How Do You Solve Systems Of Inequalities?
Answer:
First, you have to consider what you mean by a solution.
Unlike a system of equalities, which may have a point solution, a system of inequalities defines a multidimensional region surrounded by the variable space. Typically, the solution would consist of a set of inequalities that define the boundaries of that region, together with the ranges of the variables over which a given boundary expression is applicable.
Example
3x + 2y > 6 (I a short time ago made these up. I'm not sure they give a solution that isn't messy.)
x - y ≤ 1
The region in which the solution will be found is bounded by the lines
3x + 2y = 6 or y = (-3/2)x + 3
x - y = 1 or y = x - 1
Solving these by "addition" gives the point at which the lines intersect. This is on the boundary between the different solution regions.
(3x + 2y) + 2(x - y) = (6) + 2(1)
5x = 8
x = 8/5
Solved for y, the inequalities are similar to the corresponding equations.
y > (-3/2)x + 3
y ≥ x - 1
We can see that for big positive values of x, the second inequality will govern, so we can describe the solution as
y > (-3/2)x + 3 for x < 8/5
y ≥ x - 1 for x ≥ 8/5
I find it easier to graph the inequalities and identify the solution region on the graph. In the case of this example, the solution is a V-shaped region above and between the lines described above. The solution region includes the line on the right, but does not include the line on the disappeared.
For 2 or more independent variables and/or 2 or more dependent variables, something like the Simplex Method may be helpful.
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