GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

 It is currently 23 Jan 2020, 08:05 ### 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

Author Message
TAGS:

### Hide Tags

Math Expert V
Joined: 02 Sep 2009
Posts: 60627

### Show Tags

36
194

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.

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.

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: 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.
_________________
Manager  Joined: 11 Oct 2013
Posts: 97
Concentration: Marketing, General Management
GMAT 1: 600 Q41 V31
Re: Inequalities: Tips and hints  [#permalink]

### Show Tags

1
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..
Manager  Status: Not afraid of failures, disappointments, and falls.
Joined: 20 Jan 2010
Posts: 243
Concentration: Technology, Entrepreneurship
WE: Operations (Telecommunications)
Re: Inequalities: Tips and hints  [#permalink]

### Show Tags

Top Contributor
Is this part of GMAT Math Book? And, if not; can it be included in the book?
IIMA, IIMC School Moderator V
Joined: 04 Sep 2016
Posts: 1381
Location: India
WE: Engineering (Other)
Re: Inequalities: Tips and hints  [#permalink]

### Show Tags

1
1
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?
_________________
It's the journey that brings us happiness not the destination.

Feeling stressed, you are not alone!!
Math Expert V
Joined: 02 Aug 2009
Posts: 8335
Re: Inequalities: Tips and hints  [#permalink]

### Show Tags

4
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
_________________
Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 10008
Location: Pune, India
Re: Inequalities: Tips and hints  [#permalink]

### Show Tags

4
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?

Or say to yourself - Always add, always same sign

If the signs of the inequalities are not the same, make them same by multiplying one inequality by -1 and then add.
_________________
Karishma
Veritas Prep GMAT Instructor

IIMA, IIMC School Moderator V
Joined: 04 Sep 2016
Posts: 1381
Location: India
WE: Engineering (Other)
Re: Inequalities: Tips and hints  [#permalink]

### Show Tags

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?
_________________
It's the journey that brings us happiness not the destination.

Feeling stressed, you are not alone!!
Math Expert V
Joined: 02 Sep 2009
Posts: 60627
Re: Inequalities: Tips and hints  [#permalink]

### Show Tags

1
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  D
Joined: 17 May 2015
Posts: 245
Re: Inequalities: Tips and hints  [#permalink]

### Show Tags

1
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.
Non-Human User Joined: 09 Sep 2013
Posts: 13999
Re: I'm very confused with cross multiplying inequalities. Is  [#permalink]

### Show Tags

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: I'm very confused with cross multiplying inequalities. Is   [#permalink] 29 Jan 2019, 05:11
Display posts from previous: Sort by

# Inequalities: Tips and hints  