We study systems of relations of the form $Ax\,��\,b$, where $��$ is a vector of binary relations with the components "$=$", "$\geq$" and "$\leq$", and the parameters (elements of the matrix $A$ and right-hand side vector $b$) can take values from prescribed intervals. What is considered to be the set of its solutions depends on which logical quantifier is associated with each interval-valued parameter and what is the order of the quantifier prefixes for certain parameters. For solution sets that correspond to the quantifier prefix of a general form, we present equivalent quantifier-free descriptions in the classical interval arithmetic, in Kaucher complete interval arithmetic and in the usual real arithmetic.

The work is a revised and extended English translation of the original Russian work - Sharaya, I.A. Quantifier-free descriptions of interval-quantifier linear systems // Trudy Instituta Matematiki i Mekhaniki UrO RAN (Proceedings of the Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences), Vol. 20 (2014), No. 2, pp. 311-323