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# Confirmation on some Inequalities

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Senior Manager
Joined: 12 Mar 2010
Posts: 384
Concentration: Marketing, Entrepreneurship
GMAT 1: 680 Q49 V34
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Kudos [?]: 144 [0], given: 87

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07 Feb 2012, 01:59
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Hi

I need some confirmation on my thought process regarding some inequalities.

This is primarily to do with an inequality between a and b and how the inequality changes b/w a^n and b^n.

If n is an odd integer:
If a > b then a^n > b^n holds true irrespective of a and b being positive or negative, if n is positive odd integer
If a > b then a^n < b^n holds true irrespective of a and b being positive or negative, if n is negative odd integer

If n is an even integer:
If a > b and if a, b, and n are positive integers then a^n > b^n. If n is negative then it will be the opp. ineqlty.
If a > b and if a, b, are negative integers then a^n < b^n for positive n. If n is negative then it will be the opp. ineqlty.

If a and b are not of the same sign,then for even n we cannot say anything about the relation.
Because 4>-2 and 4>-8 will result in two different inequalities when raised to even powers.

Kindly confirm if I am correct in the above.

Also, please let me know how do we handle the above if 'n' is a real number.

Thanks
Sai
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Joined: 02 Sep 2009
Posts: 34817
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Re: Confirmation on some Inequalities [#permalink]

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07 Feb 2012, 04:14
bsaikrishna wrote:
Hi

I need some confirmation on my thought process regarding some inequalities.

This is primarily to do with an inequality between a and b and how the inequality changes b/w a^n and b^n.

If n is an odd integer:
If a > b then a^n > b^n holds true irrespective of a and b being positive or negative, if n is positive odd integer
If a > b then a^n < b^n holds true irrespective of a and b being positive or negative, if n is negative odd integer

If n is an even integer:
If a > b and if a, b, and n are positive integers then a^n > b^n. If n is negative then it will be the opp. ineqlty.
If a > b and if a, b, are negative integers then a^n < b^n for positive n. If n is negative then it will be the opp. ineqlty.

If a and b are not of the same sign,then for even n we cannot say anything about the relation.
Because 4>-2 and 4>-8 will result in two different inequalities when raised to even powers.

Kindly confirm if I am correct in the above.

Also, please let me know how do we handle the above if 'n' is a real number.

Thanks
Sai

Seems right.

Below is what you need to know for the GMAT about raising inequalities into a power:

A. 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 can not square both sides and write $$x^2>y^2$$ if we are not certain that both $$x$$ and $$y$$ are non-negative.

B. 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 $$-5<1$$ --> $$-5^2=-125<1=1^3$$;
$$x<y$$ --> we can raise both sides to third power and write: $$x^3<y^3$$.

So for our question we can not square x/|x|< x as we don't know the sign of either of side.

Hope it helps.
_________________
Re: Confirmation on some Inequalities   [#permalink] 07 Feb 2012, 04:14
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