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

Responses

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: 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.

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.

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.

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, we need to look for a situation where the regions defined by the inequalities do not overlap at all.

  1. Look for Dashed Lines: If there are dashed lines, they typically represent inequalities (e.g., "<" or ">"). However, if they do not intersect or overlap, it could indicate no solutions.

  2. Solid Lines with Arrows: Solid lines with arrows indicate boundaries for inequalities (e.g., "≤" or "≥"), which means the areas can overlap.

  3. No Overlapping Regions: The critical component is identifying if there are two regions that are defined by inequalities, which do not intersect at any point.

Given the descriptions of the four options, the first and the latter options involving dashed lines appear to be potential candidates for representing inequalities that have no solution if their respective regions do not overlap.

Assuming from the provided choices:

  • The first option has two dashed lines and points plotted. If those lines separate the regions defined by the inequalities completely, it could indicate no solution.
  • The other options with solid lines suggest that there may be a region where the solutions can exist.

In summary, if the first option depicts two dashed lines that separate areas of the coordinate plane without overlap, it indicates a system of inequalities with no solutions. If the dashed lines present non-overlapping regions, then that choice would be the answer.