91Ó°ÊÓ

A system of inequalities consists of multiple inequalities that are considered simultaneously. In the context of the problem, we are dealing with two linear inequalities:

• \(2x - y \geq 4\)
• \(4x - 2y < -2\)

Each inequality describes a region in the coordinate plane, and the solution to the system of inequalities is the region that satisfies all the inequalities at once. To solve such systems, we graph each inequality and find the overlapping area that satisfies all the conditions.
When working with systems of inequalities, it is crucial to interpret not just where individual solutions might lie, but also how they interact together on the graph. Sometimes, these systems have overlapping regions, and other times, as in this case, the solutions do not overlap, meaning the system has no solution.

When graphing inequalities, the solution can be either bounded or unbounded. A bounded region is a closed and finite area, meaning the solution set is entirely contained within some boundary. For example, picture a square or a rectangle on a graph.

• Bounded Regions: These are often encountered when all the inequalities in the system intersect to form a closed shape.
• Unbounded Regions: These appear when the inequalities do not confine the solution to a finite space.

In our specific system, when we graph and analyze the inequalities, we find that there is no intersection point within the feasible regions. Therefore, the solution is unbounded because there is no contained solution region; in fact, the system has no solution at all because the shaded areas do not overlap on the graph.

The graphical solution method involves plotting each inequality on the coordinate plane as a boundary line. Begin by turning inequalities into equations by replacing inequality signs with equal signs to graph their corresponding lines.
The process can be broken down as follows:

• Plot the Boundaries: Convert the inequalities to equalities, find their slope-intercept form (\(y = mx + b\)), and draw the straight lines on the graph.
• Shading Regions: After plotting, select a test point that is not on the line (commonly the origin (0,0) unless it lies on the line) to determine which side of each line is part of the solution. Shade the appropriate side that satisfies the inequality.
• Find Overlaps: The final solution is where all shaded regions intersect on the graph.

In the given exercise, due to the nature of the inequalities, the shading did not yield an overlapping region for the intersection of both inequalities, resulting in no solution for the system.