Defining Solutions to Systems of Linear Inequalities

Learning Outcomes

  • Identify solutions to systems of linear inequalities

Previously, we learned how we can plug an ordered pair into a system of equations to determine whether it is a solution to the system.  We can use this same method to determine whether a point is a solution to a system of linear inequalities.

Determine whether an ordered pair is a solution to a system of linear inequalities

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On the graph above, you can see that the points B and N are solutions for the system because their coordinates will make both inequalities true statements.

In contrast, points M and A both lie outside the solution region (purple). While point M is a solution for the inequality [latex]y>−x[/latex] and point A is a solution for the inequality [latex]y<2x+5[/latex], neither point is a solution for the system. The following example shows how to test a point to see whether it is a solution to a system of inequalities.

Example

Is the point [latex](2, 1)[/latex] a solution of the system [latex]x+y>1[/latex] and [latex]2x+y<8[/latex]?

Here is a graph of the system in the example above. Notice that [latex](2, 1)[/latex] lies in the purple area, which is the overlapping area for the two inequalities.

Two dotted lines, one red and one blue. The region below the blue dotted line is shaded and labeled 2x+y is less than 8. The region above the dotted red line is shaded and labeled x+y is greater than 1. The overlapping shaded region is purple and is labeled x+y is greater than 1 and 2x+y is less than 8. The point (2,1) is in the overlapping purple region.

Example

Is the point [latex](2, 1)[/latex] a solution of the system [latex]x+y>1[/latex] and [latex]3x+y<4[/latex]?

Here is a graph of this system. Notice that [latex](2, 1)[/latex] is not in the purple area, which is the overlapping area; it is a solution for one inequality (the red region), but it is not a solution for the second inequality (the blue region).

A downward-sloping bue dotted line with the region below shaded and labeled 3x+y is less than 4. A downward-sloping red dotted line with the region above it shaded and labeled x+y is greater than 1. An overlapping purple shaded region is labeled x+y is greater than 1 and 3x+y is less than 4. A point (2,1) is in the red shaded region, but not the blue or overlapping purple shaded region.

You can watch the video below for another example of how to verify whether an ordered pair is a solution to a system of linear inequalities.

Summary

  • Solutions to systems of linear inequalities are entire regions of points.
  • You can verify whether a point is a solution to a system of linear inequalities in the same way you verify whether a point is a solution to a system of equations.