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.
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
--> 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
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
--> 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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