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The feasible region of a system of inequalities is the area of the graph containing the points that satisfy all the inequalities at once. Watch this lesson to learn how to graph one.
The Feasible Region
The Feasible Region. It sounds like the title of a futuristic spy flick where everything is green and black and everyone wears sci-fi clothing with no pockets. In a world where global warming has left most of the Earth uninhabitable, a corrupt dictatorship controls the Feasible Region, the only place left that can support life… or is it?
But as fun as that movie might be to watch, in math, the feasible region is actually something else. The feasible region of a system of inequalities is the area of the graph showing all the possible points that satisfy all inequalities.
OK, it's not as dramatic, even if you also put in a green-and-black futuristic color scheme. But knowing how to find the feasible region of a system of inequalities is really useful, so in this lesson we'll walk through how to do it.
Graphing a System of Inequalities
To start, let's review how to graph one inequality. First, replace the inequality sign with an equals sign and graph the line. Then shade the region above or below the line, depending on which values satisfy the original inequality.
Here's a quick example. Let's say we want to graph y > x . First, we'll replace the inequality sign with an equals sign and graph the line y = x .
On one side of this line will be all the points where y is less than x, and on the other side will be all the points where y is greater than x. We just need to figure out which side is which so we'll plug in a test point from each side.
We can see that below the line, y < x and above the line, y > x. So to graph the inequality y > x, we'll shade the area above the line. It's a little bit hard to see, but we'll also make the original line a dashed line, to show that y = x is technically not part of the solution.
When you graph a system of inequalities, you're basically doing that exact same process for several lines on the same graph. So instead of graphing just the points that fit y > x, you might want to graph the points that fit all of the following:
y > x
x > -5
y < 6
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To do this, you'll just graph all the inequalities on the graph at the same time and see where all three regions overlap. If you graph all the inequalities on one graph, you get this.
To help you see the overlap more clearly, the area of y > x is shown in solid red, the area of x > -5 is shown by the blue stripes, and the area of y < 6 is shown by the green dots. The place where all three overlap is the feasible region.
Testing the Feasible Region
The feasible region of this system contains all possible points that satisfy all the inequalities. We can test this by plugging in a bunch of points to see where they fall.
You can plug in as many points as you like, but you'll find the same thing for all of them: the purple region contains all the points that satisfy all three inequalities and only points that satisfy all three of them.
In this lesson, you learned how to find the feasible region of a system of inequalities. The feasible region is the region of the graph containing all the points that satisfy all the inequalities in a system. To graph the feasible region, first graph every inequality in the system. Then find the area where all the graphs overlap. That's the feasible region.
You can check this mathematically. Points that satisfy all the inequalities in the system will be inside the feasible region, but points that don't satisfy all of them will be outside of it. And hey, if you get really good at math, maybe you could make a futuristic crime thriller out of that.
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