When Do You Flip an Inequality Sign?

Date: 12/19/2005 at 21:28:17
From: Oscar
Subject: what does it mean to change the sense of inequality?

What does it mean to change the sense of an inequality?  I just
started studying about inequalities, and I can't grasp the concept of
that question I found in the book.  I would really like to know if
anyone can help out please.

Date: 12/19/2005 at 22:57:13
From: Doctor Peterson
Subject: Re: what does it mean to change the sense of inequality?

Hi, Oscar.

Your book probably defined what they mean by "the sense of an
inequality", but it may be hard to find or hard to follow without some
examples.

The "sense" of an inequality, also called its "direction", means
whether it is < or >. We change its sense when we change > into < or
vice versa.

We can do that for two reasons, which are alternative ways to solve
certain inequalities. One is when we just want to rewrite an
inequality so that it reads more naturally:

  3 < x   becomes   x > 3

These mean the same thing, but the first sounds like it is telling you
where 3 is (3 is less than x), while the second sounds like a proper
answer to the question "where is x?" (x is greater than 3). We've
flipped the whole inequality around, including changing the direction
of the "<".

The other reason for changing the sense is when we multiply both sides
of an inequality by a negative number. Notice that

  -2 < 3

but when we multiply both sides by -2, the order changes:

  4 > -6

We have sort of flipped the whole number line around backward when we
did the multiplication, so that the number that was on the left is now
on the right. That means we have to change the sense when we multiply
(or divide, which is a form of multiplication) by a negative number.

Does that help? If you have any further questions, feel free to write
back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/