Absolute Value

Introduction

The absolute value of a number is its value, or magnitute, without respect to sign. It's the "amount" you are working with expressed as a positive number, ignoring any negative sign. Or, it's the number's distance from 0 on a number line. For example, the number 9 is 9 units away from 0. Therefore its absolute value is 9. The number -9 is the same distance from zero, so its absolute value is also just 9. In both cases the magnitude, or absolute value, of your number is just plain old "9" because you've removed any negative sign that might have existed.

The absolute value leaves a positive unchanged, and makes a negative positive.

How do I write it?

An absolute value is written like this: |x|, and is read as "the absolute value of x." Note: In certain places, such as calculator and computer programs, you may see it written as abs(x), which naturally means "the absolute value of x," but |x| is the common way your teacher probably wants you to write it on your homework and tests.

Applications

Another use of the absolute value bars is actually to force a number to be negative, by writing -|x|. This takes the number, makes it positive, and then negates it. Why do we have to do it like that? Well, remember -- just putting a negative sign in front of a number doesn't always make it negative. If the number was already negative then you just made it positive! Using the absolute value guarantees we have a positive value inside the bars, so the negative sign will definitely make it negative.

Examples

|4| = 4
|-4| = 4
|4+3| = 7
|-4-3| = 7
|3-4| = 1
-|4| = -4
-|-4| = -4

The absolute value sign can be used in equations as well:
|-8| = x, thus x=8
|x| = 8, thus x=8 or x=-8. Remember that |-8| is also 8 so there are two solutions here!
|x| = -8, there are no solutions because the absolute value can never be negative.

Taking the Absolute Value of an algebraic expression

Absolute values are easy enough to compute when they contain constants (regular numbers), but absolute value equations containing variables are more difficult. Suppose we are given the following equation, and asked to solve for x:
|x+2| = 9

We can not assume that x+2 is positive or negative, so we can not simply "drop the bars." If x+2 were indeed negative, the absolute value of x+2 would really be -(x+2), since a negative times a negative equals a positive. We will solve using cases.

The first case, or possibility, is that x+2 is positive. Taking the absolute value of a positive does not change the outcome.
First Case: x + 2 = 9

The second case is that x+2 is negative. To get the absolute value of a negative, you have to negate it (which makes it positive again). Therefore |x+2| = -(x+2).
Second Case: -(x + 2) = 9

Here we can solve both cases for x.

x+2 = 9
x = 7

or

-(x+2) = 9
-x -2 = 9
-x = 11
x = -11

Our two solutions for |x+2|=9 are 7 and -11. Try them. They both work.

More complicated equations can usually be solved the same way, by splitting the absolute value into two cases. You should check that you answers match the case, however. If you get a possible answer of 8 from the negative case, that can't be right.

If you have time you should plug your answers back into the original equation to check for correctness.

That's about it for a simple introduction of the absolute value. You may not use it that often, but it is vital to understand later in math. For more information, try searching Google for "absolute value." You may also want to check out another lesson on absolute value provided by PurpleMath, or perhaps this one from Wikipedia. If you want to play around with other numbers and check your own work, try this absolute value calculator -- enter any number and get the absolute value.

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