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: Begin by analyzing the system of inequalities provided in the problem. A system of inequalities typically consists of two or more inequalities that need to be solved simultaneously. If the image is not visible, refer to the textbook for the specific inequalities and their corresponding graphs.
2. Step 2: Rewrite each inequality in slope-intercept form (y = mx + b) if necessary. This will make it easier to graph each inequality. For example, if an inequality is given as Ax + By ≤ C, solve for y to express it in terms of x.
3. Step 3: Graph each inequality on the coordinate plane. Use a solid line for inequalities that include equality (≤ or ≥) and a dashed line for strict inequalities (< or >). Shade the region of the graph that satisfies the inequality. For example, if the inequality is y > mx + b, shade above the line.
4. Step 4: Identify the overlapping shaded regions from all the inequalities in the system. The solution set of the system is the region where all shaded areas intersect. If there is no overlapping region, the system has no solution.
5. Step 5: Verify the solution by selecting a test point within the overlapping region (if it exists) and substituting it into all inequalities to ensure it satisfies the system. If the test point works for all inequalities, the solution set is correct.

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.