GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 24 Sep 2018, 02:43

### 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

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

# Solving Inequalities

Author Message
TAGS:

### Hide Tags

Current Student
Joined: 29 Mar 2012
Posts: 317
Location: India
GMAT 1: 640 Q50 V26
GMAT 2: 660 Q50 V28
GMAT 3: 730 Q50 V38

### Show Tags

Updated on: 16 May 2013, 00:24
14
31

Solving Inequalities

I was going through the posts on inequalities and found that many good concepts are explained here, but still people do have trouble solving the question using these concept.
In these posts, there were quadratic equations, curves, graphs and other mathematical stuff. With this post, I am trying to provide a simple method to solve such questions quickly. I won't be writing the concepts behind it.
Remember this is the same OLD concept, it's just presented differently.

Case 1: Multiplication

for example: $$(x-1)(x-2)(x-3)(x-7) \leq 0$$

To check the intervals in which this inequality holds true, we need to pick only one value from the number line.
Lets say x = 10, then (9)(8)(7)(3) > 0, in every alternate interval the sign would be + for the above expression

---(+)-----1---(-)---2---(+)---3-----------(-)----------7----(+)------

Thus, inequality would hold true in the intervals:
$$1 \leq x \leq 2$$
$$3 \leq x \leq 7$$,

Note that intervals are inclusive of 3 & 7

Case 2: Division

In case of division:
$$\frac{(x-1)(x-2)}{(x-3)(x-7)} \leq 0$$
Using the same approach as above;
$$1 \leq x \leq 2$$
$$3 < x < 7,$$ $$(x\neq3,$$ $$7)$$

Since (x-3)(x-7) is in denominator, its value can't be 0.

Following things to be kept in mind while using above method:
1. Cofficient of x should be positive: for ex - $$(x-a)(b-x)>0$$, can be written as $$(x-a)(x-b)<0$$
2. Even powers: for ex - $$(x-9)^2(x+3) \geq 0$$, $$(x-9)^2$$ is always greater than 0, so, it should be only considered to check the equality (=0)
3. Odd powers: $$(x-a)^3(x-b)^5>0$$, will be same as $$(x-a)(x-b)>0$$
4. Cancelling the common terms:
for ex - $$\frac {(x^2+x-6)(x-11)}{(x+3)} >0$$, it can be simplified as (x-2)(x-11)>0
or, ---(+)-----2---(-)-------------11----(+)------
thus $$x <2$$ or $$x>11$$, but since at x = -3 (in the original expression), we get undefined form, so, $$x \neq -3$$

A question for you:
For what values of x, does the following inequality holds true?
$$(x-a)(x-b)...(x-n)...(x-z) \geq 0$$, where {a, b, c,...} are integers.
Spoiler: :: Solution
The expression has (x-x), thus, it is always 0=0 for every value of x.

Reference post:
http://gmatclub.com/forum/inequalities-trick-91482.html

PS: I hope you find this post useful, please provide feedback to improve the quality of the post.

Thanks,

Originally posted by cyberjadugar on 19 Jun 2012, 03:34.
Last edited by cyberjadugar on 16 May 2013, 00:24, edited 1 time in total.
Senior Manager
Affiliations: UWC
Joined: 09 May 2012
Posts: 380
GMAT 1: 620 Q42 V33
GMAT 2: 680 Q44 V38
GPA: 3.43
WE: Engineering (Entertainment and Sports)

### Show Tags

19 Jun 2012, 05:21
1
This one is a really useful trick cyberjadugar. Kudos to you and gurpreetsingh.

Example for practice from OG13 PS229:

How many of the integers that satisfy the inequality $$\frac{{(x+2)(x+3)}}{{x-2}}x\geq{0}$$ are less than 5?

A 1
B 2
C 3
D 4
E 5
Current Student
Joined: 29 Mar 2012
Posts: 317
Location: India
GMAT 1: 640 Q50 V26
GMAT 2: 660 Q50 V28
GMAT 3: 730 Q50 V38

### Show Tags

19 Jun 2012, 05:59
macjas wrote:
This one is a really useful trick cyberjadugar. Kudos to you and gurpreetsingh.

Example for practice from OG13 PS229:

How many of the integers that satisfy the inequality $$\frac{{(x+2)(x+3)}}{{x-2}}x\geq{0}$$ are less than 5?

