NEW HERE? LEARN MORE
closeclose
Last visit was: 07 May 2024, 23:22 It is currently 07 May 2024, 23:22
Decision Tracker
My Rewards
New posts
New comers' posts

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

Help With Add/Subtract/Mult/Divid Multiple Inequalities

SORT BY:
Date
Kudos
Tags:
Find Similar Topics 
Show Tags
Hide Tags
User avatar
amargius
Intern
Intern
Joined: 08 Jun 2013
Posts: 3
Own Kudos [?]: 27 [27]
Given Kudos: 0
Send PM
Help With Add/Subtract/Mult/Divid Multiple Inequalities [#permalink] New post   02 Jul 2013, 22:08
27
Bookmarks
Trying to remember the rules. Anyone know a good comprehensive place to find the rules?

I remember that you can add them so long as you keep the inequality sign the same way, but can't remember the other rules.
Most Helpful Reply
User avatar
L
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 93086
Own Kudos [?]: 622052 [148]
Given Kudos: 81798
Send PM
CAT Tests Forum Quiz Mode
Help With Add/Subtract/Mult/Divid Multiple Inequalities [#permalink] New post   02 Jul 2013, 22:27
27
Kudos
120
Bookmarks
Expert Reply
amargius wrote:
Trying to remember the rules. Anyone know a good comprehensive place to find the rules?

I remember that you can add them so long as you keep the inequality sign the same way, but can't remember the other rules.


ADDING/SUBTRACTING INEQUALITIES:


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 \(2<5\) --> \(3+2<4+5\).

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:


A. We can raise both parts of an inequality to a positive 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 can not square both sides and write \(x^2>y^2\) if we are not certain that both \(x\) and \(y\) are non-negative.

B. We can always raise both parts of an inequality to a positive 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 \(-5<1\) --> \(-5^2=-125<1=1^3\);
\(x<y\) --> we can raise both sides to third power and write: \(x^3<y^3\).

THEORY ON INEQUALITIES:


https://gmatclub.com/forum/x2-4x-94661.html#p731476
https://gmatclub.com/forum/inequalities-trick-91482.html
https://gmatclub.com/forum/data-suff-ine ... 09078.html
https://gmatclub.com/forum/range-for-var ... 09468.html
https://gmatclub.com/forum/everything-is ... 08884.html
https://gmatclub.com/forum/graphic-appro ... 68037.html
https://gmatclub.com/forum/inequations-i ... 54664.html
https://gmatclub.com/forum/inequations-i ... 54738.html

QUESTIONS:


All DS Inequalities Problems to practice: https://gmatclub.com/forum/search.php?se ... tag_id=184
All PS Inequalities Problems to practice: https://gmatclub.com/forum/search.php?se ... tag_id=189

700+ Inequalities problems: https://gmatclub.com/forum/inequality-an ... 86939.html

Hope it helps.
User avatar
L
KarishmaB
Tutor
Joined: 16 Oct 2010
Posts: 14883
Own Kudos [?]: 65164 [17]
Given Kudos: 431
Location: Pune, India
Send PM
Forum Quiz Mode
Re: Multiplying Two Inequalities [#permalink] New post   03 Jul 2013, 20:39
7
Kudos
10
Bookmarks
Expert Reply
alphabeta1234 wrote:
Hi,

Is there any general rule for multiplication or division of two inequalities?

If
a<x<b and c<y<d
then
a+c<x+y<b+d
a-d<x-y<b-c

Now is there any rule to find the relationships for xy or x/y?

Thank you in advance!



Multiplication - Yes but very constrained.
If both sides of both inequalities are positive and the inequalities have the same sign, you can multiply them.

x < a
y < b
xy < ab
Given x, y, a, b are all positive.

Otherwise
-2 < -1
10 < 30
Multiply: -20 < -30 (Not correct)
or
-2 < 7
-8 < 1
Multiply: 16 < 7 (Not correct)

For division, this may not hold.
e.g
2 < 10
4 < 40
Divide: 1/2 < 1/4

But if both sides of both the inequalities are positive and the signs of the inequality are opposite, then you can divide them

x < a
y > b

x/y < a/b (given all x, y, a, b are positive)
The final inequality takes the sign of the numerator. Take examples.
General Discussion
User avatar
alphabeta1234
Manager
Manager
Joined: 12 Feb 2012
Posts: 105
Own Kudos [?]: 218 [0]
Given Kudos: 28
Send PM
Multiplying Two Inequalities [#permalink] New post   03 Jul 2013, 15:19
Hi,

Is there any general rule for multiplication or division of two inequalities?

