Exercise 69 Page 155

The system's solution set will be the intersection of the shaded regions in the graphs of (I) and (II).

Boundary Lines

We can tell a lot of information about the boundary lines from the inequalities given in the system.

• Exchanging the inequality symbols for equals signs gives us the boundary line equations.
• Observing the inequality symbols tells us whether the inequalities are strict.
• Writing the equation in slope-intercept form will help us highlight the slopes m and y-intercepts b of the boundary lines.

Let's find each of these key pieces of information for the inequalities in the system.

Great! With all of this information, we can plot the boundary lines.

Shading the Solution Sets

We need to determine on which side of the plane the solution set of each inequality lies. To do that, we will need a test point that does not lie on either boundary line.

It looks like the point ( 0, 0) would be a good test point. We will substitute this point for x and y in the given inequalities and simplify. If the substitution creates a true statement, we shade the same region as the test point. Otherwise, we shade the opposite region.

For Inequality (I) we will shade the region opposite our test point, or above the boundary line. For Inequality (II), however, we will shade the region containing the test point, or below the boundary line.

To identify a solution to this system of inequalities, we can choose any point that lies within the overlapping shaded region. Above we have selected the lattice point (0,3). Note that this is only one of infinitely many possible coordinate pairs that are solutions to the system.