To determine which additional inequality can be added to the system without changing the solution set, we need to analyze the given inequalities and understand their solution regions.
Given inequalities:
1. [tex]\( y \leq 0.5x + 2 \)[/tex]
2. [tex]\( y > 3x - 3 \)[/tex]
Let's analyze these step-by-step:
### Step 1: Graphing the Inequalities
Inequality 1: [tex]\( y \leq 0.5x + 2 \)[/tex]
This is a linear inequality with a slope of 0.5 and a y-intercept of 2. This line divides the plane into two regions, and we are interested in the region below the line (as indicated by [tex]\( \leq \)[/tex]).
Inequality 2: [tex]\( y > 3x - 3 \)[/tex]
This is another linear inequality with a steeper slope of 3 and a y-intercept of -3. This line also divides the plane into two regions, and we are interested in the region above this line (as indicated by [tex]\( > \)[/tex]).
### Step 2: Finding the Intersection
The solution set of the system is the region where the shaded areas of the two inequalities overlap.
### Step 3: Considering Additional Inequalities
We need to analyze how each of the proposed inequalities interacts with the intersection of the given inequalities.
Option 1: [tex]\( y > 2 \)[/tex]
- This inequality would exclude part of the shaded region already defined by [tex]\( y \leq 0.5x + 2 \)[/tex], specifically the portion where [tex]\( y \leq 2 \)[/tex].
- Therefore, adding [tex]\( y > 2 \)[/tex] would change the solution set.
Option 2: [tex]\( y < 3 \)[/tex]
- This inequality encompasses the entire solution region of the given system because the y-values within the intersection of [tex]\( y \leq 0.5x + 2 \)[/tex] and [tex]\( y > 3x - 3 \)[/tex] naturally fall below [tex]\( y < 3 \)[/tex].
- Therefore, [tex]\( y < 3 \)[/tex] would not alter the solution set.
Option 3: [tex]\( y < 2 \)[/tex]
- This inequality would exclude parts of the solution set that fall between [tex]\( y = 2 \)[/tex] and [tex]\( y = 0.5x + 2 \)[/tex].
- Therefore, adding [tex]\( y < 2 \)[/tex] would change the solution set.
Option 4: [tex]\( y = 3 \)[/tex]
- This equality would restrict the solution set to only the points on the line [tex]\( y = 3 \)[/tex], which is not part of the original solution set since it does not lie in the overlapping region.
- Therefore, adding [tex]\( y = 3 \)[/tex] would change the solution set.
### Conclusion
The inequality that can be added to the system without changing the solution set is:
[tex]\[ y < 3 \][/tex]
Therefore, the correct answer is [tex]\( \boxed{y < 3} \)[/tex]
