Problem 42 - Algebra and Trigonometry

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} {x+y>3} \\ {x+y<-2} \end{array}\right. $$

Step-by-Step Solution

Answer

The system of inequalities has no solution because there is no region in the plane that satisfies both inequalities concurrently.

Step 1: Graph the first inequality

To graph the inequality \(x + y > 3\), first consider the equal part, i.e., the line \(x + y = 3\). This line divides the plane into two parts. Test a point not on the line (e.g., the origin (0,0)) in the original inequality. The origin doesn't satisfy the inequality, so the solution set for \(x + y > 3\) is the region of the plane on the side of the line opposite to the origin.

Step 2: Graph the second inequality

Similarly, to graph the inequality \(x + y < -2\), first consider the equal part, i.e., the line \(x + y = -2\). This line also divides the plane into two parts. Test a point not on the line (e.g., the origin (0,0)) in the original inequality. The origin satisfies the inequality, so the solution set for \(x + y < -2\) is the region of the plane on the same side of the line as the origin.

Step 3: Identify the intersection

The solution to the system of inequalities is the region where the solution sets to both inequalities overlap. However, in this case, the two regions identified in Step 1 and Step 2 do not intersect. The regions are on opposite sides of the lines corresponding to the inequalities.

Key Concepts

Graphing Inequalities

Understanding how to graph inequalities is the first step toward solving a system of inequalities. Each inequality splits the coordinate plane into two parts. Take, for example, the inequality \(x + y > 3\). This inequality can be initially graphed as the line \(x + y = 3\). This line is crucial because it forms the boundary of the solution region. However, the solution also involves determining which side of the line contains all the points that satisfy the inequality.
To do this, select a test point; a common choice is the origin point \((0,0)\) because it simplifies calculations. Plug in this point to see if it satisfies the inequality. If it doesn’t, then the solution set does not include the side of the line where the origin is located. Hence, for \(x + y > 3\), the solution region is above the line \(x + y = 3\). The line itself generally isn't part of the solution; thus, we use a dashed line to depict this in the graph.

In the same way, graphing \(x + y < -2\) involves first drawing the line \(x + y = -2\). Conduct a similar test using the origin. If it satisfies the inequality, the solution set is on the same side as the origin. For \(x + y < -2\), this means below the line. Again, use a dashed line for the boundary since it is not part of the inequality.

Solution Set

The solution set of a system of inequalities encompasses all the possible points that satisfy all inequalities simultaneously. If you have just one inequality, the solution set is straightforward: it is the region that is on one side of the line you have graphed based on your inequality. Once you have more than one inequality, finding the solution set involves determining where these individual regions overlap after they have been graphed.
For instance, consider our inequalities \(x + y > 3\) and \(x + y < -2\). Each one has its own region of solutions:

• \(x + y > 3\) has a solution region above its line.
• \(x + y < -2\) has its solution below its line.

These solution sets are visualized on the graph as shaded regions. The final solution set is ideally where these two shaded regions overlap. This overlap is crucially important as it represents the points satisfying all conditions in our inequality system.

Inequality Intersection

When solving a system of inequalities, finding the intersection of the solution sets is the main goal. This intersection is where all given inequalities share common ground and are simultaneously true. If there is no intersection, the system has no solution, meaning there's no set of values meeting all inequalities.

With our example system \(x + y > 3\) and \(x + y < -2\), visualize the graph. Each inequality region faces away from the other, creating areas that are mutually exclusive:

• In this case, the above-the-line region for \(x + y > 3\) never reaches the below-the-line region for \(x + y < -2\).
• Thus, we have no overlap.

This signifies that the system of inequalities lacks a solution region, as their graphically represented solution areas do not intersect. Understanding the non-intersection concept is just as valuable as finding an intersection in a system because it helps confirm no shared solutions exist.