Let's analyze each system of inequalities one by one to determine which one has no solution.
1. System 1:
[tex]\[
\begin{array}{l}
x+3 y \geq 0.5 \\
x+3 y \leq 2.5
\end{array}
\][/tex]
This system of inequalities represents the region of the plane that is bounded by the two lines [tex]\(x + 3y = 0.5\)[/tex] and [tex]\(x + 3y = 2.5\)[/tex]. The inequalities indicate that we are considering the points lying between or on these two lines. There is indeed an overlapping region between these two lines. Therefore, this system has solutions.
2. System 2:
[tex]\[
\begin{array}{l}
x+3 y \geq 0.5 \\
x+3 y \geq 2.5
\end{array}
\][/tex]
Here, we have two inequalities [tex]\(x + 3y \geq 0.5\)[/tex] and [tex]\(x + 3y \geq 2.5\)[/tex]. This means we are looking for the region where both conditions are true simultaneously. Given that [tex]\(x + 3y \geq 2.5\)[/tex] is more restrictive, it encompasses the region already covered by [tex]\(x + 3y \geq 0.5\)[/tex] and therefore reduces the feasible set to [tex]\(x + 3y \geq 2.5\)[/tex]. There are indeed solutions that satisfy both inequalities. Thus, this system has solutions as well.
3. System 3:
[tex]\[
\begin{aligned}
x+3 y & \leq 0.5 \\
x+3 y & \geq 2.5
\end{aligned}
\][/tex]
This system is asking for values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both [tex]\(x + 3y \leq 0.5\)[/tex] and [tex]\(x + 3y \geq 2.5\)[/tex]. However, these two inequalities are contradictory. The region where [tex]\(x + 3y \leq 0.5\)[/tex] lies on one side of the line [tex]\(x + 3y = 0.5\)[/tex], while the region where [tex]\(x + 3y \geq 2.5\)[/tex] lies on the other side of the line [tex]\(x + 3y = 2.5\)[/tex]. There is no overlap between these two regions since [tex]\( x + 3y \)[/tex] cannot simultaneously be less than or equal to 0.5 and greater than or equal to 2.5. Therefore, there is no solution that satisfies both inequalities together.
Thus, the system of inequalities that has no solution is:
[tex]\[
\begin{aligned}
x + 3y & \leq 0.5 \\
x + 3y & \geq 2.5
\end{aligned}
\][/tex]
So, the correct answer is
[tex]\[
3
\][/tex]