A 1
B 2
C 3
D 4
E 5

Hi,

Check here:
http://gmatclub.com/forum/how-many-of-the-integers-that-satisfy-the-inequality-x-2-x-134194.html#p1094729
You can get the range for x using the mentioned method.

Regards,
Senior Manager
Affiliations: UWC
Joined: 09 May 2012
Posts: 380
GMAT 1: 620 Q42 V33
GMAT 2: 680 Q44 V38
GPA: 3.43
WE: Engineering (Entertainment and Sports)

### Show Tags

19 Jun 2012, 06:09
macjas wrote:
This one is a really useful trick cyberjadugar. Kudos to you and gurpreetsingh.

Example for practice from OG13 PS229:

How many of the integers that satisfy the inequality $$\frac{{(x+2)(x+3)}}{{x-2}}x\geq{0}$$ are less than 5?

A 1
B 2
C 3
D 4
E 5

Hi,

Check here:
http://gmatclub.com/forum/how-many-of-the-integers-that-satisfy-the-inequality-x-2-x-134194.html#p1094729
You can get the range for x using the mentioned method.

Regards,

haha I know; I started that thread. I reposted this problem to link a real GMAT question to this technique to add value to this thread...
Senior Manager
Affiliations: UWC
Joined: 09 May 2012
Posts: 380
GMAT 1: 620 Q42 V33
GMAT 2: 680 Q44 V38
GPA: 3.43
WE: Engineering (Entertainment and Sports)

### Show Tags

20 Jun 2012, 01:24
So that problem that you posted, let me give it a try:

A question for you:
For what values of x, does the following inequality holds true?
$$(x-a)(x-b)...(x-n)...(x-z) \geq 0$$, where {a, b, c,...} are integers.

Roots are: a,b,c,d...z
The condition will hold true for these intervals:
$$x\geq{z}$$
$$x =x$$
$${x}\leq{x}\leq{y}$$
$${v}\leq{x\leq{w}$$
$${s}\leq{x\leq{t}$$
$${p}\leq{x\leq{t}$$
$${m}\leq{x\leq{n}$$
$${j}\leq{x\leq{k}$$
$${g}\leq{x\leq{h}$$
$${d}\leq{x\leq{e}$$
$${a}\leq{x\leq{b}$$

Is this correct??
Current Student
Joined: 29 Mar 2012
Posts: 317
Location: India
GMAT 1: 640 Q50 V26
GMAT 2: 660 Q50 V28
GMAT 3: 730 Q50 V38

### Show Tags

20 Jun 2012, 01:33
macjas wrote:
So that problem that you posted, let me give it a try:

A question for you:
For what values of x, does the following inequality holds true?
$$(x-a)(x-b)...(x-n)...(x-z) \geq 0$$, where {a, b, c,...} are integers.

Roots are: a,b,c,d...z
The condition will hold true for these intervals:
$$x\geq{z}$$
$$x =x$$
$${x}\leq{x}\leq{y}$$
$${v}\leq{x\leq{w}$$
$${s}\leq{x\leq{t}$$
$${p}\leq{x\leq{t}$$
$${m}\leq{x\leq{n}$$
$${j}\leq{x\leq{k}$$
$${g}\leq{x\leq{h}$$
$${d}\leq{x\leq{e}$$
$${a}\leq{x\leq{b}$$

Is this correct??

Hi,

You have identified everything, but give a close tought again. The answer is pretty much straight forward.

Regards,
Senior Manager
Affiliations: UWC
Joined: 09 May 2012
Posts: 380
GMAT 1: 620 Q42 V33
GMAT 2: 680 Q44 V38
GPA: 3.43
WE: Engineering (Entertainment and Sports)

### Show Tags

20 Jun 2012, 01:48
macjas wrote:
So that problem that you posted, let me give it a try:

A question for you:
For what values of x, does the following inequality holds true?
$$(x-a)(x-b)...(x-n)...(x-z) \geq 0$$, where {a, b, c,...} are integers.

