General
The absolute value of a number tells you how far away that number is from 0 on the number line, so it is a measure of distance. And distance, as we know, is always a positive quantity.
The absolute value of a number is denoted by | |, with the number being referred to inside the bars. Say, for example, we wanted to know how far away from 0 the number negative three is. The notation for that, then, would be |-3|. And since -3 is three units away from 0, the absolute value of -3 is 3. Formally, |-3|=3. Similarly, we know that the number 3 is also three units away from 0, so |3| = 3 also. From this we can see that additive opposites have the same absolute value, because they are the same distance away from 0 on the number line.
The formal definition of absolute value is given by :
When working with absolute value we often see equations such as:
.
These equations will have two solutions provided a > 0; If a = 0 it will have only one solution; and if a < 0 it will have no solutions. We can solve these equations simply using the definition of absolute value:
or 
Giving us:
or 
This approach works, but we usually use a modified form of this:
, means :
So lets try a problem:
As for the problem in the practice test this should be done using the definition of absolute value:
If x < 5 then x - 5 < 0 (subtracting five from both sides)
Thus | x - 5| = - ( x - 5) = - x + 5
More information on absolute value is found in the next few pages.
Next Page: Absolute Value and Inequalities