Reversing the Inequality

Date: 09/21/98 at 22:12:51
From: Jonathan Lawrence Dulaney
Subject: Greater or less than symbols

I need to find out why the greater than or less than symbol changes 
when you're solving for inequalities.

Date: 09/22/98 at 12:16:28
From: Doctor Peterson
Subject: Re: Greater or less than symbols

Hi, Jonathan. I assume you're referring to the fact the if we multiply:

    x > 5

by a negative number, say -1, we have to reverse the symbol to get:

    -x < -5

The answer depends on whether you need to convince yourself or a 
mathematician. Here's how I like to convince kids:

Just as an equation can be thought of as a balanced scale:

    2 = 2           2               2
                    -----------------
                            A

an inequality can be thought of as an unbalanced scale:

    2 > 1                           1
                                    /
                                /
                            /
                    2   /   A
                    /

Multiplying it by -1 is like reversing gravity, so that things fall 
up, and the heavier object is now pulled more strongly up rather than 
down. If you do that, the scale will reverse:

    -2 < -1         2
                    \
                        \
                            \
                            A   \   1
                                    \

(Maybe you can think of this as replacing lead with helium, so the 
bigger helium balloon goes up, while the bigger lead balloon goes 
down.)

Another way to see it is with a number line. "Less than" means to the 
left on the number line:

              -2  -1   0   1   2   3
    1 < 2    --+---+---+---+---+---+--
                           1 < 2

Multiplying by -1 reflects points through the origin:

              -2  -1   0   1   2   3
    -1 > -2  --+---+---+---+---+---+--
              -2 < -1

That means left and right switch places, and the less than relation 
reverses with it.

If you need a more formal proof, let me know.

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