Triangle inequality

Statement

For all real numbers $a$ and $b$, $$|a + b| \le |a| + |b|,$$ where $|\cdot|$ is the absolute value.

Remarks

Proof: $-|a| \le a \le |a|$ and $-|b| \le b \le |b|$. Add these inequalities to get $-(|a| + |b|) \le a + b \le |a| + |b|$. Equality holds exactly when $a b \ge 0$. For distances, it gives $|x - z| \le |x - y| + |y - z|$. The reverse triangle inequality is a consequence.

Explanations

L1 human

The name comes from geometry: one side of a triangle is never longer than the other two sides together. For numbers, $|a + b|$ can be smaller than $|a| + |b|$ when $a$ and $b$ have opposite signs, for example $|5 + (-3)| = 2 < 8$.