Roots are: a,b,c,d...z
The condition will hold true for these intervals:
$$x\geq{z}$$
$$x =x$$
$${x}\leq{x}\leq{y}$$
$${v}\leq{x\leq{w}$$
$${s}\leq{x\leq{t}$$
$${p}\leq{x\leq{t}$$
$${m}\leq{x\leq{n}$$
$${j}\leq{x\leq{k}$$
$${g}\leq{x\leq{h}$$
$${d}\leq{x\leq{e}$$
$${a}\leq{x\leq{b}$$

Is this correct??

Hi,

You have identified everything, but give a close tought again. The answer is pretty much straight forward.

Regards,

I have no idea... what am I missing here??
Senior Manager
Affiliations: UWC
Joined: 09 May 2012
Posts: 380
GMAT 1: 620 Q42 V33
GMAT 2: 680 Q44 V38
GPA: 3.43
WE: Engineering (Entertainment and Sports)

### Show Tags

21 Jun 2012, 11:08
hey cj, looking forward to your Solving Set theory post...
Manager
Joined: 02 Jun 2011
Posts: 132

### Show Tags

25 Jun 2012, 05:05
nice one - and a great link .. inequalities

kudos to cyberjadugar as well as gurpreet singh, veritas karishma...

is the answer to macjas quest is - 4 ( -2, -3, 3,4) -? kindly correct with explaination.

and to the solution to cyberjadugar - is it as follows?
x<=a = +ve
a>=x>=b = -ve
b>=x>=n = +ve
n>=x>=z = -ve
x>=z = +ve

i may have done above wrong. pls do correct.
and if the above is right , how the whole thing can be put together?
like is it possible to write - z<=x<=a ot like z<=x<=a is +ve.
Manager
Joined: 15 Sep 2009
Posts: 225

### Show Tags

25 Jun 2012, 08:38
-3,-2,3,4

Therefore, there are 4 integers less than 5 that satisfy the inequality.

Cheers,
Der alte Fritz.
_________________

+1 Kudos me - I'm half Irish, half Prussian.

Intern
Joined: 07 Apr 2010
Posts: 6
Concentration: Entrepreneurship, Technology
GMAT 1: 740 Q49 V41
GPA: 3.84
WE: Research (Computer Hardware)

### Show Tags

02 Nov 2012, 14:36
Edit: The 'x' before the inequality caused the confusion. I checked and that 'x' doe not exist.

What about '0' and '-4', '-5' '-6'...?
0<5, -4<5 ...

macjas wrote:
This one is a really useful trick cyberjadugar. Kudos to you and gurpreetsingh.

Example for practice from OG13 PS229:

How many of the integers that satisfy the inequality $$\frac{{(x+2)(x+3)}}{{x-2}}x\geq{0}$$ are less than 5?

A 1
B 2
C 3
D 4
E 5
OldFritz wrote:
-3,-2,3,4

Therefore, there are 4 integers less than 5 that satisfy the inequality.

Cheers,
Der alte Fritz.
Intern
Status: Fighting to kill GMAT
Joined: 23 Sep 2012
Posts: 30
Location: United States
Schools: Duke '16
GPA: 3.8
WE: General Management (Other)

### Show Tags

05 Nov 2012, 06:23
gpk wrote:
Edit: The 'x' before the inequality caused the confusion. I checked and that 'x' doe not exist.

What about '0' and '-4', '-5' '-6'...?
0<5, -4<5 ...

macjas wrote:
This one is a really useful trick cyberjadugar. Kudos to you and gurpreetsingh.

Example for practice from OG13 PS229:

How many of the integers that satisfy the inequality $$\frac{{(x+2)(x+3)}}{{x-2}}x\geq{0}$$ are less than 5?

A 1
B 2
C 3
D 4
E 5
OldFritz wrote:
-3,-2,3,4

Therefore, there are 4 integers less than 5 that satisfy the inequality.

Cheers,
Der alte Fritz.

Yes, the extra 'x' is a typo. Of course, none of the options match if that x was still present in the inequality.
_________________

Kudos is the currency of appreciation.

