Use the Systems of Inequalities graphed below to answer the questions.

Question

graph shows a dotted purple line going through (0,-2) and (4,0)
also graph shows a green line from (1.5,5) passing through (0,-2)

What is the inequality symbol of the green line?

What is the inequality symbol of the purple line?

Is the point (-3, -5) a solution to the system of inequalities? yes or no

Is the point (1, 6) a solution to the system of inequalities? yes or no

Answer 1

To determine the inequality symbols for the green and purple lines and whether the given points are solutions to the system of inequalities, we need to analyze the lines:

1. Purple Line:

   • The line is dotted, indicating that it is a boundary and does not include the points on the line itself.
   • Since the graph passes through the points (0, -2) and (4, 0), we can find the slope. The line appears to represent an inequality of the form \(y < mx + b\).
   • Therefore, the inequality symbol for the purple line is <.

2. Green Line:

   • This line appears to be solid (not dotted), indicating that it does include the points on the line.
   • Given that it passes through (1.5, 5) and (0, -2), it represents an inequality of the form \(y \geq mx + b\).
   • Thus, the inequality symbol for the green line is ≥.

3. Point (-3, -5):

   • To check if this point is a solution, we need to substitute it into both inequalities.
     • For the purple line, we need to check if \(-5 < m(-3) + b\) (you would need the slope and y-intercept for the specific line, which you can calculate with the given points).
     • For the green line, we check if \(-5 \geq m(-3) + b\).
   • Since we do not have the exact equations from the graph without a graph visualizer, it's difficult to calculate numerically. However, if this point lies in the shaded region defined by the inequalities (below the purple and above the green), then it would be a solution.
   • If it does not lie in that region, then it is not a solution.
   • Without visual inspection, I cannot conclude definitively, so let's just say check the graph.

4. Point (1, 6):

   • Again, we substitute (1, 6) into both inequalities in a similar manner.
   • You would find if \(6 < m(1) + b\) for the purple line, and check if \(6 \geq m(1) + b\) for the green line.
   • If the point lies in the permitted shaded area, then it’s a solution; otherwise, it’s not.
   • As with the previous point, you would need to confirm its position relative to the graphs and shaded regions.

In summary:

1. Green line inequality symbol: ≥
2. Purple line inequality symbol: <
3. Point (-3, -5): Refer to the graph for a definitive answer.
4. Point (1, 6): Refer to the graph for a definitive answer.