A theorem of the alternatives for the equation |Ax| − |B||x| = b

A theorem of the alternatives for the equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${|Ax|-|B||x|=b\ (A,B\in{\mathbb{R}}^{n\times n},\, b\in{\mathbb{R}}^n)}$$\end{document} is proved and several consequences are drawn. In particular, a class of matrices A, B is identified for which the equation has exactly 2n solutions for each positive right-hand side b.