If
a<x<b and c<y<d
then
a+c<x+y<b+d
a-d<x-y<b-c

Now is there any rule to find the relationships for xy or x/y?

Thank you in advance!
User avatar
L
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 93086
Own Kudos [?]: 622052 [0]
Given Kudos: 81798
Send PM
CAT Tests Forum Quiz Mode
Re: Help With Add/Subtract/Mult/Divid Multiple Inequalities [#permalink] New post   03 Jul 2013, 23:24
Expert Reply
_____________
Merging topics.
User avatar
S
Supermaverick
Intern
Intern
Joined: 16 Apr 2017
Posts: 35
Own Kudos [?]: 138 [0]
Given Kudos: 82
Send PM
Help With Add/Subtract/Mult/Divid Multiple Inequalities [#permalink] New post   09 Jul 2017, 03:29
VeritasPrepKarishma wrote:
alphabeta1234 wrote:
Hi,

Is there any general rule for multiplication or division of two inequalities?

If
a<x<b and c<y<d
then
a+c<x+y<b+d
a-d<x-y<b-c

Now is there any rule to find the relationships for xy or x/y?

Thank you in advance!



Multiplication - Yes but very constrained.
If both sides of both inequalities are positive and the inequalities have the same sign, you can multiply them.

x < a
y < b
xy < ab
Given x, y, a, b are all positive.

Otherwise
-2 < -1
10 < 30
Multiply: -20 < -30 (Not correct)
or
-2 < 7
-8 < 1
Multiply: 16 < 7 (Not correct)

For division, this may not hold.
e.g
2 < 10
4 < 40
Divide: 1/2 < 1/4

But if both sides of both the inequalities are positive and the signs of the inequality are opposite, then you can divide them

x < a
y > b

x/y < a/b (given all x, y, a, b are positive)
The final inequality takes the sign of the numerator. Take examples.



We can divide two negative inequalities as well provided that all variables have negative signs:
Given : x ,y, a and b are all negative

x < a
y > b

x/y > a/b ( always)

The final inequality takes the sign of the DENOMINATOR. Take examples
User avatar
L
IanStewart
GMAT Tutor
Joined: 24 Jun 2008
Posts: 4129
Own Kudos [?]: 9266 [1]
Given Kudos: 91
 Q51  V47
Send PM
Re: Help With Add/Subtract/Mult/Divid Multiple Inequalities [#permalink] New post   16 Oct 2020, 07:43
1
Kudos
Expert Reply
Bunuel wrote:

RAISING INEQUALITIES TO EVEN/ODD POWER:


A. 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: \(x^2<y^2\);


I generally don't recommend memorizing the more obscure inequality rules, since they're very easy to misapply. It's usually best just to think conceptually about unusual inequality situations, because if you aren't aware of all of the exceptions, you'll very likely make a mistake.

For these rules about powers, we don't only need to know whether our bases are positive or negative. We also need to know if the exponent is positive or negative. So what I quote above is not strictly correct. It's true, for example, that 3 > 2. If we raise both sides to a positive integer power, we'll get something true; 3^2 > 2^2, for example. But if we raise both sides to the even power 0, what we get is no longer true: 3^0 = 1 is not greater than 2^0 = 1. Similarly, if we raise both sides to a negative power, we'll get something untrue: 3^(-2) = 1/9 is not greater than 2^(-2) = 1/4.

But as I said above, I don't suggest studying inequalities by learning all of these 'rules' and their various exceptions.
avatar
yppaul
Intern
Intern
Joined: 25 Jul 2020
Posts: 5
Own Kudos [?]: 0 [0]
Given Kudos: 1
Send PM
Re: Help With Add/Subtract/Mult/Divid Multiple Inequalities [#permalink] New post   17 Oct 2020, 02:45
I always go to the GBAT Club chat sheets download and keep those handy :)
User avatar
bumpbot
Non-Human User
Joined: 09 Sep 2013
Posts: 32828
Own Kudos [?]: 827 [0]
Given Kudos: 0
Send PM
Re: Help With Add/Subtract/Mult/Divid Multiple Inequalities [#permalink] New post   26 Oct 2023, 11:29
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 Club Bot
gmatclubot
Re: Help With Add/Subtract/Mult/Divid Multiple Inequalities [#permalink]
26 Oct 2023, 11:29
Moderator:
Bunuel
Math Expert
93086 posts

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne
Main navigation
GMAT Resources
Partners
MBA Resources
www.gmac.com|www.mba.com | | GMAT Club Rules | Terms and Conditions