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

The classical identity √(−Y)² = |Y| is universally valid for all real Y, arising from the principal square rootand 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 rigorousanalytical examination of the inequality and demonstrates that it holds if and only if Y ≥ 0. For Y < 0 theinequality fails due to the non-negativity constraint imposed by the principal square root. The resultshighlight that transforming universally valid equalities into inequalities introduces implicit logicalconstraints not visible in the original formulation. The findings underscore the importance of explicitdomain awareness in algebraic reasoning, inequality analysis, and pedagogical practice. Please check the attachment for details