## 1 Introduction
Figure 1: Illustration of the solution to the equation \([2,3]x=[3,6]\) with the M1 method which transforms it into two inequalities \([2,3]x\leq[3,6]\) and \([2,3]x\geq[3,6]\). \[[2,3]x=[3,6] \tag{2}\]0 Visualization of [PERSON]’s interval equation anomaly. As the LIE anomaly shows, solving interval equations is much more difficult than precise equations. The explanation of this anomaly will be presented in next sections. However, the motivation of this paper is not only to explain the LIE anomaly, but above all to present a new method of determining the tolerance solution (TS) of the basic equation \([\underline{a},\overline{a}]x=[\underline{b},\overline{b}]\). Due to the limitation of the volume of the paper, the method of solving the system of many interval equations (IES) will not be presented here. We will cover this in our next article. The equation \([\underline{a},\overline{a}]X=[\underline{b},\overline{b}]\) is the basic element of the IESs studied in the literature since at least 1964 [8; 9; 10]. A special case of IESs are the interval linear systems (ILSs). A significant contribution to the development of ILSs was made by [PERSON] [3; 4; 5; 6] in the 1990's. Since then, the terminology used in solving ILSs has been almost unchanged, which can be checked in the latest publications [7; 12; 17; 18; 19; 20; 21]. The mathematical model for the deterministic linear static system is the linear algebraic Equation (10). \[[2,3]x=[3,6] \tag{2}\]1  
In real problems, we can assume that we only know that the coefficients of (10) may independently vary within the intervals \([\underline{a},\overline{a}]=\mathbf{a}\) and \([\underline{b},\overline{b}]=\mathbf{b}\) respectively. Thus, we formally have the interval linear algebraic equation. \[[2,3]x=[3,6] \tag{2}\]2  
The solution set of (11) can be defined in a variety of ways [6]. We can find the united solution set (12)  
\[[2,3]x=[3,6] \tag{2}\]3  
or the _tolerable solution set_ formed by all values \(x\) such that the product \(ax\) falls into \(\mathbf{b}\) for any \(a\in\mathbf{a}\), i.e., the set (13). \[[2,3]x=[3,6] \tag{2}\]4  
The definitions of different types of ILSs solutions provided by [PERSON] are still valid and are used in subsequent research papers, the authors of which are looking for ILSs solutions based on new ideas, new types of IA [19], and for special cases of ILSs, e.g., two-sided ILS [20; 21]. One of the most important articles on ILSs is [7], in which [PERSON] and [PERSON] emphasize the importance of IA for fuzzy linear systems. They present various problems related to IA, and most of all, the problem with some methods providing results that are improper intervals. In [7], they present their own \"unified\" computation method that will provide solutions in the form of proper or empty intervals but \"never\" improper ones. The authors of [7], taking into account the existence of various forms of ILSs solutions, propose a model of this system in the form (14):  
\[[2,3]x=[3,6] \tag{2}\]5  
\"where \([\mathbf{A}]\) is a matrix whose entries are intervals...and \(\approx\) is a relation between intervals...\". They define the case of the multidimensional tolerance solution set as Shary (15), but they call this set \"robust\". \[[2,3]x=[3,6] \tag{2}\]6  
The authors of [7] analyze in detail one-variable interval linear equation, where \([A]=[a]=[\underline{a},\overline{a}]\), \(b=[b]=[\underline{b},\overline{b}]\), and they define the tolerance solution in the form (16). \[[2,3]x=[3,6] \tag{2}\]7The definition (16) is supplemented with the statement that we require  
\[[2,3]x=[3,6] \tag{2}\]8  
[PERSON] and [PERSON] analyze in [7] all possible variants of 1V-ILSs with different combinations of signs for \(\underline{a}\), \(\overline{a}\), \(\underline{b}\), \(\overline{b}\) and give ready-made solutions for these variants. For example, for 1V-ILS (18):  
\[[2,3]x=[3,6] \tag{2}\]9  
with positive values of \(\underline{a}\), \(\overline{a}\), \(\underline{b}\), \(\overline{b}\) the tolerance solution set is calculated from Formula (19).