Solving Systems of Inequalities

We recently learned how to solve systems of inequalities; as opposed to systems of equations, whose solution was the intersection of the lines involved, the solution of a system of inequalities is the graph itself.

Here is a very simple example:

x > 3

y > 5

We can see that x is greater than 3 and y is greater than 5.

To graph this, we can locate the point on the x-axis where x = 3 and draw a vertical line through it. Then we can locate the point on the y-axis where y = 3 and draw a horizontal line through it.

The type of line we need to draw is dependent on the inequality.

1. If the inequality has a less than or greater to sign, we need to draw a dashed line because the region does not include the line itself. 
2. If the inequality has a less than or equal to sign or a greater than or equal to sign, we need to draw a solid line because the region does include the line itself.

Since both of the inequalities contain a greater than sign, both lines will be dashed.

To determine which side of the line to shade, we can substitute any ordered pair into the inequality, and, if the ordered pair satisfies it, we shade that side. If it does not, we shade the opposite side.

The easiest ordered pair to substitute is usually (0, 0), so that is what we will use.

0 > 3 (false)

0 > 5 (false)

Now we see that neither of these inequalities have been satisfied, so we must shade the side that does not include (0, 0).

Points of intersection of the lines are called vertices, and these will be useful in linear programming. Nevertheless, the solution to the system of inequalities is the region which all inequalities in the system include.
Any point within this region is a valid solution to the system.
Therefore, for our example problem, any point whose x position is greater than 3 and whose y position is greater than 5 will be considered a solution.