Concept guide
Absolute value: nonnegativity, the zero case and |a−b|=|b−a| from the piecewise definition
Using only the piecewise definition (|x|=x when x≥0, |x|=−x when x<0) together with the order and addition of the reals, prove for all reals x, a, b: (1) |x|≥0; (2) |x|=0 ⇔ x=0; (3) |−x|=|x|; (4) |a−b|=|b−a|. Then compute |−3|, |2−7| and |7−2|, and show that '|x|=x for every x' is false.