To identify the graph that represents the given system of inequalities, graph each inequality separately and find the overlapping region. Two ordered pairs that are solutions to the system are (0, 3) and (2, 7).
The system of inequalities y ≤ x + 5 and y ≤ 2x + 3 represents shaded areas below the lines y = x + 5 and y = 2x + 3, respectively.
The graph of the system would show the shaded region where both inequalities are simultaneously true, which is the area below both lines shaded in common.
Here are two ordered pairs that satisfy this system:
- For y ≤ x + 5 and y ≤ 2x + 3, one solution is the point of intersection between the lines y = x + 5 and y = 2x + 3.
- Solving x + 5 = 2x + 3:
x = 2
- Substituting x = 2 into y = x + 5:
y = 2 + 5 = 7
- So, one solution is (2, 7).
- Another point that satisfies both inequalities is a point in the shaded region below both lines, for instance, (0, 3), which lies in the shaded region where both y ≤ x + 5 and y ≤ 2x + 3 are true.
Question:
Identify the graph that represents the given system of inequalities. Also identify two ordered pairs that are solutions to the system.
y ≥ x + 5
y ≥ 2x + 3