You can have anything you want if you want it badly enough. You can be anything you want to be and do anything you set out to accomplish, if you hold to that desire with the singleness of purpose. ~William Adams

Many of life's failures are people who did not realize how close to success they were when they gave up. ~Thomas A. Edison

Wir müssen wissen, Wir werden wissen. (We must know, we will know.) ~Hilbert

Intern
Status: Fighting to kill GMAT
Joined: 23 Sep 2012
Posts: 30
Location: United States
Schools: Duke '16
GPA: 3.8
WE: General Management (Other)

### Show Tags

05 Nov 2012, 06:33
2. Even powers: for ex - $$(x-9)^2(x+3) \geq 0$$, $$(x-9)^2$$ is always greater than 0, so, it should be only considered to check the equality (=0)
3. Odd powers: $$(x-a)^3(x-b)^5>0$$, will be same as $$(x-a)(x-b)>0$$

Could you illustrate (2) and (3) with examples. Though (3) is kind of clear, I do not understand what exactly you mean in (2).
_________________

Kudos is the currency of appreciation.

You can have anything you want if you want it badly enough. You can be anything you want to be and do anything you set out to accomplish, if you hold to that desire with the singleness of purpose. ~William Adams

Many of life's failures are people who did not realize how close to success they were when they gave up. ~Thomas A. Edison

Wir müssen wissen, Wir werden wissen. (We must know, we will know.) ~Hilbert

Manager
Joined: 23 Jan 2013
Posts: 156
Concentration: Technology, Other
Schools: Berkeley Haas
GMAT Date: 01-14-2015
WE: Information Technology (Computer Software)

### Show Tags

20 Apr 2013, 20:16
closed271 wrote:
2. Even powers: for ex - $$(x-9)^2(x+3) \geq 0$$, $$(x-9)^2$$ is always greater than 0, so, it should be only considered to check the equality (=0)
3. Odd powers: $$(x-a)^3(x-b)^5>0$$, will be same as $$(x-a)(x-b)>0$$

Could you illustrate (2) and (3) with examples. Though (3) is kind of clear, I do not understand what exactly you mean in (2).

Could some one please explain point 2 ??
Current Student
Joined: 29 Mar 2012
Posts: 317
Location: India
GMAT 1: 640 Q50 V26
GMAT 2: 660 Q50 V28
GMAT 3: 730 Q50 V38

### Show Tags

15 May 2013, 23:55
shelrod007 wrote:
closed271 wrote:
2. Even powers: for ex - $$(x-9)^2(x+3) \geq 0$$, $$(x-9)^2$$ is always greater than 0, so, it should be only considered to check the equality (=0)
3. Odd powers: $$(x-a)^3(x-b)^5>0$$, will be same as $$(x-a)(x-b)>0$$

Could you illustrate (2) and (3) with examples. Though (3) is kind of clear, I do not understand what exactly you mean in (2).

Could some one please explain point 2 ??

Hi,

For even powers, such as $$(x-9)^2$$, if you check for various values of x,
for example,
$$x = 1, (x-9)^2 = 64(>0)$$
$$x=-1, (x-9)^2 = 100(>0)$$
$$x =10, (x-9)^2=1(>0)$$
but for $$x = 9, (x-9)^2=0$$
so, for every value of x, the even powers will always be greater or equal to 0, i.e. the sign of the expression doesn't change from positive to negative.

Let me know if you need further clarification.

Regards,
Intern
Status: K... M. G...
Joined: 22 Oct 2012
Posts: 36
GMAT Date: 08-27-2013
GPA: 3.8

### Show Tags

24 Jul 2013, 07:13

Solving Inequalities

I was going through the posts on inequalities and found that many good concepts are explained here, but still people do have trouble solving the question using these concept.
In these posts, there were quadratic equations, curves, graphs and other mathematical stuff. With this post, I am trying to provide a simple method to solve such questions quickly. I won't be writing the concepts behind it.
Remember this is the same OLD concept, it's just presented differently.

