It is currently 20 Sep 2017, 04:24

### 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

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# Inequalities trick

Author Message
TAGS:

### Hide Tags

CEO
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2764

Kudos [?]: 1853 [91], given: 235

Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35

### Show Tags

16 Mar 2010, 10:11
91
KUDOS
280
This post was
BOOKMARKED
I learnt this trick while I was in school and yesterday while solving one question I recalled.
Its good if you guys use it 1-2 times to get used to it.

Suppose you have the inequality

f(x) = (x-a)(x-b)(x-c)(x-d) < 0

Just arrange them in order as shown in the picture and draw curve starting from + from right.

now if f(x) < 0 consider curve having "-" inside and if f(x) > 0 consider curve having "+" and combined solution will be the final solution. I m sure I have recalled it fully but if you guys find any issue on that do let me know, this is very helpful.

Don't forget to arrange then in ascending order from left to right. a<b<c<d

So for f(x) < 0 consider "-" curves and the ans is : (a < x < b) , (c < x < d)
and for f(x) > 0 consider "+" curves and the ans is : (x < a), (b < x < c) , (d < x)

If f(x) has three factors then the graph will have - + - +
If f(x) has four factors then the graph will have + - + - +

If you can not figure out how and why, just remember it.
Try to analyze that the function will have number of roots = number of factors and every time the graph will touch the x axis.

For the highest factor d if x>d then the whole f(x) > 0 and after every interval of the roots the signs will change alternatively.
Attachments

1.jpg [ 6.73 KiB | Viewed 54013 times ]

_________________

Fight for your dreams :For all those who fear from Verbal- lets give it a fight

Money Saved is the Money Earned

Jo Bole So Nihaal , Sat Shri Akaal

GMAT Club Premium Membership - big benefits and savings

Gmat test review :
http://gmatclub.com/forum/670-to-710-a-long-journey-without-destination-still-happy-141642.html

Kudos [?]: 1853 [91], given: 235

Intern
Joined: 08 Mar 2010
Posts: 6

Kudos [?]: [0], given: 4

### Show Tags

16 Mar 2010, 10:46
Can u plz explainn the backgoround of this & then the explanation.

Thanks

Kudos [?]: [0], given: 4

Manager
Joined: 13 Dec 2009
Posts: 249

Kudos [?]: 233 [0], given: 13

### Show Tags

19 Mar 2010, 12:48
I have applied this trick and it seemed to be quite useful.
_________________

My debrief: done-and-dusted-730-q49-v40

Kudos [?]: 233 [0], given: 13

CEO
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2764

Kudos [?]: 1853 [1], given: 235

Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35

### Show Tags

19 Mar 2010, 12:59
1
KUDOS
ttks10 wrote:
Can u plz explainn the backgoround of this & then the explanation.

Thanks

i m sorry i dont have any background for it, you just re-read it again and try to implement whenever you get such question and I will help you out in any issue.

sidhu4u wrote:
I have applied this trick and it seemed to be quite useful.

Nice to hear this....good luck.
_________________

Fight for your dreams :For all those who fear from Verbal- lets give it a fight

Money Saved is the Money Earned

Jo Bole So Nihaal , Sat Shri Akaal

GMAT Club Premium Membership - big benefits and savings

Gmat test review :
http://gmatclub.com/forum/670-to-710-a-long-journey-without-destination-still-happy-141642.html

Kudos [?]: 1853 [1], given: 235

Senior Manager
Status: Upset about the verbal score - SC, CR and RC are going to be my friend
Joined: 30 Jun 2010
Posts: 316

Kudos [?]: 23 [0], given: 6

### Show Tags

17 Oct 2010, 16:14
gurpreetsingh wrote:
I learnt this trick while I was in school and yesterday while solving one question I recalled.
Its good if you guys use it 1-2 times to get used to it.

