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Which of the following graphs represents the solution set of the system of inequalities?

−12x+y≥−1

y<2x

Use the image to answer the question.

(1 point)
Responses

The coordinate grid has x and y axes labeled from negative 5 to 5.
Image with alt text: The coordinate grid has x and y axes labeled from negative 5 to 5.

The coordinate grid has x and y axes labeled from negative 5 to 5.
Image with alt text: The coordinate grid has x and y axes labeled from negative 5 to 5.

The coordinate grid has x- and y axes labeled from negative 5 to 5.
Image with alt text: The coordinate grid has x- and y axes labeled from negative 5 to 5.

The coordinate grid has x and y axes labeled from negative 5 to 5.
Image with alt text: The coordinate grid has x and y axes labeled from negative 5 to 5.
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1 answer

To determine which graph represents the solution set of the system of inequalities, we can analyze each inequality:

  1. For the inequality \( −12x + y \geq −1 \):

    • Rearranging gives us \( y \geq 12x - 1 \).
    • This line has a slope of \( 12 \) and a y-intercept of \( -1 \).
    • The region above (or on) this line will be included in the solution set (since it's a "greater than or equal to" inequality).

  2. For the inequality \( y < 2x \):

    • This line has a slope of \( 2 \) and passes through the origin (0,0).
    • The region below this line will be included in the solution set (since it’s a "less than" inequality).

Steps to Find the Intersection:

  • Plot the line \( y = 12x - 1 \):
    • Find two points:
      • When \( x = 0 \), \( y = -1 \) (point (0, -1)).
      • When \( x = 1 \), \( y = 11 \) (point (1, 11)).
  • Plot the line \( y = 2x \):
    • Find two points:
      • When \( x = 0 \), \( y = 0 \) (point (0, 0)).
      • When \( x = 1 \), \( y = 2 \) (point (1, 2)).

Identify the Valid Region:

  • Shade the area above the line \( y = 12x - 1 \).
  • Shade the area below the line \( y = 2x \).
  • The solution set will be in the region where the shading overlaps.

Conclusion:

Look at the provided graphs to find one that has a shaded area above the line \( y = 12x - 1 \) and below the line \( y = 2x \). Since you indicated that there are images but did not provide the actual visuals for me to assess, you should choose the graph that matches this description.

If you provide temperature-specific details about each graph, I could help you narrow it down even further!