How to check if a system of linear inequalities is solvable or not?I have the variables x 1 , x 2 , . . . , x n and the following relations: x 1 ρ 1 x 2 x 2 ρ 2 x 3 ... x n − 1 ρ n − 1 x n ,where the relations ρ i ∈ [ = , &lt; , &gt; ] are known. How to check if the system of inequalities above are consistent or not?

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How to check if a system of linear inequalities is solvable or not?I have the variables       x      1    ,      x      2    ,  .  .  .  ,      x      n   and the following relations:      x      1        ρ      1        x      2        x      2        ρ      2        x      3  ...      x          n      −      1            ρ          n      −      1            x      n    ,where the relations       ρ      i    ∈  [  =  ,  &amp;lt;  ,  &amp;gt;  ] are known. How to check if the system of inequalities above are consistent or not?

Braedon Rivas · Accepted Answer

The way you describe the system, it is always solvable, and one solution is given by:      x    1    =  0        x          k      +      1        =      {                                        x            k                                     if                       ρ                      k                    =&quot;=&quot;                                                  x            k                    +          1                           if                       ρ                      k                    =&quot;&amp;lt;&quot;                                                  x            k                    −          1                           if                       ρ                      k                    =&quot;&amp;gt;&quot;

taghdh9 · Accepted Answer

You can also see that you could take any tuple   (      a    1    ,      a    2    …      a          n      −      1        ) of   n  −  1 positive numbers, where       a    k    =  0 if       ρ          k      −      1        =&quot;=&quot;, and define      x    1    =      a    1          x          k      +      1        =      {                                        x            k                                     if                       ρ                      k                    =&quot;=&quot;                                                  x            k                    +                      a                          k              +              1                                                 if                       ρ                      k                    =&quot;&amp;lt;&quot;                                                  x            k                    −                      a                          k              +              1                                                 if                       ρ                      k                    =&quot;&amp;gt;&quot;                        to get an infinite set of possible solutions.

How to check if a system of linear inequalities is solvable or not? I have the variables x

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