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Lesson 134: Graphing Systems of Linear Inequalities

In this very important lesson we'll learn more about graphing systems of linear inequalities.  All we have to do is graph two different inequalities on the same coordinate plane, and see where they intersect.  It is easy once you know what to do.

Before starting this lesson, make sure that you review all of the previous lessons on graphing linear equations and graphing linear inequalities.  Make sure that you fully understand them, or you will have difficulty with this lesson.   

Very often we are asked to graph a system of linear inequalities.  Sometimes this is referred to as solving a system of linear inequalities graphically.  We're given two linear inequalities, just like the kind we've been working with.  What we're asked to do is graph both of them on the same coordinate plane, and determine the region in which they intersect.  All of the coordinates of that region satisfy both inequalities simultaneously.  The x and y values will work in both inequalities.  Sometimes we refer to these problems as a system of simultaneous inequalities.  We to see the region of x and y values that work in both inequalities at the same time. 

Graphing Systems of Linear InequalitiesOn the graph at left, I've graphed two different linear inequalities like we've learned to do.  In blue we have y > 3x - 2, and in red we have y ≤ (-1/2)x + 1.  Review the previous lessons on graphing linear inequalities if you're not sure how I graphed these lines. 

Now that they are graphed, I can see the region in which they intersect, and I've labeled it with a big S, for "solution set".  That means that any point in the solution set will work in both inequalities.  If you substitute a point in that region into each inequality, you will see that it works. 

Note that if you are using graph paper and pencil to do your graphing, you'll need to be very careful.  It certainly can get a bit messy looking, and it's harder if you don't have colored pencils.  Just shade in the area of one inequality using lines in one direction, and the area of the other inequality using lines in the other direction, just like I've done.  The "cross-hatched" area will be your solution set, which you should label with a big S. 

There is a special case that we need to be aware of.  It is possible that we could have two inequalities whose graphs do not overlap at all.  For example, if it easy to visualize that the graphs of y > 4 and y < 1 will not intersect.  There is no possible value of y that at the same time is both greater than 4, and less than 1. We can say that the solution to such a system of inequalities is the empty set, often written as { }.  In this case, there would be no large S on your graph.

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