To determine which graph represents a system of inequalities with no solutions, we should look for overlapping regions where the inequalities contradict each other.
-
Graphs with Solid Lines: Inequalities that have solid lines (typically "≤" or "≥") indicate that points on the line are part of the solution set. Therefore, at least part of the solution region is included.
-
Graphs with Dashed Lines: Inequalities that have dashed lines (typically "<" or ">") mean that the points on the line itself are not part of the solution set. The solution could either be above or below the dashed lines.
In a system of inequalities with no solutions, the regions defined by the inequalities must be entirely separate, with no overlap whatsoever.
From the given options (which generally describe systems with various types of lines), look for:
- Two lines that intersect but define opposite regions (one line shading above the line while the other shades below). This could indicate no solutions as there can be no point that satisfies both inequalities simultaneously.
- If the graphs only show separate sections without any possibility of overlap, those would also imply no solutions.
Without images or further details about which regions are shaded in each graph, look for these patterns of contradiction:
- Opposing shade areas (one area shaded above one line and one area shaded below another line).
- Clear separation of the possible solution regions indicated by the dashed or solid lines.
Given this analysis, you should choose the graph that clearly has inequalities that lead to mutually exclusive regions—indicating that no point can satisfy both inequalities, meaning there is no solution to that system.