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Consider the solution to the system of inequalities below.

[tex]\[
\left\{\begin{array}{l}
y \leq 0.5x + 2 \\
y \ \textgreater \ 3x - 3
\end{array}\right.
\][/tex]

Which of the following functions could be added to the system without changing the solution set?

A. [tex]\( y \ \textgreater \ 2 \)[/tex]
B. [tex]\( y \ \textless \ 3 \)[/tex]
C. [tex]\( y \ \textless \ 2 \)[/tex]
D. [tex]\( y = 3 \)[/tex]

Sagot :

GinnyAnsweridn
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]
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