[phpBB Debug] PHP Notice: in file /includes/check_new_recommended_questions.php on line 37: Undefined array key "last_recommended_questions_epoch"
[phpBB Debug] PHP Notice: in file /includes/check_new_recommended_questions.php on line 41: Undefined array key "last_recommended_questions_epoch"
Inequalities: Tips and hints : Quantitative Questions
 Last visit was: 20 Jul 2024, 03:40 It is currently 20 Jul 2024, 03:40
Toolkit
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
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

# Inequalities: Tips and hints

SORT BY:
Tags:
Show Tags
Hide Tags
Math Expert
Joined: 02 Sep 2009
Posts: 94428
Own Kudos [?]: 642460 [367]
Given Kudos: 86449
Tutor
Joined: 16 Oct 2010
Posts: 15125
Own Kudos [?]: 66742 [12]
Given Kudos: 436
Location: Pune, India
General Discussion
Manager
Joined: 11 Oct 2013
Posts: 70
Own Kudos [?]: 288 [1]
Given Kudos: 137
Concentration: Marketing, General Management
GMAT 1: 600 Q41 V31
Manager
Joined: 20 Jan 2010
Status:Not afraid of failures, disappointments, and falls.
Posts: 217
Own Kudos [?]: 450 [1]
Given Kudos: 260
Concentration: Technology, Entrepreneurship
WE:Operations (Telecommunications)
Re: Inequalities: Tips and hints [#permalink]
Top Contributor
Is this part of GMAT Math Book? And, if not; can it be included in the book?
IIM School Moderator
Joined: 04 Sep 2016
Posts: 1253
Own Kudos [?]: 1265 [2]
Given Kudos: 1207
Location: India
WE:Engineering (Other)
Re: Inequalities: Tips and hints [#permalink]
1
Kudos
1
Bookmarks
Bunuel VeritasPrepKarishma chetan2u

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

Quote:
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 $$5>1$$ --> $$3-5<4-1$$.

Any alternative way to memorize highlighted text under time crunch other than picking numbers?
RC & DI Moderator
Joined: 02 Aug 2009
Status:Math and DI Expert
Posts: 11476
Own Kudos [?]: 34458 [5]
Given Kudos: 322
Re: Inequalities: Tips and hints [#permalink]
5
Kudos
Bunuel VeritasPrepKarishma chetan2u

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

Quote:
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 $$5>1$$ --> $$3-5<4-1$$.

Any alternative way to memorize highlighted text under time crunch other than picking numbers?

Just remember that you can add INEQUALITIES by adding the terms on same side of INEQUALITY..
So if a>b and c<d...c<d is same as d>c..
So we have a>b and d>c...
Add the same sides of INEQUALITY..
a+d>b+c.......a>b+c-d.....a-c>b-d...
Same as what you are trying to remember about SUBTRACTION
IIM School Moderator
Joined: 04 Sep 2016
Posts: 1253
Own Kudos [?]: 1265 [0]
Given Kudos: 1207
Location: India
WE:Engineering (Other)
Re: Inequalities: Tips and hints [#permalink]
Bunuel chetan2u VeritasPrepKarishma niks18

Let us say, I am given a SINGLE inequality:

a - b > a + b

Given: a and b are integers.

Can I add / subtract an integer with unknown sign (ie positive or negative)
to both sides of inequality WITHOUT knowing existing sign of another variable?

Eg. Here, can I subtract a from both sides, without knowing sign of b?
Math Expert
Joined: 02 Sep 2009
Posts: 94428
Own Kudos [?]: 642460 [1]
Given Kudos: 86449
Re: Inequalities: Tips and hints [#permalink]
1
Kudos
Bunuel chetan2u VeritasPrepKarishma niks18

Let us say, I am given a SINGLE inequality:

a - b > a + b

Given: a and b are integers.

Can I add / subtract an integer with unknown sign (ie positive or negative)
to both sides of inequality WITHOUT knowing existing sign of another variable?

