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

Hailie Blevins

Hailie Blevins

Answered question

2022-06-08

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 [ = , < , > ] are known. How to check if the system of inequalities above are consistent or not?

Answer & Explanation

Braedon Rivas

Braedon Rivas

Beginner2022-06-09Added 24 answers

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 ="=" x k + 1  if  ρ k ="<" x k 1  if  ρ k =">"
taghdh9

taghdh9

Beginner2022-06-10Added 6 answers

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 ="=", and define
x 1 = a 1 x k + 1 = { x k  if  ρ k ="=" x k + a k + 1  if  ρ k ="<" x k a k + 1  if  ρ k =">"
to get an infinite set of possible solutions.

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