The paper focuses on methodology on an application of interval methods to the specific problem of spectral analysis. The subject of the study is the mathematical formulation of the problem and its solution by the methods of interval analysis in a linear case, that is, the Linear Interval Tolerance Problem. The main methodological goal of the work is to demonstrate the use of the recognizing functional as a method of investigation. We suggest the way to construct the right-hand side of the interval system of linear algebraic equations (ISLAE) without involving models of the background shape under the peak, "natural intervalization". The forms of the united solution sets for the problem are considered. The sign of the recognizing functional serves for a study of the solvability of ISLAE. The study demonstrates that in the initial formulation the problem is unsolvable. Various methods for achieving solvability are considered: by the reducing the tolerances of the right-hand side of the system and the modification of the ISLAE matrix. The results of using both approaches are discussed.