5. Must x < 0 be true?

x < 0 is true for tons of values you'll never have (like -2, -0.5), but every single x < -5 value is also below 0. So yes - it must be true. The extra values x < 0 allows are irrelevant; what matters is that your whole given range lives inside it.

Same logic drives statement II. Your ranges were right - just check containment, not matching.

Answer: D

Shlok02

I had a query regarding Statement 2
We get 2 conditions- x>-1 and x<-5
since the second range satisfies the inequality given, but the first one does not, how can we say that (2) MUST BE TRUE??
   1   2 
Moderator:
Math Expert
113061 posts