Q&A: What is the Easiest Way to Memorize Absolute Value Inequalities?

by Justin Skycak on

Cross-posted from here.


What is the easiest way to memorize the following absolute value inequalities?

$$\begin{align*} |x+y| &\leq |x| + |y| \\[5pt] |x-y| &\leq |x| + |y| \\[5pt] |x| - |y| &\leq |x-y| \\[5pt] | \,\, |x| - |y| \,\, | &\leq |x-y| \end{align*}$$


However, in this case, pictures are not the best memory aid. You have to remember the pictures and think critically about them in order to translate them into symbols. The more critical thinking you have to do, the harder it is to memorize.

The easiest way to remember these inequalities is to combine them into a single compound inequality that looks so remarkably intuitive that it just “feels” right and is easy to reconstruct based on simple patterns, without having to engage in critical thinking.

$$\begin{align*} \begin{matrix} \\ |x| - |y|, \\ | \,\, |x| - |y| \,\, | \phantom{,} \end{matrix} \, \leq \, \begin{matrix} \\ |x+y|, \\ |x-y| \phantom{,} \\ \end{matrix} \, \leq |x| + |y| \\ \\ \end{align*}$$

Here’s how to memorize the inequality:

  1. Put $\,\, \begin{matrix} \vert x+y \vert, \\ \vert x-y \vert \phantom{,} \\ \end{matrix} \,\,$ in the middle.
  2. "Distribute" the absolute value to get the top LHS and RHS. (LHS becomes "-" because left is negative direction; RHS changes to "+" because right is positive direction.)
  3. $$\begin{align*} |x| - |y| \, \leq \, \begin{matrix} \\ |x+y|, \\ |x-y| \phantom{,} \\ \end{matrix} \, \leq |x| + |y| \\ \\ \end{align*}$$

  4. If you put another set of absolute value bars around the LHS, the inequality also holds true. (You can do this to the RHS too; it's just not noteworthy because it's mathematically equivalent to the existing RHS.)
  5. $$\begin{align*} \begin{matrix} \\ |x| - |y|, \\ | \,\, |x| - |y| \,\, | \phantom{,} \end{matrix} \, \leq \, \begin{matrix} \\ |x+y|, \\ |x-y| \phantom{,} \\ \end{matrix} \, \leq |x| + |y| \\ \\ \end{align*}$$