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 must be satisfied simultaneously. If the image is not visible, refer to the textbook for the specific inequalities.
2. Step 2: Rewrite each inequality in slope-intercept form (y = mx + b) if necessary. This makes it easier to graph the inequalities. 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 that satisfies each inequality. The shading direction depends on the inequality sign: above the line for '>' or '≥', and below the line for '<' or '≤'.
4. Step 4: Identify the overlapping shaded region, which represents the solution set of the system of inequalities. If there is no overlapping region, the system has no solution.
5. Step 5: If the problem involves a piecewise function, graph each segment of the function separately based on its domain restrictions. Ensure that the graph accurately reflects the behavior of the function within each specified interval.

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 to visualize their solution sets. This includes determining boundary lines, using dashed or solid lines to indicate whether points on the line are included, and shading the appropriate regions. Mastery of these techniques is essential for accurately representing the solution set.

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 affect the boundaries of the solution set in a system of inequalities. Recognizing the conditions under which each piece applies is key to solving related problems.