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Graphing the Feasible Region of a System of Inequalities

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Instructor: Elizabeth Foster

Elizabeth has been involved with tutoring since high school and has a B.A. in Classics.

The feasible region of a system of inequalities is graphed by including all inequalities located within the system. Learn about the feasible region, how to graph a system of inequalities, and learn how to test the feasible region. Updated: 11/23/2021

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 lifeā€¦ 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.

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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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Practice:
Graphing the Feasible Region of a System of Inequalities Quiz

Instructions: Choose an answer and click 'Next'. You will receive your score and answers at the end.

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The lines x = 10, y = -4, and y = x are shown on the graph below. Which of the colored regions represents the feasible region of the three inequalities x < 10, y > -4, and y < x? Choose the answer that represents the ENTIRE feasible region, but does not cover any areas outside the feasible region.

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