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Determine which of the graphs is the graph of a system of inequalities with no solutions.(1 point)

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Image with alt text: Two dashed lines and four points are graphed on a coordinate plane. The x-axis ranges from negative 6 to 30 in increments of 2. The y-axis ranges from negative 5 to 15 in increments of 1.

Image with alt text: A Cartesian coordinate plane's axes range from negative 4 to 4, both by 1-unit increments. 2 solid lines with arrows at both ends are plotted, and various regions of the graph are marked.

Image with alt text: A Cartesian coordinate plane's axes range from negative 4 to 4, both by 1-unit increments. 2 solid lines with arrows at both ends are plotted, and various regions of the graph are marked.

Image with alt text: A Cartesian coordinate plane's axes range from negative 4 to 4, both by 1-unit increments. A solid line with arrows at both ends is plotted, and various regions of the graph are marked.

1 answer

To determine which graph represents a system of inequalities with no solutions, you need to look for a graph where the regions defined by the inequalities do not overlap at all.

In a system of inequalities, if there is no region where the solutions overlap (meaning there is no set of points that satisfies all the inequalities), then the system has no solution. This can often be observed when:

  1. Two or more inequalities define regions that are completely separated.
  2. The inequalities might be strict and oriented in such a way that they don't intersect.

Review each graph based on these criteria:

  • Graph with two dashed lines and points: If the regions defined by the dashed lines overlap, there may still be solutions.

  • First graph with two solid lines: If the inequalities represented by these lines are such that they create non-overlapping regions, this might be the graph with no solutions.

  • Second graph with two solid lines: Similar to the first, check for overlapping regions.

  • Graph with a single solid line: This graph will likely have solutions, as it may define a single direction defined by one inequality.

Given this reasoning, inspect the images closely, especially the first and second graphs with two solid lines, for any separation. If either shows no possible overlap for the regions defined by the inequalities, that would be your answer. Generally, a system of inequalities with no solutions is characterized by non-overlapping shaded areas created by the visual representation of the inequalities.

Without directly seeing the images, it is difficult to definitively identify the correct graph, but following this logic should help you narrow it down to the one indicating no solutions. If any graph shows two distinct regions completely isolated from each other, that would likely be the correct choice.