The modulus |x| measures distance from zero: |x| = x if x >= 0, |x| = -x if x < 0. Key algebraic properties: |xy|=|x||y|, |x/y|=|x|/|y|, |x|^2=$x^2$. The triangle inequality |a+b| <= |a|+|b| provides upper bounds, while ||a|-|b|| <= |a-b| provides lower bounds. These appear in maximum-minimum problems and range estimation. Graphically, y=|f(x)| reflects the negative part of f above the x-axis, while y=f(|x|) makes the graph symmetric about the y-axis. These transformations help visualize solutions without algebraic case-splitting.