Graphing Systems of Inequalities
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If you can graph an inequality, you can graph a system of inequalities!

Simply graph each inequality separately on the same set of axes.  The area where the shadings overlap is the solution to the system of inequalities. 
 

To Graph a System of  Inequalities

1.  Graph each inequality separately.  The directions for graphing inequalities can be found in "Graphing Inequalities".

2.  The solution to the system, will be the area where
the shadings from each inequality overlap one
another.

Graph the following  system of inequalities 

 

Graph the system

  Graph each inequality as if it was stated in "y=" form.

  If the inequality is < or > a dotted line must be used to represent the line.  If the inequality is < or >, a solid line is used.

  Choose a test point to determine which side of the line needs to be shaded.  The test point for this problem was (0,0).  Always pick a point that is easy to work with.

For the test point (0,0),

0 < 2(0)-3  False

0 (-2/3)0+2 False
Since both equations were false, shading occurred on the other side of the line, not covering the point.

The solution, S, is where the two shadings overlap one another.

 



Applied Example Using a System of Linear Inequalities
The "We Sell CDs" website plans to purchase ads in a local newspaper advertising their site.  Their operating budget will allow them to spend at most $2200 on this advertising adventure.  They plan to run at most 20 ads.  An ad will cost $50 to appear in the weekday paper and $200 to appear in a weekend edition.  Prepare a graph that will represent all of the possible combinations of ads under these conditions.
Solution:
Let x = the number of weekend ads
Let y = the number of weekday ads

x + y < 20  (there will be at most 20 ads)
200x + 50y < 2200 
(the cost of the ads at most $2200)

For this problem x and y cannot be negative numbers, so the answer will be in the first quadrant only.
Solve each of the inequalities above for the y value.
Using your graphing calculator, graph
Y1 = 20 - (with icon set to "shade below")
Y2 = (2200-200x)/50  ("shade below" icon set)

Set the window to view only the first quadrant.
Use the intersect option to find the vertices of the quadrilateral whose area makes up the pool of answers (the overlapping shaded region).
 

 

See how to use your
TI-83+/84+ graphing calculator  with systems of inequalities.
Click calculator.