Triangle Inequality¶

Proposition¶

For all real numbers $x$ and $y\text{,}$

Interpretation¶

The triangle inequality bounds the size of a sum by the sizes of its parts: $|x+y|$ is never more than $|x|+|y|$$|y|$. It formalizes the idea that direct paths are never longer than indirect ones: walking $x$ and then $y$ ends at distance $|x+y|$ from $0$, after covering a total length of $|x|+|y|$$|y|$.

In the top diagram, $x$ and $y$ have the same sign, so $|x+y|=|x|+|y|$$|y|$. In the bottom diagram their signs are opposite, so the displacements partly cancel. In general, equality holds precisely when $xy\ge0$$0$, including the case where either value is zero.

Applied to differences, the inequality becomes a statement about three points on the line. Writing $x-z=(x-y)+(y-z)$ gives

so the distance from $x$ to $z$ is at most the distance via a third point $y$. This is the form used most often: to show that two numbers are close, compare both with a third number. Applying the inequality to $x=(x-y)+y$ and to $y=(y-x)+x$ also gives the reverse triangle inequality $\bigl||x|-|y|\bigr|\le|x-y|$$y|$.

A common slip is to expect $|x-y|\le|x|-|y|$$|y|$; it fails for $x=0$ and $y=1$$1$, where it would say $1\le-1$$-1$. The correct bounds are $\bigl||x|-|y|\bigr|\le|x-y|\le|x|+|y|$$|y|$. The name becomes literal later, for the side lengths of a triangle formed by vectors in Math II.

Key Results¶

• Absolute values interact subadditively under addition.
• The inequality provides fundamental bounds in analysis.

Examples¶

• $|3+(-1)|\le |3|+|{-1}|\text{.}$
• $|x-y|\le |x|+|y|$ for all real numbers $x,y\text{.}$

Prerequisites¶

• Absolute Value

Related Concepts¶

• Triangle Inequality for Vector Norms

Sources¶

• Math I: Calculus · Lecture 2

Last updated: 20 October 2026 03:02