So for f(x) < 0 consider "-" curves and the ans is : (a < x < b) , (c < x < d)
and for f(x) < 0 consider "+" curves and the ans is : (x < a), (b < x < c) , (d < x)

I don't understand this part alone. Can you please explain?
_________________

My gmat story
MGMAT1 - 630 Q44V32
MGMAT2 - 650 Q41V38
MGMAT3 - 680 Q44V37
GMATPrep1 - 660 Q49V31
Knewton1 - 550 Q40V27

Kudos [?]: 23 [0], given: 6

CEO
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2764

Kudos [?]: 1853 [0], given: 235

Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35

### Show Tags

17 Oct 2010, 17:19
Dreamy wrote:
gurpreetsingh wrote:
I learnt this trick while I was in school and yesterday while solving one question I recalled.
Its good if you guys use it 1-2 times to get used to it.

So for f(x) < 0 consider "-" curves and the ans is : (a < x < b) , (c < x < d)
and for f(x) < 0 consider "+" curves and the ans is : (x < a), (b < x < c) , (d < x)

I don't understand this part alone. Can you please explain?

Suppose you have the inequality

f(x) = (x-a)(x-b)(x-c)(x-d) < 0 you will consider the curve with -ve inside it.. check the attached image.

f(x) = (x-a)(x-b)(x-c)(x-d) > 0 consider the +ve of the curve
_________________

Fight for your dreams :For all those who fear from Verbal- lets give it a fight

Money Saved is the Money Earned

Jo Bole So Nihaal , Sat Shri Akaal

GMAT Club Premium Membership - big benefits and savings

Gmat test review :
http://gmatclub.com/forum/670-to-710-a-long-journey-without-destination-still-happy-141642.html

Kudos [?]: 1853 [0], given: 235

Intern
Joined: 12 Oct 2010
Posts: 9

Kudos [?]: [0], given: 11

### Show Tags

18 Oct 2010, 09:23
hey i remember having used this in my Boards ?(School exams) . Thanks Gurpreet . This makes life and its inequalities a lil easier . +1.

Kudos [?]: [0], given: 11

Manager
Joined: 03 Jun 2010
Posts: 173

Kudos [?]: 70 [0], given: 40

Location: United States (MI)
Concentration: Marketing, General Management

### Show Tags

19 Oct 2010, 04:09
if we have smth like (x-a)^2(x-b)
we don't need to change a sign when jump over "a".

Kudos [?]: 70 [0], given: 40

CEO
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2764

Kudos [?]: 1853 [0], given: 235

Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35

### Show Tags

19 Oct 2010, 12:56
ulm wrote:
if we have smth like (x-a)^2(x-b)
we don't need to change a sign when jump over "a".

yes even powers wont contribute to the inequality sign. But be wary of the root value of x=a
_________________

Fight for your dreams :For all those who fear from Verbal- lets give it a fight

Money Saved is the Money Earned

Jo Bole So Nihaal , Sat Shri Akaal

GMAT Club Premium Membership - big benefits and savings

Gmat test review :
http://gmatclub.com/forum/670-to-710-a-long-journey-without-destination-still-happy-141642.html

Kudos [?]: 1853 [0], given: 235

Senior Manager
Status: Upset about the verbal score - SC, CR and RC are going to be my friend
Joined: 30 Jun 2010
Posts: 316

Kudos [?]: 23 [0], given: 6

### Show Tags

21 Oct 2010, 10:52
gurpreetsingh wrote:
Dreamy wrote:
gurpreetsingh wrote:
I learnt this trick while I was in school and yesterday while solving one question I recalled.
Its good if you guys use it 1-2 times to get used to it.

So for f(x) < 0 consider "-" curves and the ans is : (a < x < b) , (c < x < d)
and for f(x) < 0 consider "+" curves and the ans is : (x < a), (b < x < c) , (d < x)

I don't understand this part alone. Can you please explain?

