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.