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Which of these systems of inequalities is graphed below?

A solid line passing through the points (0, 5), (-5, 0) and another line passing through the points (-10, -5), (5, 10). The intersection of these lines is at (0, 5). The area below (0, 5) and between the lines is shaded.
A.
y ≤ x + 5 and y ≤ 5x + 5

B.
y ≤ x + 5 and y ≥ 5x + 5

C.
y ≥ x + 5 and y ≤ 5x + 5

D.
y ≥ x + 5 and y ≥ 5x + 5

1 answer

To determine which system of inequalities is represented by the graph you described, we need to analyze the information given:

  1. Lines and Points:

    • A solid line passes through the points (0, 5) and (-5, 0). This line can be described by the equation \(y = x + 5\).
    • Another line passes through the points (-10, -5) and (5, 10). This line can be described by the equation \(y = 5x + 5\).

  2. Shading:

    • The area below (0, 5) and between the two lines is shaded. This means that the region of interest is below the first line (which corresponds to a ≤ inequality) and below the second line.

  3. Analyzing the Options:

    • Option A: \(y ≤ x + 5\) and \(y ≤ 5x + 5\) – This matches since it says to shade below both lines.
    • Option B: \(y ≤ x + 5\) and \(y ≥ 5x + 5\) – This would shade below the first line but above the second; does not match.
    • Option C: \(y ≥ x + 5\) and \(y ≤ 5x + 5\) – This would shade above the first line, does not match.
    • Option D: \(y ≥ x + 5\) and \(y ≥ 5x + 5\) – This would shade above both lines; does not match.

Given this analysis, the correct system of inequalities that corresponds to the described graph is:

A. y ≤ x + 5 and y ≤ 5x + 5