Final answer:
The correct function that could be added to the system of inequalities without changing the solution set is Option (a) y > 2, which sets a horizontal boundary below the existing inequality boundaries.
Step-by-step explanation:
The student asked which function could be added to the system of inequalities 1. 0.5x + 2 ≤ y and 2. y > 3x - 3 without changing the solution set. To discern this, we must consider the shape and position of the potential boundaries added by these new functions compared to the existing system of inequalities.
Option (a) y > 2 provides a horizontal line which is always below lines with positive slopes, like 0.5x + 2 for x > 0 and 3x - 3 for x > 1. Since the original system requires y to be greater than both these lines, option (a) does not alter the solution set significantly.
In contrast, Option (b) y < 3 would add a horizontal line above the point (0,2) but below (0,3) which may exclude some solutions of the original system.
Option (c) y < 2 imposes a line below the existing boundary 0.5x + 2 ≤ y, potentially excluding all current solutions above y = 2. This would change the solution set.
Finally, Option (d) y = 3 imposes a hard equality constraint that would significantly alter the solution set that currently allows y to be above a certain value determined by the original inequalities.
Therefore, the correct option that can be added to the system without altering the solution set is Option (a) y > 2.