Find all School-related info fast with the new School-Specific MBA Forum

It is currently 30 Aug 2014, 06:30

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.

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

Inequalities trick

  Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:
38 KUDOS received
CEO
CEO
User avatar
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2793
Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35
Followers: 177

Kudos [?]: 931 [38] , given: 235

GMAT Tests User Reviews Badge
Inequalities trick [#permalink] New post 16 Mar 2010, 09:11
38
This post received
KUDOS
61
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
1.jpg [ 6.73 KiB | Viewed 21268 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

:thanks Support GMAT Club by putting a GMAT Club badge on your blog/Facebook :thanks

Get the best GMAT Prep Resources with GMAT Club Premium Membership

Gmat test review :
670-to-710-a-long-journey-without-destination-still-happy-141642.html

Expert Post
36 KUDOS received
Veritas Prep GMAT Instructor
User avatar
Joined: 16 Oct 2010
Posts: 4688
Location: Pune, India
Followers: 1084

Kudos [?]: 4871 [36] , given: 163

Re: Inequalities trick [#permalink] New post 22 Oct 2010, 05:33
36
This post received
KUDOS
Expert's post
13
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
doc.jpg [ 7.9 KiB | Viewed 20580 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

Save $100 on Veritas Prep GMAT Courses And Admissions Consulting
Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options.

Veritas Prep Reviews

11 KUDOS received
Math Forum Moderator
avatar
Joined: 20 Dec 2010
Posts: 2047
Followers: 128

Kudos [?]: 904 [11] , given: 376

GMAT Tests User
Re: Inequalities trick [#permalink] New post 11 Mar 2011, 05:49
11
This post received
KUDOS
4
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.
_________________

~fluke

Get the best GMAT Prep Resources with GMAT Club Premium Membership

8 KUDOS received
Manager
Manager
avatar
Joined: 29 Sep 2008
Posts: 154
Followers: 2

Kudos [?]: 30 [8] , given: 1

GMAT Tests User
Re: Inequalities trick [#permalink] New post 22 Oct 2010, 10:45
8
This post received
KUDOS
3
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
4 KUDOS received
Manager
Manager
User avatar
Status: On...
Joined: 16 Jan 2011
Posts: 189
Followers: 3

Kudos [?]: 35 [4] , given: 62

GMAT Tests User
Re: Inequalities trick [#permalink] New post 10 Aug 2011, 16:01
4
This post received
KUDOS
2
This post was
BOOKMARKED
WoW - This is a cool thread with so many thing on inequalities....I have compiled it together with some of my own ideas...It should help.

1) CORE CONCEPT
@gurpreetsingh -
Suppose you have the inequality

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

Arrange the NUMBERS in ascending order from left to right. a<b<c<d
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.

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.

Note: Make sure that the factors are of the form (ax - b), not (b - ax)...

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

Convert this to: (x+2)(x-1)(x-7)>0 (Multiply both sides by '-1')
Now solve in the usual way. Assign '+' to the rightmost region and then alternate with '-'
Since you are looking for positive value of the expression, every region where you put a '+' will be the region where the expression will be greater than 0.

2) Variation - ODD/EVEN POWER
@ulm/Karishma -
if we have even powers like (x-a)^2(x-b)
we don't need to change a sign when jump over "a".
This will be same as (x-b)


We can ignore squares BUT SHOULD consider ODD powers
example -
2.a
(x-a)^3(x-b)<0 is the same as (x-a)(x-b) <0
2.b
(x - a)(x - b)/(x - c)(x - d) < 0 ==> (x - a)(x - b)(x-c)^-1(x-d)^-1 <0
is the same as (x - a)(x - b)(x - c)(x - d) < 0

3) Variation <= in FRACTION
@mrinal2100 -
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

BUT if it is a fraction the denominator in the solution will not have = SIGN

example -
3.a
(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

4) Variation - ROOTS
@Karishma -
As for roots, you have to keep in mind that given \sqrt{x}, x cannot be negative.

