There is a very fundamental property of absolute value operations that we haven’t explored yet. Read the box Definition of Absolute Value on p11.  Now, this “piecewise” definition looks cryptic because of the following issue:

Question: is it always true that |a| = a?

Answer:  No.  It depends on whether a is positive or negative.

If we know the number within the absolute value sign, for example: |-3|, then it is straightforward to say  |-3| = |3|.

However, if we don’t know the number within the absolute value sign, for example, |a|, then |a| is NOT necessarily equal to -a.  Take a look at Example 7b).

In other words, if   is a negative number,    . 

Instead, if    is a negative number,   (because ).

In short, when we evaluate an absolute value expression and if we know that the expression within the absolute value sign is negative (e.g. Example 7d), then what you need to do is to put a negative sign in front of the expression e.g. .