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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