Last visit was: 24 Apr 2026, 06:56 It is currently 24 Apr 2026, 06:56
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)|   Algebra|   Inequalities|      
User avatar
EgmatQuantExpert
User avatar
e-GMAT Representative
Joined: 04 Jan 2015
Last visit: 02 Apr 2024
Posts: 3,657
Own Kudos:
20,878
 [36]
Given Kudos: 165
Expert
Expert reply
Posts: 3,657
Kudos: 20,878
 [36]
How many non-positive integers satisfy the inequality (9 x^2)(x +1) Updated on: 31 Jan 2026, 01:15
Originally posted by EgmatQuantExpert on 11 Jan 2019, 06:59.
Last edited by Bunuel on 31 Jan 2026, 01:15, edited 2 times in total.
2
Kudos
Add Kudos
33
Bookmarks
Bookmark this Post
00:00
Start Timer
Pause Timer
Resume Timer
Show Answer
Hide
Show
History
C
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:

31% (02:10) correct 69% (02:28) wrong based on 414 sessions

History

Date
Time
Result
Not Attempted Yet
How many non-positive integers satisfy the inequality \(\frac{(9 – x^2)(x +1)(x + 2)^2}{(x + 3)(x – 2)} ≤ 0\)?

A. 1
B. 2
C. 3
D. 4
E. Infinite

Show Spoiler

e-GMAT Question of the Week #31

ShowHide Answer
Official Answer
C
User avatar
Archit3110
User avatar
Major Poster
Joined: 18 Aug 2017
Last visit: 24 Apr 2026
Posts: 8,629
Own Kudos:
5,190
 [6]
Given Kudos: 243
Status:You learn more from failure than from success.
Location: India
Concentration: Sustainability, Marketing
GMAT Focus 1: 545 Q79 V79 DI73
GMAT Focus 2: 645 Q83 V82 DI81
GPA: 4
WE:Marketing (Energy)
Products:
GMAT Club Tests Forum Quiz
GMAT Focus 2: 645 Q83 V82 DI81
Posts: 8,629
Kudos: 5,190
 [6]
How many non-positive integers satisfy the inequality (9 x^2)(x +1) 14 Jan 2019, 09:59
2
Kudos
Add Kudos
2
Bookmarks
Bookmark this Post
EgmatQuantExpert
How many non-positive integers satisfy the inequality \(\frac{(9 – x^2)(x +1)(x + 2)^2}{(x + 3)(x – 2)} ≤ 0\)?

    A. 1
    B. 2
    C. 3
    D. 4
    E. Infinite

Show more

Solution attached

Posted from my mobile device
Attachments

IMG_20190114_222835001.jpg
IMG_20190114_222835001.jpg [ 3.23 MiB | Viewed 8275 times ]

User avatar
Kem12
User avatar
Joined: 18 Apr 2018
Last visit: 01 Feb 2021
Posts: 67
Own Kudos:
34
Given Kudos: 211
Posts: 67
Kudos: 34
How many non-positive integers satisfy the inequality (9 x^2)(x +1) 14 Jan 2019, 14:20
Kudos
Add Kudos
Bookmarks
Bookmark this Post
hi chetan2u Pls please help me out with solving this using wavy line method
I have tried but i still keep getting B as my answer
Thanks
Also in the Explanation in the picture the writer wrote (9-x)^2 but the question says (9-x^2) they are not the same thing
User avatar
chetan2u
User avatar
GMAT Expert
Joined: 02 Aug 2009
Last visit: 24 Apr 2026
Posts: 11,229
Own Kudos:
45,007
 [4]
Given Kudos: 335
Status:Math and DI Expert
Location: India
Concentration: Human Resources, General Management
GMAT Focus 1: 735 Q90 V89 DI81
Products:
My Reviews
Expert
Expert reply
GMAT Focus 1: 735 Q90 V89 DI81
Posts: 11,229
Kudos: 45,007
 [4]
How many non-positive integers satisfy the inequality (9 x^2)(x +1) 14 Jan 2019, 19:50
4
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Kem12
hi chetan2u Pls please help me out with solving this using wavy line method
I have tried but i still keep getting B as my answer
Thanks
Also in the Explanation in the picture the writer wrote (9-x)^2 but the question says (9-x^2) they are not the same thing
Show more

\(\frac{(9 – x^2)(x +1)(x + 2)^2}{(x + 3)(x – 2)} ≤ 0\)...

Hi..

You can get your answer by just looking at the extreme values of X for which the equation becomes 0..
Here the values will be 3 and -3, but since we are looking at only non positive integers, see the effect on the entire equation beyond x less than -3..
So substitute X as -4..
\(\frac{(9 – (-4)^2)(-4+1)(-4 + 2)^2}{(-4+ 3)(-4– 2)} ≤ 0......\frac{(-7)*(-3)*4}{(-1)*(-6)}=14>0\)..
So beyond x<-3, the equation will be positive, so discard all the values..