Suppose you have the inequality

f(x) = (x-a)(x-b)(x-c)(x-d) < 0 you will consider the curve with -ve inside it.. check the attached image.

f(x) = (x-a)(x-b)(x-c)(x-d) > 0 consider the +ve of the curve

Thank you gurpreet. I understand it now. You need fix the initial post. The second f(x) is also having < 0.
_________________

My gmat story
MGMAT1 - 630 Q44V32
MGMAT2 - 650 Q41V38
MGMAT3 - 680 Q44V37
GMATPrep1 - 660 Q49V31
Knewton1 - 550 Q40V27

Kudos [?]: 23 [0], given: 6

CEO
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2764

Kudos [?]: 1853 [0], given: 235

Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35

### Show Tags

21 Oct 2010, 10:57
Thanks typo fixed.
_________________

Fight for your dreams :For all those who fear from Verbal- lets give it a fight

Money Saved is the Money Earned

Jo Bole So Nihaal , Sat Shri Akaal

GMAT Club Premium Membership - big benefits and savings

Gmat test review :
http://gmatclub.com/forum/670-to-710-a-long-journey-without-destination-still-happy-141642.html

Kudos [?]: 1853 [0], given: 235

Current Student
Joined: 15 Jul 2010
Posts: 250

Kudos [?]: 238 [0], given: 65

GMAT 1: 750 Q49 V42

### Show Tags

21 Oct 2010, 23:12
Quite interesting.

Do you have any sample questions we can use this method on?
_________________

Consider KUDOS if my post was helpful.

My Debrief: http://gmatclub.com/forum/750-q49v42-105591.html#p825487

Kudos [?]: 238 [0], given: 65

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7608

Kudos [?]: 16884 [69], given: 230

Location: Pune, India

### Show Tags

22 Oct 2010, 06:33
69
KUDOS
Expert's post
97
This post was
BOOKMARKED
Yes, this is a neat little way to work with inequalities where factors are multiplied or divided. And, it has a solid reasoning behind it which I will just explain.

If (x-a)(x-b)(x-c)(x-d) < 0, we can draw the points a, b, c and d on the number line.
e.g. Given (x+2)(x-1)(x-7)(x-4) < 0, draw the points -2, 1, 7 and 4 on the number line as shown.

Attachment:

doc.jpg [ 7.9 KiB | Viewed 53069 times ]

This divides the number line into 5 regions. Values of x in right most region will always give you positive value of the expression. The reason for this is that if x > 7, all factors above will be positive.

When you jump to the next region between x = 4 and x = 7, value of x here give you negative value for the entire expression because now, (x - 7) will be negative since x < 7 in this region. All other factors are still positive.

When you jump to the next region on the left between x = 1 and x = 4, expression will be positive again because now two factors (x - 7) and (x - 4) are negative, but negative x negative is positive... and so on till you reach the leftmost section.

Since we are looking for values of x where the expression is < 0, here the solution will be -2 < x < 1 or 4< x < 7

It should be obvious that it will also work in cases where factors are divided.
e.g. (x - a)(x - b)/(x - c)(x - d) < 0
(x + 2)(x - 1)/(x -4)(x - 7) < 0 will have exactly the same solution as above.

Note: If, rather than < or > sign, you have <= or >=, in division, the solution will differ slightly. I will leave it for you to figure out why and how. Feel free to get back to me if you want to confirm your conclusion.
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for \$199

Veritas Prep Reviews

Kudos [?]: 16884 [69], given: 230

Manager
Joined: 29 Sep 2008
Posts: 142

Kudos [?]: 125 [13], given: 1

### Show Tags

22 Oct 2010, 11:45
13
KUDOS
18
This post was
BOOKMARKED
if = sign is included with < then <= will be there in solution
like for (x+2)(x-1)(x-7)(x-4) <=0 the solution will be -2 <= x <= 1 or 4<= x <= 7

in case when factors are divided then the numerator will contain = sign

like for (x + 2)(x - 1)/(x -4)(x - 7) < =0
the solution will be -2 <= x <= 1 or 4< x < 7
we cant make 4<=x<=7 as it will make the solution infinite

correct me if i am wrong

Kudos [?]: 125 [13], given: 1

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7608

Kudos [?]: 16884 [4], given: 230

Location: Pune, India

### Show Tags

22 Oct 2010, 12:46
4
KUDOS
Expert's post
1
This post was
BOOKMARKED
mrinal2100: Kudos to you for excellent thinking!
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for \$199

Veritas Prep Reviews

Kudos [?]: 16884 [4], given: 230

Director
Joined: 21 Dec 2010
Posts: 626

Kudos [?]: 270 [0], given: 51

### Show Tags

31 Jan 2011, 03:28
this is good thinking , i will use it in solving problems.
_________________

What is of supreme importance in war is to attack the enemy's strategy.

