Post History

I think I would describe that as ‘the two claims $|x| = \pm x$’, not a single claim. It's comparable to saying ‘the two roots of $x^2 - 4$ are $x = \pm2$’ and not ‘the root of $x^2 - 4$ is $x = \pm2$’.

Whether those two claims are meant to be and-ed or or-ed would depend on context. In a vacuum, I don't think I would be able to unambiguously assign meaning to ‘the claim $|x| = \pm x$’.

More broadly, I think I've seen only two general conventions for $\pm$-containing equations, and they both seem fairly rigorous to me if slightly informal. The first is that an expression or equation containing one or more ‘$\pm$’ and optionally ‘$\mp$’ signs is always shorthand for two of the same: one in which each $\pm$ and $\mp$ is replaced by its upper component, and one in which the lower components are used instead. The second is that an expression or equation using $n$ ‘$\pm$’ signs is a shorthand for $2^n$ of the same in which every combination is represented. I generally expect an author to clarify which convention is in use if more than one ‘$\pm$’ is present, and as above I also expect these conventions to be used in a context in which multiple equations/expressions are expected.