Explaining Inequalities

     I find that most of my students aren't terribly concerned with why things are the way they are in mathematics. I am of the opinion that this is because when students ask why questions when they are young, they don't get answers. If they ask "why is a negative times a negative a positive" they get an answer like "just because," or, if they are home-schooled, "because I said so." I'd like to discuss some of those why topics in math, most of which I don't think are asked that often.

"Why do you switch the inequality when you multiply or divide both sides by a negative number?"

I'm glad you asked. (You're still reading this, right?)

Suppose we have the inequality -2x + 3 < 11. It is solved as follows:

-2x + 3 < 11
        -3   -3                         Subtract 3 from both sides
-2x < 8
-2x  >  8                              Divide both sides by -2 and switch the sign

 -2      -2
x > -4

The solution x > -4 works if you check numbers larger than -4 (like 0).

The inequality, much like the equals sign, is a statement of fact. The inequality 2 > 5 is stating the fact that two is greater than five, and the inequality -2x + 3 < 11 is stating that the left side is less than the right half.

The reason that inequalities switch when you divide by negative is, honestly, because that's just the way it is. Here's what I do with my students:

I put up two numbers and ask which sign goes in-between them.

4          8

and they tell me that a "less than" squeezes in there.

4    <    8

I start doing things to both sides, and check to see if the inequality is still true

4    <    8
+12      +12

16   <   20                   still true, 16 is less than 20, so we subtract 30
-30       -30

-14   <   -10                still true, so we multiply by 5
x5            x5


-70   <   -50                still true, so we divide by 10
/10          /10


-7     <     -5                still true, so we multiply by -1
x(-1)          x(-1)

7       <     5                 this is no longer true. Seven is, in fact, not less than 5. The sign needs to switch.
                                   
The sign didn't need switching with multipling or dividing positive numbers, and it didn't matter if we added or subtracted when positive or negative. It only mattered when we multiplied by a negative number. The same would be true if we divided by a negative number (like negative one).

Another way to describe this is with a number line. I have tried like a trillion ways to create a number line in this space that I can work with, but I am having almost no luck. Here's the best I can do:

-5  <  -3                      but                      5   >  3   

     Imagine having two points on that number line, a point at -3 and a point at -5.

     If we multiply both of those points by -1, we get the new points 3 and 5.

     Well, for -3 to travel to three, it has to travel a total of 6 spaces, but for -5 to get to 5, it has to travel 10 spaces. In that farther distance to travel, the -5 "passes" the -3 and goes from less than to greater than.

That is why the sign switches. Most people don't care. I never had it explained to me, and when I was teaching it one day I thought to myself "why does that happen?" So I went home, figured it out, and started teaching it. I know there are a few students in every class that appreciate seeing that. Hopefully it helps.

I know that no one is still reading this, so I would like to take this opportunity to let the world know that today I wiped my face with a towel that was on the carpet, and my wife told me we had placed that towel under the baby when changing him. Thanks, wife, for waiting until I had finished.