Instead of solving the equality Ax=b, I want to solve the inequality Ax>=b. In general, how can I so

preityk7t

preityk7t

Answered question

2022-06-16

Instead of solving the equality Ax=b, I want to solve the inequality Ax>=b. In general, how can I solve this problem with n unknowns, or when A has n columns?
When A has 1 column (or, when each equation in this system has only 1 unknown x) I can solve it easily by isolating x in each inequality and combining the inequalities to yield 1 inequality that bounds x. When choosing a satisfactory x, I simply refer to the yielded inequality.
When A has 2 columns (or, when each equations has 2 unknowns x and y), I graph the constraints and find the shaded regions, and use the appropriate equations for different x intervals (i.e., for x = 0 to x = 10 use x + y <= 50 and for x = 11 to x = 20 use 10 x +10 y <= 30). When choosing a satisfactory pair (x , y), I plug in x to the equation appropriate for the x interval, and I can choose any y value within the bounds for y.

Answer & Explanation

Dwayne James

Dwayne James

Beginner2022-06-17Added 18 answers

In 3 dimensions, the region is the intersection of three closed halfspaces. Generically, they form a triangular "pyramid without a base" (i.e., a triangular cone), but there are exceptional cases such as "triangular cylinder", if the rank of A is 2. I think your attempt to rewrite A x b in a "simpler" form is not going very far in higher dimensions. You are trading one system of inequalities for another, condition-laden one.
So, answer is: it's good that you worked through this in two-dimensional case, but do not try to make A x b more "explicit" in higher dimensions: it's not going to work.

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