Kudos [?]: 270 [0], given: 51

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7608

Kudos [?]: 16884 [0], given: 230

Location: Pune, India

### Show Tags

11 Mar 2011, 05:40
Expert's post
1
This post was
BOOKMARKED
vjsharma25 wrote:

How you have decided on the first sign of the graph?Why it is -ve if it has three factors and +ve when four factors?

Check out my post above for explanation.
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for \$199

Veritas Prep Reviews

Kudos [?]: 16884 [0], given: 230

Manager
Joined: 10 Nov 2010
Posts: 158

Kudos [?]: 325 [0], given: 6

### Show Tags

11 Mar 2011, 06:29
VeritasPrepKarishma wrote:
vjsharma25 wrote:

How you have decided on the first sign of the graph?Why it is -ve if it has three factors and +ve when four factors?

Check out my post above for explanation.

I understand the concept but not the starting point of the graph.How you decide about the graph to be a sine or cosine waveform?Meaning graph starts from the +ve Y-axis for four values and starts from -ve Y-axis for three values.
What if the equation you mentioned is (x+2)(x-1)(x-7)<0,will the last two ranges be excluded or the graph will also change?

Kudos [?]: 325 [0], given: 6

Math Forum Moderator
Joined: 20 Dec 2010
Posts: 1970

Kudos [?]: 2004 [20], given: 376

### Show Tags

11 Mar 2011, 06:49
20
KUDOS
11
This post was
BOOKMARKED
vjsharma25 wrote:
VeritasPrepKarishma wrote:
vjsharma25 wrote:

How you have decided on the first sign of the graph?Why it is -ve if it has three factors and +ve when four factors?

Check out my post above for explanation.

I understand the concept but not the starting point of the graph.How you decide about the graph to be a sine or cosine waveform?Meaning graph starts from the +ve Y-axis for four values and starts from -ve Y-axis for three values.
What if the equation you mentioned is (x+2)(x-1)(x-7)<0,will the last two ranges be excluded or the graph will also change?

I always struggle with this as well!!!

There is a trick Bunuel suggested;

(x+2)(x-1)(x-7) < 0

Here the roots are; -2,1,7
Arrange them in ascending order;

-2,1,7; These are three points where the wave will alternate.

The ranges are;
x<-2
-2<x<1
1<x<7
x>7

Take a big value of x; say 1000; you see the inequality will be positive for that.
(1000+2)(1000-1)(1000-7) is +ve. Thus the last range(x>7) is on the positive side.

Graph is +ve after 7.
Between 1 and 7-> -ve
between -2 and 1-> +ve
Before -2 -> -ve

Since the inequality has the less than sign; consider only the -ve side of the graph;

1<x<7 or x<-2 is the complete range of x that satisfies the inequality.
_________________

Kudos [?]: 2004 [20], given: 376

Director
Status: Impossible is not a fact. It's an opinion. It's a dare. Impossible is nothing.
Affiliations: University of Chicago Booth School of Business
Joined: 03 Feb 2011
Posts: 872

Kudos [?]: 389 [0], given: 123

### Show Tags

11 Mar 2011, 07:37
1
This post was
BOOKMARKED
Once algebra teacher told me - signs alternate between the roots. I said whatever and now I know why Watching this article is a stroll down the memory lane.

Kudos [?]: 389 [0], given: 123

Re: Inequalities trick   [#permalink] 11 Mar 2011, 07:37

Go to page    1   2   3   4   5   6   7   8    Next  [ 146 posts ]

Similar topics Replies Last post
Similar
Topics:
an inequality 1 04 Oct 2010, 09:27
2 Inequality 4 29 Jan 2012, 22:18
31 Dec 1969, 17:00
31 Dec 1969, 17:00
198 Tips and Tricks: Inequalities 29 14 Jun 2017, 07:34
Display posts from previous: Sort by