|a+b| <= |a|+|b| with equality iff a,b have the same sign (or one is zero). |a-b| >= ||a|-|b|| with equality iff a,b have the same sign. These are crucial for finding: (1) Maximum of |f+g|: achieved when f,g have same sign. (2) Minimum of |f|+|g| over a variable: use geometric interpretation. (3) Minimum of |x-a|+|x-b| = |a-b|, achieved when x is between a and b.