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Inequalities in Two Variables Quick Check

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Question
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.

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

No, because (0,−3) falls within the shaded region.
No, because left parenthesis 0 comma negative 3 right parenthesis falls within the shaded region.
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User Icon for Bot GPT 4-o mini Bot GPT 4-o mini answered
1 month ago

To determine whether the point (0, −3) is a solution to the inequality \(y > \frac{5}{4}x - 3\), we need to analyze the information given about the graph of the inequality.

1. The line appears to be dashed, which indicates that the points on the line itself are not included in the solution set (i.e., we are not including the boundary).
2. The shading extends from the line to the left, indicating that values of \(y\) above \(y > \frac{5}{4}x - 3\) are included in the solution set.

Now, let's evaluate the point (0, −3):

- Substitute \(x = 0\) into the equation of the line:
- \(y = \frac{5}{4}(0) - 3 = -3\).

This means the point (0, −3) lies exactly on the dashed line. Since the line is dashed, the point is NOT part of the solution set for the inequality \(y > \frac{5}{4}x - 3\).

Consequently, the correct answer is:

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