Case 1: Multiplication

for example: $$(x-1)(x-2)(x-3)(x-7) \leq 0$$

To check the intervals in which this inequality holds true, we need to pick only one value from the number line.
Lets say x = 10, then (9)(8)(7)(3) > 0, in every alternate interval the sign would be + for the above expression

---(+)-----1---(-)---2---(+)---3-----------(-)----------7----(+)------

Thus, inequality would hold true in the intervals:
$$1 \leq x \leq 2$$
$$3 \leq x \leq 7$$,

Note that intervals are inclusive of 3 & 7

Case 2: Division

In case of division:
$$\frac{(x-1)(x-2)}{(x-3)(x-7)} \leq 0$$
Using the same approach as above;
$$1 \leq x \leq 2$$
$$3 < x < 7,$$ $$(x\neq3,$$ $$7)$$

Since (x-3)(x-7) is in denominator, its value can't be 0.

Following things to be kept in mind while using above method:
1. Cofficient of x should be positive: for ex - $$(x-a)(b-x)>0$$, can be written as $$(x-a)(x-b)<0$$
2. Even powers: for ex - $$(x-9)^2(x+3) \geq 0$$, $$(x-9)^2$$ is always greater than 0, so, it should be only considered to check the equality (=0)
3. Odd powers: $$(x-a)^3(x-b)^5>0$$, will be same as $$(x-a)(x-b)>0$$
4. Cancelling the common terms:
for ex - $$\frac {(x^2+x-6)(x-11)}{(x+3)} >0$$, it can be simplified as (x-2)(x-11)>0
or, ---(+)-----2---(-)-------------11----(+)------
thus $$x <2$$ or $$x>11$$, but since at x = -3 (in the original expression), we get undefined form, so, $$x \neq -3$$

A question for you:
For what values of x, does the following inequality holds true?
$$(x-a)(x-b)...(x-n)...(x-z) \geq 0$$, where {a, b, c,...} are integers.
Spoiler: :: Solution
The expression has (x-x), thus, it is always 0=0 for every value of x.

Reference post:
http://gmatclub.com/forum/inequalities-trick-91482.html

PS: I hope you find this post useful, please provide feedback to improve the quality of the post.

Thanks,

Hi,

I have question with the below one,

[list]for ex - $$\frac {(x^2+x-6)(x-11)}{(x+3)} >0$$, it can be simplified as (x-2)(x-11)>0
or, ---(+)-----2---(-)-------------11----(+)------
thus $$x <2$$ or $$x>11$$, but since at x = -3 (in the original expression), we get undefined form, so, [m]x

using the plot of + ,- , how to identify x<2 or x>2 . I have a impression like if > is the used then it should be always x>2 & x>11 but the answer u have mentioned again confused me. Please help
VP
Status: Far, far away!
Joined: 02 Sep 2012
Posts: 1086
Location: Italy
Concentration: Finance, Entrepreneurship
GPA: 3.8

### Show Tags

24 Jul 2013, 07:17
Hi there,

I wrote a post about how to choose the correct interval(s): tips-and-tricks-inequalities-150873.html#p1211920.

Hope it helps
_________________

It is beyond a doubt that all our knowledge that begins with experience.

Kant , Critique of Pure Reason

Tips and tricks: Inequalities , Mixture | Review: MGMAT workshop
Strategy: SmartGMAT v1.0 | Questions: Verbal challenge SC I-II- CR New SC set out !! , My Quant

Rules for Posting in the Verbal Forum - Rules for Posting in the Quant Forum[/size][/color][/b]

Math Expert
Joined: 02 Sep 2009
Posts: 49381

### Show Tags

09 Oct 2015, 01:25
1
More on inequalities: Inequalities Made Easy!
_________________
Intern
Joined: 12 Jul 2017
Posts: 3

### Show Tags

19 Aug 2017, 23:10
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

Posted from my mobile device
Math Expert
Joined: 02 Sep 2009
Posts: 49381

### Show Tags

20 Aug 2017, 01:27
artisood17 wrote:
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

Posted from my mobile device

Discussed here: https://gmatclub.com/forum/if-4-7-x-3-w ... 68681.html

Please follow the rules (https://gmatclub.com/forum/rules-for-po ... 33935.html) when posting a question. Thank you.
_________________
Re: Solving Inequalities &nbs [#permalink] 20 Aug 2017, 01:27

Go to page    1   2    Next  [ 21 posts ]

Display posts from previous: Sort by

# Solving Inequalities

## Events & Promotions

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.