The term “Absolute Value” refers to the magnitude of a quantity without regard to sign. In other words, its distance from zero expressed as a positive number.
The notation used to indicate absolute value is a pair of vertical bars surrounding the quantity, sort of like a straight set of parentheses. These bars mean: evaluate what is inside and, if the final result (once the entire expression inside the absolute value signs has been evaluated) is negative, change its sign to make it positive and drop the bars; if the final result inside the bars is zero or positive, you may drop the bars without making any changes:
Another example is:
Note that absolute value signs do not instruct you to make “all” quantities inside them positive. Only the final result, after evaluating the entire expression inside the absolute value signs, should be made positive.
Absolute Value expressions that contain variables
Just as with parentheses, absolute value symbols serve as grouping symbols: the expression inside the bars must be evaluated and expressed as either zero or a positive quantity before the bars may be dropped. While this is fairly straightforward when working with constant values, as shown above, what happens when a pair of absolute value signs contains a variable?
While many relationships in our world can be described using a single mathematical function or relation, there are also many that require either more or less than what one equation describes. The behavior being described might start at a specific time, or its nature changes at one or more points in time. Two examples of such situations could be:
Acceleration up to a speed limit
Free fall then controlled descent
In the graph on the left, note that the blue line starts at the origin. It does not appear to the left of the origin at all. Furthermore, when x = 3 the blue line stops and the green line begins – but with a different slope.
In the graph on the right, note that the blue curve starts at x = 0. It does not appear of the left of the vertical axis at all. And when x = 3 the blue parabola turns into a green line with a very different slope. And the green line stops at x = 5.5, just as it reaches the horizontal axis.
These graphs do not seem to follow all the rules you were taught for graphing lines or parabolas. Instead of being defined over all Real values of x, they start and stop at specific values. The graphs also show (in this case) two very different functions, but in a way that makes them look as though they are meant to represent a single, more complex function. Both of these graphs are examples of “Piecewise Functions”, functions defined in a way that lets you mix and match as many separate components (pieces) as you wish.