\[ \newcommand{\abs}[1]{|#1|} \newcommand{\sgn}{\text{sgn}} \]
The definition of absolute value inferred the fundamental properties:
\[ \begin{align} \abs a \le b ~~~~ \Leftrightarrow ~~~~ & -b \le a \le b \\ \abs a \ge b > 0 ~~~~ \Leftrightarrow ~~~~ & a > b ~~ \text{or} ~~ a < -b \end{align} \]
Triangle inequality of complex numbers can be inferred from the physical interpretation of modulus of complex number in complex plane: \[ |z_1 + z_2| \le |z_1| + |z_2| \] where \(z_1, z_2\) are complex numbers. Specially, for the real numbers in real axis, we have \[ |a + b| \le |a| + |b| \] When the two variables are both real numbers, there are only two different cases: \[ \left\{\begin{array}{ll} |a+b| = |a|+|b|, ~~~~ab\ge 0 \\ |a+b| = ||a|-|b|| < ||a| + |b|| = |a| + |b|, ~~~~ ab<0& \end{array}\right. \] This might be the most important inequality of absolute value, which leads us to many significant results.