It is currently 21 Nov 2017, 21:10

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

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

Inequalities: Tips and hints

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Expert Post
11 KUDOS received
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 42287

Kudos [?]: 132984 [11], given: 12392

Inequalities: Tips and hints [#permalink]

Show Tags

New post 02 Jul 2014, 04:33
11
This post received
KUDOS
Expert's post
101
This post was
BOOKMARKED

Inequalities: Tips and hints



!
This post is a part of the Quant Tips and Hints by Topic Directory focusing on Quant topics and providing examples of how to approach them. Most of the questions are above average difficulty.

ADDING/SUBTRACTING INEQUALITIES

1. You can only add inequalities when their signs are in the same direction:

If \(a>b\) and \(c>d\) (signs in same direction: \(>\) and \(>\)) --> \(a+c>b+d\).
Example: \(3<4\) and \(2<5\) --> \(3+2<4+5\).

2. You can only apply subtraction when their signs are in the opposite directions:

If \(a>b\) and \(c<d\) (signs in opposite direction: \(>\) and \(<\)) --> \(a-c>b-d\) (take the sign of the inequality you subtract from).
Example: \(3<4\) and \(5>1\) --> \(3-5<4-1\).

RAISING INEQUALITIES TO EVEN/ODD POWER

1. We can raise both parts of an inequality to an even power if we know that both parts of an inequality are non-negative (the same for taking an even root of both sides of an inequality).
For example:
\(2<4\) --> we can square both sides and write: \(2^2<4^2\);
\(0\leq{x}<{y}\) --> we can square both sides and write: \(x^2<y^2\);

But if either of side is negative then raising to even power doesn't always work.
For example: \(1>-2\) if we square we'll get \(1>4\) which is not right. So if given that \(x>y\) then we cannot square both sides and write \(x^2>y^2\) if we are not certain that both \(x\) and \(y\) are non-negative.

2. We can always raise both parts of an inequality to an odd power (the same for taking an odd root of both sides of an inequality).
For example:
\(-2<-1\) --> we can raise both sides to third power and write: \(-2^3=-8<-1=-1^3\) or \(-5<1\) --> \(-5^3=-125<1=1^3\);
\(x<y\) --> we can raise both sides to third power and write: \(x^3<y^3\).

MULTIPLYING/DIVIDING TWO INEQUALITIES

1. If both sides of both inequalities are positive and the inequalities have the same sign, you can multiply them.
For example, for positive \(x\), \(y\), \(a\), \(b\), if \(x < a\) and \(y < b\), then \(xy < ab\).

2. If both sides of both inequalities are positive and the signs of the inequality are opposite, then you can divide them.
For example, for positive \(x\), \(y\), \(a\), \(b\), if \(x < a\) and \(y > b\), then \(\frac{x}{y} < \frac{a}{b}\) (The final inequality takes the sign of the numerator).

MULTIPLYING/DIVIDING AN INEQUALITY BY A NUMBER

1. Whenever you multiply or divide an inequality by a positive number, you must keep the inequality sign.
2. Whenever you multiply or divide an inequality by a negative number, you must flip the inequality sign.
3. Never multiply (or reduce) an inequality by a variable (or the expression with a variable) if you don't know the sign of it or are not certain that variable (or the expression with a variable) doesn't equal to zero.

SOLVING QUADRATIC INEQUALITIES: GRAPHIC APPROACH

Say we need to find the ranges of \(x\) for \(x^2-4x+3<0\). \(x^2-4x+3=0\) is the graph of a parabola and it look likes this:
Image

Intersection points are the roots of the equation \(x^2-4x+3=0\), which are \(x_1=1\) and \(x_2=3\). "<" sign means in which range of \(x\) the graph is below x-axis. Answer is \(1<x<3\) (between the roots).

If the sign were ">": \(x^2-4x+3>0\). First find the roots (\(x_1=1\) and \(x_2=3\)). ">" sign means in which range of \(x\) the graph is above x-axis. Answer is \(x<1\) and \(x>3\) (to the left of the smaller root and to the right of the bigger root).


This approach works for any quadratic inequality. For example: \(-x^2-x+12>0\), first rewrite this as \(x^2+x-12<0\) (so that the coefficient of x^2 to be positive. It's possible to solve without rewriting, but easier to master one specific pattern).

\(x^2+x-12<0\). Roots are \(x_1=-4\) and \(x_1=3\) --> below ("<") the x-axis is the range for \(-4<x<3\) (between the roots).

Again if it were \(x^2+x-12>0\), then the answer would be \(x<-4\) and \(x>3\) (to the left of the smaller root and to the right of the bigger root).



Please share your Inequality properties tips below and get kudos point. Thank you.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

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?
Extra-hard Quant Tests with Brilliant Analytics

Kudos [?]: 132984 [11], given: 12392

Current Student
User avatar
Joined: 11 Oct 2013
Posts: 121

Kudos [?]: 67 [0], given: 137

Concentration: Marketing, General Management
GMAT 1: 600 Q41 V31
GMAT ToolKit User
Re: Inequalities: Tips and hints [#permalink]

Show Tags

New post 14 Dec 2015, 07:38
The graphical approach is awesome! Changes the way you look at the question. You can easily manage the signs just by looking at the equation!
Thanks Bunuel!
_________________

Its not over..

Kudos [?]: 67 [0], given: 137

Top Contributor
Senior Manager
Senior Manager
User avatar
Status: Not afraid of failures, disappointments, and falls.
Joined: 20 Jan 2010
Posts: 290

Kudos [?]: 266 [0], given: 260

Concentration: Technology, Entrepreneurship
WE: Operations (Telecommunications)
GMAT ToolKit User Premium Member
Re: Inequalities: Tips and hints [#permalink]

Show Tags

New post 06 Aug 2016, 00:46
Top Contributor
Is this part of GMAT Math Book? And, if not; can it be included in the book?
_________________

"I choose to rise after every fall"
Target=770
http://challengemba.blogspot.com
Kudos??

Kudos [?]: 266 [0], given: 260

Non-Human User
User avatar
Joined: 09 Sep 2013
Posts: 15581

Kudos [?]: 283 [0], given: 0

Premium Member
Re: Inequalities: Tips and hints [#permalink]

Show Tags

New post 13 Aug 2017, 04:38
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

GMAT Books | GMAT Club Tests | Best Prices on GMAT Courses | GMAT Mobile App | Math Resources | Verbal Resources

Kudos [?]: 283 [0], given: 0

Re: Inequalities: Tips and hints   [#permalink] 13 Aug 2017, 04:38
Display posts from previous: Sort by

Inequalities: Tips and hints

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  


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

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®.