Associated Topics || Dr. Math Home || Search Dr. Math

### Raising Both Sides of an Inequality to a Power

```Date: 06/19/2006 at 05:52:52
From: Antti
Subject: When to turn < or > around when raising to nth power

First I have to say your website is marvelous.  Thank you for that.  I

You know when -2 < 1 and both sides are multiplied by -1, the < has to
be changed to > to get the right answer, 2 > -1.  What if we raise
both sides of -2 < 1 to the 2nd power?  Then we also have to turn the
< around to get the right answer (-2)^2 > 1^2 = 4 > 1.

How do I know when the < or > has to be turned around when raising to
powers?  Is there a rule for raising to the 3rd, 4th, or nth power?

```

```
Date: 06/19/2006 at 10:18:41
From: Doctor Peterson
Subject: Re: When to turn < or > around when raising to nth power

Hi, Antti.

The inequality has to be turned around when you multiply by a negative
number, since that essentially flips the whole number line around,
reversing the order of everything.  But squaring or raising to a
higher power doesn't work that way; you can't always obtain an
equivalent inequality at all.

For example, suppose you know that a < b.  Perhaps a is -2 and b is 1;
or perhaps a is -1 and b is 2.  What happens when we square each of
these numbers?

-2 < 1  becomes  4 > 1;
the square of -2 is GREATER than the square of 1

-1 < 2  becomes  1 < 4;
the square of -1 is LESS than the square of 2

So whether the inequality is reversed depends on the specific numbers
you have, in particular on their absolute values.  We can't say
something like

if a < b, then a^2 < b^2

All we can say is

if |a| < |b|, then a^2 < b^2

So your question is in a sense invalid: yes, we can determine whether
the square of this is less than the square of that, but the result is
not derived in any way from the given inequality.  We aren't really
turning anything around at all.

Now, with an ODD power, we CAN say, for example, that

if a < b, then a^3 < b^3

You can see this if you look at the graph of y=x^3.  The graph is
always increasing (rising to the right), so if you pick two numbers a
and b, with a < b, then the cube of the number on the right, b, is
greater than the cube of the number on the left, a.

That doesn't work with squaring (or any even power).  Look at the
graph of y=x^2; it falls for negative x, then rises for positive x.
So if you pick a and b with a < b, you have no guarantee that a^2 <
b^2 or not.

The lesson for you:  when you're working with inequalities, DON'T
raise both sides to an even power.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Equations
Middle School Exponents
Middle School Negative Numbers

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search