<p>　　 Inequality is one of the topics that high school students find difficulties to learn in Japan.  Some previous studies investigated the difficulties of learning inequality through identifying and considering, in terms of its teaching, different approaches to solve inequalities (Ito, 2002; Hattori, 2010, 2011).  Looking at inequalities in mathematics textbooks of Japanese secondary schools, one may find that the notion of inequality appears in different places and in different ways: as an object to be solved, as an object to be proved, as a method to show the magnitude relationship, etc.  We then consider that student’s difficulties result from its complicated nature that should be primarily uncovered.</p><p>　　 The aim of this paper is therefore to advance understanding on the nature of inequality with a view to clarify the origins of student’s difficulties in its learning.  We try to find an answer to the question “what is inequality?”  Relying on Anthropological theory of the didactic (ATD) developed by Chevallard (2006), our question in this paper is formulated as a task of elaborating a reference epistemological model of inequality by means of the notion of praxeology.  In order to accomplish this task, we first identify how the inequality is dealt with in different kinds of mathematics such as mathematics in the history and school mathematics in Japan, and then characterize its nature by the elements of praxeology (types of tasks, techniques, technologies, and theory).</p><p>　　 As a result, we first found a cultural factor that complicates the nature of inequality, that is, we use two Japanese terms, in a way very ambiguous, that correspond to the term inequality in English: futoushiki (literally translated to the expression of inequality) often used in secondary schools denotes the expression which represents a magnitude relationship (or inequality), while daishou-kankei (literally translated to the relationship of small and large) denotes the magnitude relationship.  While this distinction is not necessarily shared in the community of mathematics education in Japan, the former is often a representation of the latter.  This nature of inequality in Japan results in a twofold praxeology as a reference epistemological model: the one on the magnitude relationship (or inequality) including two types of tasks (solve an inequality and prove an inequality) and the other on the representation including three types of tasks (represents a magnitude, an interval or range, and a set).</p>