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Use the image to answer the question.

A coordinate plane shows a graph of an upward-sloping, diagonal, dashed line, with shading extending from the line to the left. The line starts in quadrant 3, extends through quadrant 4, and ends in quadrant 1.

Is the point (0,−3) a solution to the inequality y>45x−3 , as shown on the graph?

(1 point)
Responses

No, because (0,−3) falls on the boundary line.
No, because left parenthesis 0 comma negative 3 right parenthesis falls on the boundary line.

No, because (0,−3) falls within the shaded region.
No, because left parenthesis 0 comma negative 3 right parenthesis falls within the shaded region.

Yes, because (0,−3) falls on the boundary line.
Yes, because left parenthesis 0 comma negative 3 right parenthesis falls on the boundary line.

Yes, because (0,−3) falls within the shaded region.

1 answer

To determine if the point (0, -3) is a solution to the inequality \( y > 45x - 3 \), we need to analyze both the equation and the graph described.

  1. The line \( y = 45x - 3 \) is dashed, indicating that the points on the line are not included in the solution (i.e., it is an inequality where \( y \) is greater than, not equal to).
  2. The region where \( y > 45x - 3 \) is shaded above the line, while the region below the line is shaded towards the left.

Now, let's check the coordinates of the point (0, -3):

  • Plugging \( x = 0 \) into the equation \( y = 45(0) - 3 \) gives \( y = -3 \).
  • The point (0, -3) is therefore exactly on the line \( y = 45x - 3 \).

Since the inequality excludes the boundary line (because it is dashed), (0, -3) cannot be a solution to the inequality \( y > 45x - 3 \).

Based on this reasoning, the correct response is:

No, because (0,−3) falls on the boundary line.