Problem

The key insight is that these three equal absolute-value expressions impose strong geometric constraints on where $a$ and $b$ can lie on the number line. Each equation describes a set of points equidistant from certain landmarks (like 0, 1, and −1), and you need to find which pairs $(a,b)$ satisfy all three simultaneously. Once you identify the valid region, minimizing $|a-b|$ becomes straightforward—look for the closest pair of distinct points that work.

The key insight is that these three equal absolute-value expressions impose strong geometric constraints on where $a$ and $b$ can lie on the number line. Each equation describes a set of points equidistant from certain landmarks (like 0, 1, and −1), and you need to find which pairs $(a,b)$ satisfy all three simultaneously. Once you identify the valid region, minimizing $|a-b|$ becomes straightforward—look for the closest pair of distinct points that work.