Although an absolute value must be non-negative, it would be short-sighted to claim that absolute values are "always positive." Remember that absolute values are essentially about the distance between two values. Say we wanted to find the distance of 0 from the origin. We could express this as:
- |0| = 0
Notice that our output here is not positive. As you continue your studies, keep the following in mind: positive (> 0) and non-negative (≥ 0) are two different concepts, and you must be careful with their usage.
As for |x| = -x, recognize that '-x' does not necessarily imply a negative number. It simply means "the opposite of x."
Imagine that 'x' is itself a negative value, like -4. In this case, we could state:
- |x| = -x
- |(-4)| = -(-4)
- |-4| = 4
As you can see, |x| = -x is a totally valid statement for cases where x ≤ 0 (or "non-positive").