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

Question

Responses

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.
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.

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.

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.

Answer 1

A system of inequalities has no solutions when the regions defined by the inequalities do not overlap at any point. This typically occurs when the constraints represented by the inequalities create separate, non-intersecting areas in the graph.

To determine which of the presented graphs represents a system of inequalities with no solutions, look for the one where two lines have been drawn in such a way that no region allowed by one inequality intersects with the region allowed by the other inequality. This can be visually evident if the lines are in direct opposition to each other, such as one line defining a region above it while the other line defines a completely separate region below it, without any shared area.

In the options presented:

1. Two solid lines - This could potentially represent overlapping regions if they intersect or if the inequalities do allow for a shared area.

2. One solid line - This usually means there is at least one region possible since a single line with inequalities will create a half-plane.

3. Two solid lines (first option and third option) - Similar to the first option, these could represent overlapping areas depending on how the inequalities are set.

4. Two dashed lines with four points - Dashed lines indicate that the lines are not included in the solution, which can often lead to regions that do not touch or overlap, possibly leading to no solutions, especially if the regions defined by the inequalities are separate.

Without the exact visual context, it seems that the graph with two dashed lines and four points is the most likely candidate for a system of inequalities with no solutions. The dashed lines suggest that the related inequalities do not allow for any overlap, especially if the dashed lines are oriented in such a way that they define regions that do not intersect.