\sqrt{x} < 10
implies 0 < \sqrt{x} < 10
Squaring, 0 < x < 100
Root questions are specific. You have to be careful. If you have a particular question in mind, send it.

Refer - inequalities-and-roots-118619.html#p959939
Some more useful tips for ROOTS....I am too lazy to consolidate

<5> THESIS -
@gmat1220 -
Once algebra teacher told me - signs alternate between the roots. I said whatever and now I know why :o Watching this article is a stroll down the memory lane.

I will save this future references....
Please add anything that you feel will help.

Anyone wants to add ABSOLUTE VALUES....That will be a value add to this post

_________________

Labor cost for typing this post >= Labor cost for pushing the Kudos Button
kudos-what-are-they-and-why-we-have-them-94812.html

Expert Post
3 KUDOS received
Veritas Prep GMAT Instructor
User avatar
Joined: 16 Oct 2010
Posts: 4688
Location: Pune, India
Followers: 1084

Kudos [?]: 4871 [3] , given: 163

Re: Inequalities trick [#permalink] New post 11 Mar 2011, 18:57
3
This post received
KUDOS
Expert's post
vjsharma25 wrote:

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?


Ok, look at this expression inequality: (x+2)(x-1)(x-7) < 0
Can I say the left hand side expression will always be positive for values greater than 7? (x+2) will be positive, (x - 1) will be positive and (x-7) will also be positive... so in the rightmost regions i.e. x > 7, all three factors will be positive. The expression will be positive when x > 7, it will be negative when 1 < x < 7, positive when -2 , x < 1 and negative when x < -2. We need the region where the expression is less than 0 i.e. negative. So either 1 < x < 7 or x < -2.

Now let me add another factor: (x+8)(x+2)(x-1)(x-7)
Can I still say that the entire expression is positive in the rightmost region i.e. x>7 because each one of the four factors is positive? Yes.

So basically, your rightmost region is always positive. You go from there and assign + and - signs to the regions. Your starting point is the rightmost region.

Note: Make sure that the factors are of the form (ax - b), not (b - ax)...

e.g. (x+2)(x-1)(7 - x)<0

Convert this to: (x+2)(x-1)(x-7)>0 (Multiply both sides by '-1')
Now solve in the usual way. Assign '+' to the rightmost region and then alternate with '-'
Since you are looking for positive value of the expression, every region where you put a '+' will be the region where the expression will be greater than 0.
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Save $100 on Veritas Prep GMAT Courses And Admissions Consulting
Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options.

Veritas Prep Reviews

Expert Post
2 KUDOS received
Veritas Prep GMAT Instructor
User avatar
Joined: 16 Oct 2010
Posts: 4688
Location: Pune, India
Followers: 1084

Kudos [?]: 4871 [2] , given: 163

Re: Inequalities trick [#permalink] New post 09 Sep 2013, 22:35
2
This post received
KUDOS
Expert's post
karannanda 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.

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.




Hi Gurpreet,
Thanks for the wonderful method.
I am trying to understand it so that i can apply it in tests.
Can you help me in applying this method to the below expression to find range of x.
x^3 – 4x^5 < 0?

I am getting the roots as -1/2, 0, 1/2 and when i plot them using this method, putting + in the rightmost region, I am not getting correct result. Not sure where i am going wrong. Can you pls help.


Before you apply the method, ensure that the factors are of the form (x - a)(x - b) etc

x^3 - 4x^5 < 0

x^3 ( 1 - 4x^2) < 0

x^3(1 - 2x) (1 + 2x) < 0

4x^3(x - 1/2)(x + 1/2) > 0 (Notice the flipped sign. We multiplied both sides by -1 to convert 1/2 - x to x - 1/2)

Now the transition points are 0, -1/2 and 1/2 so put + in the rightmost region.
The solution will be x > 1/2 or -1/2 < x< 0.

