Absolute Value

Overview

Let $x\in \mathbb{R}$. The absolute value of $x$, denoted $|x|$, is defined as $$\lvert x \rvert = \begin{cases} x & \text{if } x \geq 0 \ -x & \text{if } x \leq 0 \end{cases}$$

Triangle Inequality

Let $x,y\in \mathbb{R}$. Then the triangle inequality of $\mathbb{R}$ states $$\lvert x + y \rvert \leq \lvert x \rvert + \lvert y \rvert.$$

Geometrically speaking, any side of a triangle is less than or equal to the sum of the other two sides.

Reverse

Let $x,y\in \mathbb{R}$. Then the reverse triangle inequality of $\mathbb{R}$ states $$\lvert \lvert x \rvert - \lvert y \rvert \rvert \leq \lvert x - y \rvert.$$

Geometrically speaking, any side of a triangle is greater than or equal to the difference of the other two sides.

Bibliography

• Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
• “Triangle Inequality.” In Wikipedia, July 1, 2024. https://en.wikipedia.org/w/index.php?title=Triangle_inequality.