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.