Now, we are left with 0,-1,-2,-3
Discard -3 as the denominator becomes 0 and our equation becomes undefined..
-1 and -2 will give you a 0 as value as numerator has (x+1) and (x+2) in numerator, so ok.
When x is 0, term becomes 9*1*2/(3*(-2))=-3, so ok..

Therefore values will be 0, -1, -2--------3 values

C
avatar
SaladQueen
avatar
Joined: 08 Jan 2019
Last visit: 19 Jan 2019
Posts: 3
Own Kudos:
2
 [1]
Given Kudos: 18
Posts: 3
Kudos: 2
 [1]
How many non-positive integers satisfy the inequality (9 x^2)(x +1) 14 Jan 2019, 21:21
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
X cannot be equals to -3 or 2 since numerator is (3+X)(X-2).
Non positive integer starts from 0 all the way to negative infinity.
So just substitute X=0, X=-1, X=-2, X=-4 into the equation and see if the answer is a non positive value.
When X=0, the equation is negative.
When X=-1, the equation is negative.
When X=-2, the equation is negative.
When X>=-4, the equation is positive.
C.
User avatar
Kem12
User avatar
Joined: 18 Apr 2018
Last visit: 01 Feb 2021
Posts: 67
Own Kudos:
34
 [1]
Given Kudos: 211
Posts: 67
Kudos: 34
 [1]
How many non-positive integers satisfy the inequality (9 x^2)(x +1) 15 Jan 2019, 01:10
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Thanks so much chetan2u

Posted from my mobile device
User avatar
EgmatQuantExpert
User avatar
e-GMAT Representative
Joined: 04 Jan 2015
Last visit: 02 Apr 2024
Posts: 3,657
Own Kudos:
20,878
 [2]
Given Kudos: 165
Expert
Expert reply
Posts: 3,657
Kudos: 20,878
 [2]
How many non-positive integers satisfy the inequality (9 x^2)(x +1) 16 Jan 2019, 21:03
1
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post

Solution


Given:
    • \(\frac{(9 – x^2)(x +1)(x + 2)^2}{(x + 3)(x – 2)} ≤ 0\)

To find:
    • The number of non-positive integer values of x that satisfy the given inequality

Approach and Working:
    • \(\frac{(9 – x^2)(x +1)(x + 2)^2}{(x + 3)(x – 2)} ≤ 0\)

The denominator must not be = 0. Thus, x ≠ {-3, 2}
    • \(\frac{(3 – x)(3 + x)(x + 1)(x + 2)^2}{(x + 3)(x – 2)} ≤ 0\)

We can cancel out the common term (x + 3) in both numerator and denominator
    • \(\frac{(3 – x)(x + 1)(x + 2)^2}{(x – 2)} ≤ 0\)

And, we know that \((x + 2)^2\) is always ≥ 0. It is equal to 0, when x = -2
    • Thus, if x = -2, the inequality will be 0
    • So, we get, \(\frac{(3 – x)(x + 1)}{(x – 2)} ≤ 0\)

Now, if we multiply both numerator and denominator by (x – 2), we get,
    • \(\frac{(3 – x)(x + 1)(x - 2)}{(x – 2)^2} ≤ 0\)
      o Implies, (3 – x)(x + 1)(x - 2) ≤ 0
      o Multiplying by -1 on both sides, we get, (x - 3)(x + 1)(x - 2) ≥ 0

    • The zero points of the above inequality are x = {-1, 2, 3}

Let’s represent this on a number line and identify the regions, where this expression will give a non-negative value.



    • From the above diagram, we can see that (x - 3)(x + 1)(x - 2) ≥ 0 in the regions x ≥ 3 and -1 ≤ x ≤ 2.
    • But, we are asked for non-positive values of x, they are x = {-1, 0}

Therefore, the non-positive values of x, which satisfy the inequality, \(\frac{(9 – x^2)(x +1)(x + 2)^2}{(x + 3)(x – 2)} ≤ 0\) are x = {-2, -1, 0}

Hence the correct answer is Option C.

