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# Division and Inequalities

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Intern
Joined: 15 Oct 2009
Posts: 15

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07 Dec 2009, 12:32
After going through the inequalities section of my MGMAT book i noticed an inconsistency...with division, does the sign 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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Manager
Joined: 29 Oct 2009
Posts: 195
GMAT 1: 750 Q50 V42

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07 Dec 2009, 12:49
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.
_________________
Click below to check out some great tips and tricks to help you deal with problems on Remainders!
http://gmatclub.com/forum/compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html#p651942

1) Translating the English to Math : http://gmatclub.com/forum/word-problems-made-easy-87346.html
Intern
Joined: 15 Oct 2009
Posts: 15

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07 Dec 2009, 13:31
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
Manager
Joined: 29 Oct 2009
Posts: 195
GMAT 1: 750 Q50 V42

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07 Dec 2009, 13:39
1
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.
_________________
Click below to check out some great tips and tricks to help you deal with problems on Remainders!
http://gmatclub.com/forum/compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html#p651942

1) Translating the English to Math : http://gmatclub.com/forum/word-problems-made-easy-87346.html
Intern
Joined: 10 Dec 2009
Posts: 3

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10 Dec 2009, 12:24
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

This question will become more complicated if x can be positive, negative or zero.
Math Expert
Joined: 02 Sep 2009
Posts: 55150

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11 Dec 2009, 02:55
2
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

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)

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Intern
Joined: 18 Jul 2009
Posts: 37

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22 Dec 2009, 14:04
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.
Math Expert
Joined: 02 Sep 2009
Posts: 55150

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22 Dec 2009, 15:31
1
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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02 Sep 2017, 16:23
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Re: Division and Inequalities   [#permalink] 02 Sep 2017, 16:23
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