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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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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.

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. 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.

3 answers

To determine which graph represents a system of inequalities with no solutions, we need to identify if there are any regions where the inequalities contradict each other, leaving no area where they overlap.

Typically, graphs of systems of inequalities consist of half-planes (the regions above or below lines) defined by linear equations. If two lines are solid, they represent inequalities that include the line (e.g., ≤ or ≥). If the lines are dashed, they represent inequalities that do not include the line (e.g., < or >).

Here’s how to assess the options:

  1. A system with two lines that overlap completely but have different inequality directions would leave no possible solutions (e.g., one inequality represents the area above the line while the other represents below the same line).

  2. A graph with two non-intersecting lines (one on top of the other) would indicate that any potential solution area is entirely excluded.

From your description, the key is to look for two lines (either solid or dashed) that lead to a contradiction in potential solution areas.

Since I don't have visual access to the graphs, my suggestion would be to:

  • Identify a pair of lines (if they are present) where the regions they shade do not overlap at all.
  • Also note if there are any situations where the inequalities suggest impossible conditions (for example, one region above a line and another below the same line).

Based on your descriptions, the graph characterized by two solid lines or two dashed lines that contradict each other, possibly appearing parallel or conflicting in shading would be the graph representing a system of inequalities with no solutions.

Evaluate each option carefully based on these principles, focusing on the shading provided by the inequalities and their respective boundaries, as those are crucial in determining solution viability.

Graph One contains the following equations:
y is greater than or equal to 2x - 1
y is less than or equal to 3x - 2
Graph Two contains the following equations:
y is greater than or equal to 2x - 1
y is greater than or equal to 2x + 1
Graph Three contains the following equations:
y > x + 2
y < x - 2
Graph Four contains the following:
y is less than or equal to x + 2
y is greater than or equal to x + 2

To determine which of the given graphs corresponds to a system of inequalities with no solutions, let's analyze each system of inequalities:

  1. Graph One:

    • \( y \geq 2x - 1\) (This represents the region above or on the line \(y = 2x - 1\).)
    • \( y \leq 3x - 2\) (This represents the region below or on the line \(y = 3x - 2\).)

    These two lines intersect, and there is a region where both inequalities are satisfied (i.e., a solution exists).

  1. Graph Two:

    • \( y \geq 2x - 1\) (This represents the region above or on the line \(y = 2x - 1\).)
    • \( y \geq 2x + 1\) (This represents the region above or on the line \(y = 2x + 1\).)

    Since \(y = 2x + 1\) is above \(y = 2x - 1\), this system of inequalities has a solution set, as both inequalities indicate values above these lines. Thus, solutions exist.

  1. Graph Three:

    • \( y > x + 2\) (This represents the region above the line \(y = x + 2\).)
    • \( y < x - 2\) (This represents the region below the line \(y = x - 2\).)

    The two lines \(y = x + 2\) and \(y = x - 2\) are parallel, with the second line located below the first line. Since one inequality is about the area above the first line and the other is below the second line without overlapping, there are no solutions.

  1. Graph Four:

    • \( y \leq x + 2\) (This represents the region below or on the line \(y = x + 2\).)
    • \( y \geq x + 2\) (This represents the region above or on the line \(y = x + 2\).)

    The only point that satisfies both inequalities is the line itself (\(y = x + 2\)), which indicates there are solutions, even if it's just the boundary line.

Conclusion: Graph Three (with the inequalities \(y > x + 2\) and \(y < x - 2\)) represents a system of inequalities with no solutions, as the regions defined by these inequalities do not overlap.