Check out these posts discussing such complications:
http://www.veritasprep.com/blog/2012/06 ... e-factors/
http://www.veritasprep.com/blog/2012/07 ... ns-part-i/
http://www.veritasprep.com/blog/2012/07 ... s-part-ii/
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Save $100 on Veritas Prep GMAT Courses And Admissions Consulting
Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options.

Veritas Prep Reviews

1 KUDOS received
Senior Manager
Senior Manager
avatar
Joined: 16 Feb 2012
Posts: 259
Concentration: Finance, Economics
Followers: 4

Kudos [?]: 58 [1] , given: 107

GMAT ToolKit User
Re: Inequalities trick [#permalink] New post 22 Jul 2012, 02:03
1
This post received
KUDOS
VeritasPrepKarishma wrote:
mrinal2100: Kudos to you for excellent thinking!


Correct me if I'm wrong.
If the lower part of the equation\frac {(x+2)(x-1)}{(x-4)(x-7)} were 4\leq x \leq 7, than the lower part would be equal to zero,thus making it impossible to calculate the whole equation.
_________________

Kudos if you like the post!

Failing to plan is planning to fail.

Expert Post
1 KUDOS received
Veritas Prep GMAT Instructor
User avatar
Joined: 16 Oct 2010
Posts: 4688
Location: Pune, India
Followers: 1084

Kudos [?]: 4871 [1] , given: 163

Re: Inequalities trick [#permalink] New post 23 Jul 2012, 02:13
1
This post received
KUDOS
Expert's post
Stiv wrote:
VeritasPrepKarishma wrote:
mrinal2100: Kudos to you for excellent thinking!


Correct me if I'm wrong.
If the lower part of the equation\frac {(x+2)(x-1)}{(x-4)(x-7)} were 4\leq x \leq 7, than the lower part would be equal to zero,thus making it impossible to calculate the whole equation.


x cannot be equal to 4 or 7 because if x = 4 or x = 7, the denominator will be 0 and the expression will not be defined.
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Save $100 on Veritas Prep GMAT Courses And Admissions Consulting
Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options.

Veritas Prep Reviews

1 KUDOS received
CEO
CEO
User avatar
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2793
Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35
Followers: 177

Kudos [?]: 931 [1] , given: 235

GMAT Tests User Reviews Badge
Re: Inequalities trick [#permalink] New post 18 Oct 2012, 04:17
1
This post received
KUDOS
GMATBaumgartner wrote:
gurpreetsingh wrote:
ulm wrote:
in addition:
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



Hi Gurpreet,
Could you elaborate what exactly you meant here in highlighted text ?

Even I have a doubt as to how this can be applied for powers of the same term . like the example mentioned in the post above.


If the powers are even then the inequality won't be affected.

eg if u have to find the range of values of x satisfying (x-a)^2 *(x-b)(x-c) >0

just use (x-b)*(x-c) >0 because x-a raised to the power 2 will not affect the inequality sign. But just make sure x=a is taken care off , as it would make the inequality zero.
_________________

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

:thanks Support GMAT Club by putting a GMAT Club badge on your blog/Facebook :thanks

Get the best GMAT Prep Resources with GMAT Club Premium Membership

Gmat test review :
670-to-710-a-long-journey-without-destination-still-happy-141642.html

Expert Post
1 KUDOS received
Veritas Prep GMAT Instructor
User avatar
Joined: 16 Oct 2010
Posts: 4688
Location: Pune, India
Followers: 1084

Kudos [?]: 4871 [1] , given: 163

Re: Inequalities trick [#permalink] New post 18 Oct 2012, 09:27
1
This post received
KUDOS
Expert's post
GMATBaumgartner wrote:
Hi Gurpreet,
Could you elaborate what exactly you meant here in highlighted text ?

Even I have a doubt as to how this can be applied for powers of the same term . like the example mentioned in the post above.


