I don't understand this argument. I've heard high school teachers decree it as a mathematical truth before.
Do we all agree absolute value is a function from reals to reals? If so, the absolute value of any real number is nonnegative. It can be zero or any positive numbers.
It was my understanding that it's like velocity vs speed. Positive/Negative numbers have a directional component, that is that they are above or below zero. Absolute values, however, lack a direction. They're a value, but they don't tell you if the value is positive or negative.