How can |x| = -x?? I thought absolute value was always positive.

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").

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