In addition, you can check out this post:

http://www.veritasprep.com/blog/2012/07 ... s-part-ii/

I have discussed how to handle powers in it.
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Save $100 on Veritas Prep GMAT Courses And Admissions Consulting
Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options.

Veritas Prep Reviews

Expert Post
1 KUDOS received
Math Expert
User avatar
Joined: 02 Sep 2009
Posts: 25207
Followers: 3419

Kudos [?]: 25067 [1] , given: 2702

Re: Inequalities trick [#permalink] New post 02 Dec 2012, 06:25
1
This post received
KUDOS
Expert's post
GMATGURU1 wrote:
Going ahead in this inequalities area of GMAT, can some one has the problem numbers of inequalities in OG 11 and OG 12?

Cheers,
Danny


Search for hundreds of question with solutions by tags: viewforumtags.php

DS questions on inequalities: search.php?search_id=tag&tag_id=184
PS questions on inequalities: search.php?search_id=tag&tag_id=189
Hardest DS inequality questions with detailed solutions: inequality-and-absolute-value-questions-from-my-collection-86939.html

Hope it helps.
_________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
25 extra-hard Quant Tests

Get the best GMAT Prep Resources with GMAT Club Premium Membership

1 KUDOS received
CEO
CEO
User avatar
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2793
Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35
Followers: 177

Kudos [?]: 931 [1] , given: 235

GMAT Tests User Reviews Badge
Re: Inequalities trick [#permalink] New post 20 Dec 2012, 20:04
1
This post received
KUDOS
VeritasPrepKarishma wrote:
The entire concept is based on positive/negative factors which means <0 or >0 is a must. If the question is not in this format, you need to bring it to this format by taking the constant to the left hand side.

e.g.
(x + 2)(x + 3) < 2
x^2 + 5x + 6 - 2 < 0
x^2 + 5x + 4 < 0
(x+4)(x+1) < 0

Now use the concept.


Yes this is probable but it might not be possible always to group them. So in case you are unsure just follow the number plugging approach. But most of the times this trick would be very handy.
_________________

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

:thanks Support GMAT Club by putting a GMAT Club badge on your blog/Facebook :thanks

Get the best GMAT Prep Resources with GMAT Club Premium Membership

Gmat test review :
670-to-710-a-long-journey-without-destination-still-happy-141642.html

Expert Post
1 KUDOS received
Math Expert
User avatar
Joined: 02 Sep 2009
Posts: 25207
Followers: 3419

Kudos [?]: 25067 [1] , given: 2702

Re: Inequalities trick [#permalink] New post 11 Nov 2013, 06:51
1
This post received
KUDOS
Expert's post
anujpadia wrote:
Can you please explain the above mentioned concept in relation to the following question?
Is a > O?
(1) a^3 - 0 < 0
(2) 1- a^2 > 0

Can you please explain the scenario when (x-a)(x-B)(x-C)(x-d)>0?
Sorry, but finding it difficult to understand.


Check alternative solutions here: is-a-0-1-a-3-a-0-2-1-a-86749.html

Hope this helps.
_________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
25 extra-hard Quant Tests

Get the best GMAT Prep Resources with GMAT Club Premium Membership

Expert Post
1 KUDOS received
Veritas Prep GMAT Instructor
User avatar
Joined: 16 Oct 2010
Posts: 4688
Location: Pune, India
Followers: 1084

Kudos [?]: 4871 [1] , given: 163

Re: Inequalities trick [#permalink] New post 23 Apr 2014, 03:43
1
This post received
KUDOS
Expert's post
1
This post was
BOOKMARKED
PathFinder007 wrote:

I have a query. I have following question

x^3 - 4x^5 < 0

I can define this as (1+2x).x^3(1-2x). now I have roots -1/2, 0, 1/2. so in case of >1/2 I will always get inequality value as <0 and in case of -1/2 and 0 I will get value as 0.

So How I will define them in graph and what range I will consider for this inequality.

