Absolute Difference of Two Integers New to math. I'm looking for an explanation (proof, rule, relationship or property) that explains that the absolute value of the difference between two integers $x$ and $y$ are equal regardless of: 1. the sign of $x$ or $y$ 2. the order of subtraction. meaning regardless whether $x$ is the minuend or subtrahend $$x - y = |d| = y - x$$

What you wrote, $x-y=|d|=y-x$ is incorrect. Take x=1 an y=2 for example. $1-2\
eq2-1$

What you probably meant to say is this. ($d$ is positive)

$$|x-y|=d=|y-x|$$

Here is my explanation:

$$x-y=-(-x)+(-y)$$ $$x-y=-((-x)+y)$$ $$x-y=-(y-x)$$

Now, in simple terms, the absolute value of a a number just makes it posotive. Because of that, for any variable $n, |-n|=|n|$. Therefore:

$$|-(y-x)|=|y-x|$$ so $$|x-y|=|y-x|$$