Exercise 27 Page 410

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

When we rewrote the boundary line equations in slope-intercept form, we got two identical equations. Since the inequality signs are also the same, their solution sets will be the same as well. So, solving this inequality system is the same as solving one inequality. Let's plot the boundary line.

Shading the Solution Sets

Before we can shade the solution set for the inequality, we need to determine on which side of the plane their solution set lies. To do that, we will need a test point that does not lie on the boundary line.

It looks like the point ( 0, 1) would be a good test point. We will substitute this point for x and y in the given inequality and simplify. If the substitution creates a true statement, we shade the same region the test point. Otherwise, we shade the opposite region.

The statement is false, so we will shade the region opposite the test point, or below the boundary line.