Thanks



The factors must be of the form (x - a)(x - b) .... etc

x^3 - 4x^5 < 0
x^3 * (1 - 4x^2) < 0
x^3 * (1 - 2x) * (1 + 2x) < 0
x^3 * (2x - 1) * (2x + 1) > 0 (Note the sign flip because 1-2x was changed to 2x - 1)
x^3 * 2(x - 1/2) *2(x + 1/2) > 0

So transition points are 0, 1/2 and -1/2.

____________ - 1/2 _____ 0 ______1/2 _________

This is what it looks like on the number line.
The rightmost region is positive. We want the positive regions in the inequality.

So the desired range of x is given by x > 1/2 or -1/2 < x< 0

For more on this method, check these posts:

http://www.veritasprep.com/blog/2012/06 ... e-factors/
http://www.veritasprep.com/blog/2012/07 ... ns-part-i/
http://www.veritasprep.com/blog/2012/07 ... s-part-ii/

The links will give you the theory behind this method in detail.
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Save $100 on Veritas Prep GMAT Courses And Admissions Consulting
Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options.

Veritas Prep Reviews

Intern
Intern
avatar
Joined: 08 Mar 2010
Posts: 6
Followers: 0

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

Re: Inequalities trick [#permalink] New post 16 Mar 2010, 09:46
Can u plz explainn the backgoround of this & then the explanation.

Thanks
Senior Manager
Senior Manager
User avatar
Joined: 13 Dec 2009
Posts: 264
Followers: 10

Kudos [?]: 105 [0], given: 13

GMAT Tests User Reviews Badge
Re: Inequalities trick [#permalink] New post 19 Mar 2010, 11:48
I have applied this trick and it seemed to be quite useful.
_________________

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

CEO
CEO
User avatar
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2793
Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35
Followers: 177

Kudos [?]: 931 [0], given: 235

GMAT Tests User Reviews Badge
Re: Inequalities trick [#permalink] New post 19 Mar 2010, 11:59
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

:thanks Support GMAT Club by putting a GMAT Club badge on your blog/Facebook :thanks

Get the best GMAT Prep Resources with GMAT Club Premium Membership

Gmat test review :
670-to-710-a-long-journey-without-destination-still-happy-141642.html

Senior Manager
Senior Manager
User avatar
Status: Upset about the verbal score - SC, CR and RC are going to be my friend
Joined: 30 Jun 2010
Posts: 318
Followers: 5

Kudos [?]: 15 [0], given: 6

GMAT Tests User
Re: Inequalities trick [#permalink] New post 17 Oct 2010, 15:14
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

CEO
CEO
User avatar
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2793
Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35
Followers: 177

Kudos [?]: 931 [0], given: 235

GMAT Tests User Reviews Badge
Re: Inequalities trick [#permalink] New post 17 Oct 2010, 16:19
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

:thanks Support GMAT Club by putting a GMAT Club badge on your blog/Facebook :thanks

Get the best GMAT Prep Resources with GMAT Club Premium Membership

Gmat test review :
670-to-710-a-long-journey-without-destination-still-happy-141642.html

Re: Inequalities trick   [#permalink] 17 Oct 2010, 16:19
    Similar topics Author Replies Last post
Similar
Topics:
83 Experts publish their posts in the topic Tips and Tricks: Inequalities Zarrolou 26 14 Apr 2013, 08:20
2 To use tricks or not to use tricks..... Victor81 3 20 Apr 2008, 15:57
inequalities ugo_castelo 1 06 Aug 2006, 07:23
inequalities mand-y 3 05 Dec 2005, 11:18
inequalities mandy 2 15 Jun 2005, 06:47
Display posts from previous: Sort by

Inequalities trick

  Question banks Downloads My Bookmarks Reviews Important topics  

Go to page    1   2   3   4    Next  [ 76 posts ] 



GMAT Club MBA Forum Home| About| Privacy Policy| Terms and Conditions| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group and phpBB SEO

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®.