Eg. Here, can I subtract a from both sides, without knowing sign of b?

Yes. We are concerned about the sign of a variable when multiplying/dividing an inequality by it. However we can safely add/subtract a variable from both sides of an inequality regardless of its sign.
Manager
Joined: 17 May 2015
Posts: 199
Own Kudos [?]: 3138 [1]
Given Kudos: 85
Re: Inequalities: Tips and hints [#permalink]
1
Kudos
Bunuel chetan2u VeritasPrepKarishma niks18

Let us say, I am given a SINGLE inequality:

a - b > a + b

Given: a and b are integers.

Can I add / subtract an integer with unknown sign (ie positive or negative)
to both sides of inequality WITHOUT knowing existing sign of another variable?

Eg. Here, can I subtract a from both sides, without knowing sign of b?

Inequality presearves under following operations:

- addition or subtraction of a number from both sides.

- Multiplication or division from both sides by a positive number.

Quote:
Can I add / subtract an integer with unknown sign (ie positive or negative) to both sides of inequality WITHOUT knowing existing sign of another variable?

Yes, we can add or subtract any number (NOT just integer) from both sides without knowing the existing sign.

Now, let's consider example provided by you.
Given inequality,
$$A - B > A + B$$
Assume A = 3 , B = -5. These values will satisfy the above inequality.

Case1: Add a positive value both side i.e. add A both side:

$$2A - B > 2A + B$$ . You can verify that this inequality still holds true.

Case2: Add a negative value both side i.e add B both side:

$$A > A + 2B$$ . Still, the inequality holds true.

I hope this helps.

Thanks.
Intern
Joined: 06 Oct 2019
Posts: 43
Own Kudos [?]: 88 [0]
Given Kudos: 71
Location: India
GMAT 1: 740 Q50 V41
Re: Inequalities: Tips and hints [#permalink]

The post says "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)."

To me, the bold part means : "We can take even root of both parts of an inequality if we know that both parts of the inequality are non-negative"

However, this does not seem to hold true for the below example, can you please clarify?

Let's say : x^2 > y^4 (given)
So according to the above rule (see bold part of the excerpt), since both sides of the inequality are non-negative(as anything raised to even power is non negative), we can say:
x > y^2 (taking square root on both sides of the inequality)
But that's not necessarily true.
Consider the example :
Case 1 : X = 300, Y = 2
Case 2 : X = -300 , Y = 2
In both cases x^2 > y^4, but for case 1 : x > y^2, whereas for case 2 : x < y^2
Math Expert
Joined: 02 Sep 2009
Posts: 94428
Own Kudos [?]: 642460 [1]
Given Kudos: 86449
Re: Inequalities: Tips and hints [#permalink]
1
Kudos
Debo1988 wrote:

The post says "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)."

To me, the bold part means : "We can take even root of both parts of an inequality if we know that both parts of the inequality are non-negative"

However, this does not seem to hold true for the below example, can you please clarify?

Let's say : x^2 > y^4 (given)
So according to the above rule (see bold part of the excerpt), since both sides of the inequality are non-negative(as anything raised to even power is non negative), we can say:
x > y^2 (taking square root on both sides of the inequality)
But that's not necessarily true.
Consider the example :
Case 1 : X = 300, Y = 2
Case 2 : X = -300 , Y = 2
In both cases x^2 > y^4, but for case 1 : x > y^2, whereas for case 2 : x < y^2

The point is if you take the square root from x^2 > y^4, you get |x| > y^2, not x > y^2 (recall that $$\sqrt{x^2}=|x|$$).
Intern
Joined: 14 Jan 2013
Posts: 10
Own Kudos [?]: 8 [0]
Given Kudos: 25
Re: Inequalities: Tips and hints [#permalink]
chetan2u wrote:
Bunuel VeritasPrepKarishma chetan2u

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

Quote:
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 $$5>1$$ --> $$3-5<4-1$$.

Any alternative way to memorize highlighted text under time crunch other than picking numbers?

