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Inequalities: Tips and hints

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Inequalities: Tips and hints  [#permalink]

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New post 02 Jul 2014, 04:33
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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.
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Re: Inequalities: Tips and hints  [#permalink]

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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!
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Re: Inequalities: Tips and hints  [#permalink]

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New post 06 Aug 2016, 00:46
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Is this part of GMAT Math Book? And, if not; can it be included in the book?
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Re: Inequalities: Tips and hints  [#permalink]

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New post 04 Apr 2018, 22:30
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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?
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Re: Inequalities: Tips and hints  [#permalink]

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New post 04 Apr 2018, 22:49
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adkikani 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
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3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


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Re: Inequalities: Tips and hints  [#permalink]

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New post 05 Apr 2018, 00:52
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adkikani 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?



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.
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Re: Inequalities: Tips and hints  [#permalink]

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New post 13 Apr 2018, 23:16
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?
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Re: Inequalities: Tips and hints  [#permalink]

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New post 14 Apr 2018, 00:30
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adkikani wrote:
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.
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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.


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Re: Inequalities: Tips and hints  [#permalink]

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New post 14 Apr 2018, 00:38
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adkikani wrote:
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?

adkikani,

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.
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Re: Inequalities: Tips and hints &nbs [#permalink] 14 Apr 2018, 00:38
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Inequalities: Tips and hints

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