Triangle inequality for a metric

Statement

Let $(X, d)$ be a metric space. For all points $x, y, z \in X$, $$d(x, z) \le d(x, y) + d(y, z).$$

Remarks

This is one of the four axioms of a metric. For $X = \mathbb{R}$ with $d(x, y) = |x - y|$, it is the triangle inequality for real numbers, with $a = x - y$ and $b = y - z$. A consequence is the reverse form $|d(x, y) - d(x, z)| \le d(y, z)$ for all $x, y, z \in X$.

Explanations

L1 human

A direct path from $x$ to $z$ is never longer than a path that goes through a third point $y$. In the plane, this says that one side of a triangle is at most the sum of the other two sides.