Wednesday, August 17, 2011

How to Show Inequalities and Absolute Values on a Number Line

This will be a short post, but I wanted to quickly go over something I touched on in my previous post about properties of absolute value, which is how to graphically show absolute value and inequalities on a number line. I mentioned that there are a couple of acceptable ways to do this, so I will try to explain them here.

I want to first go over the concept of inequalities briefly. It is all very intuitive and common sense, but you will see why I want to point this out now before we put inequalities on to a number line.

If I give you the example x < 2, we know exactly what that means. In fact, the words you say when you read that mathematical expression pretty much says it all: x is less than 2. Great. Similarly, the example x < 2 is also as obvious: x is less than or equal to 2. That was the easy part, but the key is to always pay attention to what the sign is saying. Does it say "less than" or "less than or equal to" something. Let me show you why this distinction is important when it comes to graphing inequalities on a number line.

Let's take that first examples again, x < 2. To put this on a number line, there are 3 important things here: the point in question (2), the direction on the number line that makes the expression true (less than), and does the inequality contain the point, or exclude it (less than, or less than or equal to).

If the sign being used is either < or >, without the equals part, you can represent this on a number line as an empty circle at that point. Alternatively, you could show it with a round bracket. This notation implies that you are talking about points that get infinitely close to your number (2, in our example... such as 1.9999999999), but not exactly 2. In addition to that, for all the values that make the expression true you put a line over the number line, or draw a thicker line on top of the number line. If the true values go all the way to infinity, you can put an arrowhead on the line. So for our example of x < 2, we would represent it like this:

or like this:
Now, when we are talking about an expression that has an equals part to it, that is simply represented by a filled circle on the number line, or a square bracket. Note that both kinds of brackets open towards the direction that the expression is true. So, for our second example of x < 2, we could show this on a number line like this:
or like this:
I should probably apologize for the shoddy artistic skills.  ;)

By looking at these number lines, and the ways that we have represented the inequalities, it is very easy to see that we are talking about x < 2 in the first pair, and x < 2 in the second pair.

So, that is most of what I wanted to explain about inequalities and number lines. Now, to apply these ideas to solving an absolute value question, it should be straight forward. And when in doubt, just substitute in numbers for x to see if the inequality is true or false!

If we are talking about |x| < 2, we saw in my original post on absolute values that this simply means that x is 2 units away from zero. It doesn't specify only positive numbers, or negative numbers, or any type of restrictions on the question. Therefore, 2 units away from zero in both directions encompasses the interval of -2 to 2. So, to show this on a number line is simply two open circles (on -2 and 2), and a thick line connection to two points to represent that all number within that interval make that inequality true. Try it and see for yourself! 1.999934958... true. -1.21111... true. 0.00000000003... true. The absolute value of these values for x are all less than 2.

I don't think I need to give anymore examples to explain this any further. It's pretty simple stuff, right? If you do need a bit more explanation or you would like additional examples, don't hesitate to drop me a line and I'll see what I can do!

1 comment:

  1. Well done you apear to be the only person on the internet showing this correctly!


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