On the Domain-Dependent Validity of the Inequality √(−Y)2≤Y

The algebraic identity √(−Y)2=∣Y∣ is a foundational result in elementary algebra and real analysis, arising from the definition of the principal square root and the absolute value function. Although this equality is universally valid for all real numbers, its behavior under inequality reformulation has not been explicitly articulated in most standard mathematical treatments. This paper presents a rigorous analytical examination of the inequality √(−Y)2 ≤Y and establishes that its validity is inherently domain-dependent. Using formal algebraic manipulation, absolute value theory, and exhaustive case-based reasoning over the real number system, the study proves that the inequality holds if and only if Y≥0. For negative values of Y, the inequality is shown to fail due to the non-negativity constraint imposed by the principal square root operator. This analysis reveals that converting a universally valid equality into an inequality introduces implicit logical constraints that are not immediately apparent in the original formulation. The findings emphasize the critical role of domain awareness in algebraic reasoning and demonstrate how inequality-based reinterpretations of standard identities can uncover structural conditions embedded within real-number operations. The proposed analytical perspective has relevance for mathematical pedagogy, inequality theory, and foundational proof-based learning, offering a clearer conceptual link between absolute value functions, sign dependence, and domain-restricted validity...... Please check the attachment for details