In Exercises 46–55, graph the solution set of each system of inequalities or indicate that the system has no solution.
This
is a piecewise function. Refer to the textbook.

Verified step by step guidance

1. Step 1: Identify each inequality in the system and rewrite them in slope-intercept form (y = mx + b) if necessary.
2. Step 2: Graph each inequality on the same coordinate plane. Use a solid line for inequalities that include equality (≤ or ≥) and a dashed line for strict inequalities (< or >).
3. Step 3: Determine the region that satisfies each inequality by testing a point (usually the origin (0,0) if it is not on the line) and shading the appropriate side of the line.
4. Step 4: Identify the intersection of the shaded regions. This common area represents the solution set of the system of inequalities.
5. Step 5: If there is no common shaded region, then the system has no solution.

Key Concepts

Systems of Inequalities

A system of inequalities consists of two or more inequalities that share the same variables. The solution set is the region where the graphs of these inequalities overlap. Understanding how to graph each inequality and identify the feasible region is crucial for solving these systems.

Graphing Techniques

Graphing techniques involve plotting inequalities on a coordinate plane. This includes determining boundary lines, using dashed or solid lines to indicate whether points on the line are included, and shading the appropriate regions to represent the solution set. Mastery of these techniques is essential for visualizing and solving systems of inequalities.

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. Understanding how to interpret and graph piecewise functions is important, as they can represent complex relationships in systems of inequalities. Recognizing the conditions under which each piece applies helps in accurately graphing the overall function.