Answer: C

User avatar
arunbhati
User avatar
Joined: 08 Aug 2025
Last visit: 17 Apr 2026
Posts: 40
Own Kudos:
1
Given Kudos: 61
Posts: 40
Kudos: 1
How many non-positive integers satisfy the inequality (9 x^2)(x +1) 30 Jan 2026, 23:15
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Hi Sir,


How this solution is possible -1<=x<=2. How x=2 is possible ? when x=2 then denominator will be zero


EgmatQuantExpert

Solution


Given:

• \(\frac{(9 – x^2)(x +1)(x + 2)^2}{(x + 3)(x – 2)} ≤ 0\)

To find:

• The number of non-positive integer values of x that satisfy the given inequality

Approach and Working:

• \(\frac{(9 – x^2)(x +1)(x + 2)^2}{(x + 3)(x – 2)} ≤ 0\)

The denominator must not be = 0. Thus, x ≠ {-3, 2}

• \(\frac{(3 – x)(3 + x)(x + 1)(x + 2)^2}{(x + 3)(x – 2)} ≤ 0\)

We can cancel out the common term (x + 3) in both numerator and denominator

• \(\frac{(3 – x)(x + 1)(x + 2)^2}{(x – 2)} ≤ 0\)

And, we know that \((x + 2)^2\) is always ≥ 0. It is equal to 0, when x = -2

• Thus, if x = -2, the inequality will be 0
• So, we get, \(\frac{(3 – x)(x + 1)}{(x – 2)} ≤ 0\)

Now, if we multiply both numerator and denominator by (x – 2), we get,
    • \(\frac{(3 – x)(x + 1)(x - 2)}{(x – 2)^2} ≤ 0\)
    • o Implies, (3 – x)(x + 1)(x - 2) ≤ 0o Multiplying by -1 on both sides, we get, (x - 3)(x + 1)(x - 2) ≥ 0
    • The zero points of the above inequality are x = {-1, 2, 3}

Let’s represent this on a number line and identify the regions, where this expression will give a non-negative value.




• From the above diagram, we can see that (x - 3)(x + 1)(x - 2) ≥ 0 in the regions x ≥ 3 and -1 ≤ x ≤ 2.
• But, we are asked for non-positive values of x, they are x = {-1, 0}

Therefore, the non-positive values of x, which satisfy the inequality, \(\frac{(9 – x^2)(x +1)(x + 2)^2}{(x + 3)(x – 2)} ≤ 0\) are x = {-2, -1, 0}

Hence the correct answer is Option C.

Answer: C

Show more
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 24 Apr 2026
Posts: 109,813
Own Kudos:
810,996
Given Kudos: 105,870
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: 109,813
Kudos: 810,996
How many non-positive integers satisfy the inequality (9 x^2)(x +1) Updated on: 03 Feb 2026, 04:29
Originally posted by Bunuel on 31 Jan 2026, 01:19.
Last edited by Bunuel on 03 Feb 2026, 04:29, edited 1 time in total.
Kudos
Add Kudos
Bookmarks
Bookmark this Post
arunbhati
Hi Sir,


How this solution is possible -1<=x<=2. How x=2 is possible ? when x=2 then denominator will be zero



Show more

You are right, x cannot be 2. The solution to \(\frac{(9 – x^2)(x + 1)(x + 2)^2}{(x + 3)(x – 2)} ≤ 0\) is \(-1 ≤ x < 2\) or \(x ≥ 3\) or \(x = -2\).
User avatar
arunbhati
User avatar
Joined: 08 Aug 2025
Last visit: 17 Apr 2026
Posts: 40
Own Kudos:
1
Given Kudos: 61
Posts: 40
Kudos: 1
How many non-positive integers satisfy the inequality (9 x^2)(x +1) Updated on: 03 Feb 2026, 04:31
Originally posted by arunbhati on 03 Feb 2026, 02:30.
Last edited by Bunuel on 03 Feb 2026, 04:31, edited 1 time in total.
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Hi Bunuel,

if X is not equal to 2 then it should have only two non-postive solution that is -1 and 0
Bunuel


You are right, x cannot be 2. The solution to \(\frac{(9 – x^2)(x + 1)(x + 2)^2}{(x + 3)(x – 2)} ≤ 0\) is \(-1 ≤ x < 2\) or \(x ≥ 3\) or \(x = -2\) .
Show more
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 24 Apr 2026
Posts: 109,813
Own Kudos:
810,996
Given Kudos: 105,870
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: 109,813
Kudos: 810,996
How many non-positive integers satisfy the inequality (9 x^2)(x +1) 03 Feb 2026, 04:31
Kudos
Add Kudos
Bookmarks
Bookmark this Post
arunbhati
Hi Bunuel,

if X is not equal to 2 then it should have only two non-postive solution that is -1 and 0

Show more

The solution to \(\frac{(9 – x^2)(x + 1)(x + 2)^2}{(x + 3)(x – 2)} ≤ 0\) is \(-1 ≤ x < 2\) or \(x ≥ 3\) or \(x = -2\).

So, non-positive solutions are -2, -1, and 0.
Moderators:
Bunuel
Math Expert
109813 posts
Krunaal
Tuck School Moderator
853 posts