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!

Properties of Absolute Value

In my previous post, I went over the definition of absolute value and gave you a bunch of examples to show you how to work with absolute values in your math homework. In this post, I'm going to add to that by describing some more properties and rules of absolute values. Then I'm going to give you a couple of practice questions like those you would likely find in your math textbook or on a math test. By the end of it, you will be a pro at working with absolute values!

These properties are all extensions of what I taught in my last post, so if you need a refresher, head back to that one and give it a review. There's not many, so it shouldn't be too difficult for you to think about each point for a minute and see how it relates to what you already know.  Here we go:

1)  |x| > 0, where x is a real number.
This one should be straight-forward.  We talked about how absolute value is always positive, as it represents a distance on the number line between a number and zero. Therefore, we can write is this way. The absolute value of x is always greater than, OR EQUAL, to zero. (Yes, you can technically say that the absolute value of zero is zero. The distance on the number line from zero to zero is zero units. Weird, I know...)

2)  x < |x| and -x < |x|, where x is a real number.
This is, again, another way of showing that the absolute value of a number (positive or negative) will always be positive. If x is negative, the absolute value is positive, which is greater. If x is positive, the absolute value is positive, and so they're equal. Substitute in some numbers for x and you can prove for yourself that those expressions are always true.

3)  |x|^2 = x^2, where x is a real number.
The absolute value of x squared is equal to x squared. This is because whenever you square a real number, your result is positive. 2 times 2 is positive 4, but so is -2 times -2... positive 4! Again, plug in some numbers to prove it to yourself.

There are also a couple of arithmetic rules for when using absolute values. These are important to learn, and important to not confuse or get wrong! I'm not going to go in depth to explain these here, but you will learn through examples how and where to use them.

4)  |ab| = |a| |b|, and |a/b| = |a| / |b| (where b is not zero) for all real numbers.
5)  |a+b| < |a|+|b|

For this last one, refer to a number line:

6)  For two points on a number line, a and b, the distance between them is |a-b| = |b-a|.
This one is intuitively easy to understand.  Basically, it says the distance between a and b is the same as the distance between b and a. It almost goes without saying!

Also, I should mention that I will put up a post soon about representing absolute values and inequalities graphically on a number line. There are a few different ways to actually show these visually.

OK, I said I would give some practice questions, so here I will put a few different kinds of questions that all deal with absolute values.

1. Evaluate the expression: |-5-4| + |-3|.
This is |-9| + |-3|, which is 9 + 3, which is 12.

2. Rewrite the statement using absolute value notation: The distance between x and 2 is 10.
When you're talking about distance, you're talking about the difference of two points. Therefore, we can write this as |x-2| = 10.

3. Rewrite the statement using absolute value notation: The distance between x and 7 is less than 20.
Same thing here, only using a less than sign. |x-7| < 20.

Those are a couple of typical examples. Let me know if you have any questions in particular, and I'll try to address them as soon as possible!

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Absolute Value

"Absolute value" sounds like a complicated mathematical concept, but really, it is quite simple. The concept is straight-forward, and most arithmetic that uses is can be quite easy as well. It can be thought of as a tool for measuring distances between two number. "Absolute value" simply refers to how far away a value is from zero. It is easiest to understand this concept by using a number line:

If we consider the value of 3, we can easily see that it is 3 units away from zero. Simple, right?

Similarly, if we consider the value of -4, we can also easily see that it is 4 units away from zero.  It is not 'negative 4' units away. It is just 4 units away.

It doesn't matter if you're on the positive or negative side of zero, whatever value that you are considering is a positive number of units away from zero.

In mathematical notation, absolute value is denoted as two straight lines that surround the value, like this: |x|. So, for our examples above, we can state mathematically that:

|3| = 3

|-5| = 5

These expressions say the same thing as our words did above.  3 is 3 units away from zero, and -5 is 5 units away from zero.

Now, if we want to go a step further and actually perform some arithmetic with the concept of absolute value, this is quite simple to do as well. Just as you have likely learned in your Order of Operations lessons (solve expressions in this order: brackets, exponents, division/multiplication, addition/subtraction), when working with absolute value expression, you solve for the expression within the absolute value sign (similarly to solving inside the brackets first).

So, if we have the expression |2+5|, we solve inside first to get |7|, and then we just learned that |7|=7.

OK, let's try one a bit tougher. |4-9|. Solving inside first gives |-5|, for which we already saw |-5|=5.

What about |-8-12|? Solve inside to get |-20|, and then |-20|=20.

Now, let's throw in a twist.  Just like when working with brackets, where you can have brackets inside brackets and you solve them from the inside out, you can have the same thing here.

Try ||-3|+|7||.

Solving the inner values first gives: |3+7|, which gives |10|=10.

Similarly, try ||-5|-||2-4|-8||. (This is tough to type on a blog, but you would likely find in your math book that the matched sets of bars to be the same size, and inner sets would get smaller.)
||-5|-||2-4|-8||
=||-5|-||-2|-8||
= ||-5|-|2-8||
= |5-|-6||
= |5-6|
= |-1|
= 1

That's a tougher one, so if you can follow through that, you are well on your way.

Now that you understand the concept of absolute value, I will put the algebraic definition of it here:

|x| = {  x   when x > 0

          -x   when x < 0

Think about that definition, and you will see that it says exactly the things that we've been practicing. It just looks a lot fancier and more complicated. But, if you followed through all those examples and they made sense to you, then you are now and expert at working with absolute value!

In my next post, I will teach you a few more properties about absolute value, and give you more examples to think about.