Confirmation on some Inequalities

bsaikrishna · **Posted:** Tue Feb 07, 2012 1:59 am


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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.

Please confirm the following:

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

Bunuel · **Posted:** Tue Feb 07, 2012 4:14 am

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.

Please confirm the following:

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.

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Confirmation on some Inequalities

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