Domain-Dependent Validity of an Inequality Derived from a Classical Absolute Value Identity

Abstract

The classical identity √(−Y)² = |Y| is universally valid for all real Y, arising from the principal square root and absolute value definitions. However, when this identity is reformulated as an inequality—namely √(−Y)² ≤ Y—its validity becomes domain-restricted rather than universal. This paper provides a rigorous analytical examination of the inequality and demonstrates that it holds if and only if Y ≥ 0. For Y < 0 the inequality fails due to the non-negativity constraint imposed by the principal square root. The results highlight that transforming universally valid equalities into inequalities introduces implicit logical constraints not visible in the original formulation. The findings underscore the importance of explicit domain awareness in algebraic reasoning, inequality analysis, and pedagogical practice.