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Division and Inequalities

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Division and Inequalities [#permalink] New post 07 Dec 2009, 11:32
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After going through the inequalities section of my MGMAT book i noticed an inconsistency...with division, does the sign < or > flip only when you are dividing by a negative number or does it flip whenever you divide.

Two places in the book say it gets flipped only when dividing by a negative and the other says whenever there is division

(I know with multiplication, the sign gets flipped only when multiplying by a negative)

Thanks for the help!
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Re: Division and Inequalities [#permalink] New post 07 Dec 2009, 11:49
sac8513 wrote:
After going through the inequalities section of my MGMAT book i noticed an inconsistency...with division, does the sign < or > flip only when you are dividing by a negative number or does it flip whenever you divide.

Two places in the book say it gets flipped only when dividing by a negative and the other says whenever there is division

(I know with multiplication, the sign gets flipped only when multiplying by a negative)

Thanks for the help!


It only changes when you multiply or divide by a negative number.

Consider the following example : 10 < 20

Now divide both sides by 2 : 5 < 10

If we were to switch the inequality then it would read as : 5 > 10 which doesn't make sense right?

However, if we were to divide by -2 then we would have to change the inequality sign : -5 > -10

This is again obvious since if we don't change the sign, then it would read : -5 < -10 which again does not make sense.

Thus change inequality sign only why multiplying or dividing by a negative number.

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Re: Division and Inequalities [#permalink] New post 07 Dec 2009, 12:31
I might have left out a piece of my question...my fault

....When dividing extrem value equestions

So in the case of 8 divided by 'Less Than 2' - would that be straight division in which one would need to flip the sign or only with division of a negative
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Re: Division and Inequalities [#permalink] New post 07 Dec 2009, 12:39
sac8513 wrote:
I might have left out a piece of my question...my fault

....When dividing extrem value equestions

So in the case of 8 divided by 'Less Than 2' - would that be straight division in which one would need to flip the sign or only with division of a negative


Not quite sure what you mean here. Maybe if you could provide a sample question I might be better able to address the doubt you have.

Otherwise just remember the following :

1) If you are sure that the number you are multiplying or dividing by is positive, then the sign of the inequality always remains unchanged.

2) If you are sure that the number you are multiplying or dividing by is negative, then the sign of the inequality always changes.

3) If you are not sure whether the value of the number you are multiplying or dividing by is positive or negative (or 0), then NEVER multiply or divide.

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Click below to check out some great tips and tricks to help you deal with problems on Remainders!
compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html#p651942

Word Problems Made Easy!
1) Translating the English to Math : word-problems-made-easy-87346.html
2) 'Work' Problems Made Easy : work-word-problems-made-easy-87357.html
3) 'Distance/Speed/Time' Word Problems Made Easy : distance-speed-time-word-problems-made-easy-87481.html

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Re: Division and Inequalities [#permalink] New post 10 Dec 2009, 11:24
How to solve the following type of question faster i.e. without substituting values.

If x is positive, which of the following could be the correct ordering of 1/x, 2x, and x^2?

(I) X^2 < 2x < 1/x
(II) x^2 < 1/x < 2x
(III) 2x < x^2 < 1/x

(a) none
(b) I only
(c) III only
(d) I and II
(e) I, II and III

Answer is D.

This question will become more complicated if x can be positive, negative or zero.
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Re: Division and Inequalities [#permalink] New post 11 Dec 2009, 01:55
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brownybuddy wrote:
How to solve the following type of question faster i.e. without substituting values.

If x is positive, which of the following could be the correct ordering of 1/x, 2x, and x^2?

(I) X^2 < 2x < 1/x
(II) x^2 < 1/x < 2x
(III) 2x < x^2 < 1/x

(a) none
(b) I only
(c) III only
(d) I and II
(e) I, II and III

Answer is D.

This question will become more complicated if x can be positive, negative or zero.


First note that we are asked "which of the following COULD be the correct ordering" not MUST be.
Basically we should determine relationship between x, \frac{1}{x} and x^2 in three areas: 0<1<2<.

x>2

1<x<2

0<x<1

When x>2 --> x^2 is the greatest and no option is offering this, so we know that x<2.
If 1<x<2 --> 2x is greatest then comes x^2 and no option is offering this.

So, we are left with 0<x<1:
In this case x^2 is least value, so we are left with:

I. x^2<2x<\frac{1}{x} --> can 2x<\frac{1}{x}? Can \frac{2x^2-1}{x}<0, the expression 2x^2-1 can be negative or positive for 0<x<1. (You can check it either algebraically or by picking numbers)

II. x^2<\frac{1}{x}<2x --> can \frac{1}{x}<2x? The same here \frac{2x^2-1}{x}>0, the expression 2x^2-1 can be negative or positive for 0<x<1. (You can check it either algebraically or by picking numbers)

Answer: D.

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Re: Division and Inequalities [#permalink] New post 22 Dec 2009, 13:04
Bunuel can u explain how 2nd condition holds true.

x^2 < 1/x < 2x


put x=1/2 then x^2=1/4,1/x=2,2x=1.
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Re: Division and Inequalities [#permalink] New post 22 Dec 2009, 14:31
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hrish88 wrote:
Bunuel can u explain how 2nd condition holds true.

x^2 < 1/x < 2x


put x=1/2 then x^2=1/4,1/x=2,2x=1.


The question is which of the following COULD be the correct ordering not MUST be.

Put 0.9 --> x^2=0.81, 1/x=1.11, 2x=1.8 --> 0.81<1.11<1.8. Hence this COULD be the correct ordering.

Hope it's clear.

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NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

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: Division and Inequalities   [#permalink] 22 Dec 2009, 14:31
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