Just remember that you can add INEQUALITIES by adding the terms on same side of INEQUALITY..
So if a>b and c<d...c<d is same as d>c..
So we have a>b and d>c...
Add the same sides of INEQUALITY..
a+d>b+c.......a>b+c-d.....a-c>b-d...
Same as what you are trying to remember about SUBTRACTION

Hi chetan2u / Bunuel

Does this concept also work in multiplication.

Like highlighted above, we can multiple inequalities only when both sides of both inequalities are positive and the inequalities have the same sign.
Say if the signs are not the same ; can we multiply the inequality with -1 to make the sign same & then multiply ?

Like
if x<a and y>b ; then (-1)y<-b
hence on multiplying : x*(-y) < a(-b) ? will the signs cancel ; nullifying the approach or multiplication of 2 inequalities with opposite signs just can't happen ?
RC & DI Moderator
Joined: 02 Aug 2009
Status:Math and DI Expert
Posts: 11476
Own Kudos [?]: 34458 [0]
Given Kudos: 322
Re: Inequalities: Tips and hints [#permalink]
sheldoncooper wrote:
chetan2u wrote:
Bunuel VeritasPrepKarishma chetan2u

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

Quote:
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 $$5>1$$ --> $$3-5<4-1$$.

Any alternative way to memorize highlighted text under time crunch other than picking numbers?

Just remember that you can add INEQUALITIES by adding the terms on same side of INEQUALITY..
So if a>b and c<d...c<d is same as d>c..
So we have a>b and d>c...
Add the same sides of INEQUALITY..
a+d>b+c.......a>b+c-d.....a-c>b-d...
Same as what you are trying to remember about SUBTRACTION

Hi chetan2u / Bunuel

Does this concept also work in multiplication.

Like highlighted above, we can multiple inequalities only when both sides of both inequalities are positive and the inequalities have the same sign.
Say if the signs are not the same ; can we multiply the inequality with -1 to make the sign same & then multiply ?

Like
if x<a and y>b ; then (-1)y<-b
hence on multiplying : x*(-y) < a(-b) ? will the signs cancel ; nullifying the approach or multiplication of 2 inequalities with opposite signs just can't happen ?

No that will not be correct until you know the value of the variables.
For example.
1<10 and 3>2 or -3<-2......1*-3<-2*10.....NO
3>2 and 1<10 or -1>-10....1*-3>2*-10.....YES

So same numbers but different answers depending on which inequality you are multiplying by -1.
Intern
Joined: 29 Jun 2021
Posts: 1
Own Kudos [?]: 0 [0]
Given Kudos: 0
Location: Taiwan
Re: Inequalities: Tips and hints [#permalink]

I have a question.

MULTIPLYING/DIVIDING AN INEQUALITY BY A NUMBER
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.
⬆️

does "reduce" here mean divide or subtract? I am a little bit confuse.
as I understand, we could not multiply or divide a variable whose sign is unknown.
but we could add or subtract a variable whose sign is unknown without changing the sign of INEQUALITY.
RC & DI Moderator
Joined: 02 Aug 2009
Status:Math and DI Expert
Posts: 11476
Own Kudos [?]: 34458 [0]
Given Kudos: 322
Re: Inequalities: Tips and hints [#permalink]
irene727008 wrote:

I have a question.

MULTIPLYING/DIVIDING AN INEQUALITY BY A NUMBER
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.
⬆️

does "reduce" here mean divide or subtract? I am a little bit confuse.
as I understand, we could not multiply or divide a variable whose sign is unknown.
but we could add or subtract a variable whose sign is unknown without changing the sign of INEQUALITY.

Reduce here means division.
You are correct that you can add or subtract any term to both sides without changing inequality sign.
Non-Human User
Joined: 09 Sep 2013
Posts: 34039
Own Kudos [?]: 853 [0]
Given Kudos: 0
Re: Inequalities: Tips and hints [#permalink]
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.
Re: Inequalities: Tips and hints [#permalink]
Moderator:
Math Expert
94427 posts