To determine which system of inequalities has no solution, let's analyze each given pair of inequalities step-by-step.
System 1:
[tex]\[
\begin{aligned}
x + 3y \geq 0.5 \\
x + 3y \leq 2.5
\end{aligned}
\][/tex]
For this system, we are looking at values of [tex]\( x + 3y \)[/tex] that simultaneously satisfy both inequalities. Essentially, [tex]\( x + 3y \)[/tex] must be between 0.5 and 2.5 inclusive. We can find solutions that meet this condition. Therefore, this system has a solution.
System 2:
[tex]\[
\begin{aligned}
x + 3y \geq 0.5 \\
x + 3y \geq 2.5
\end{aligned}
\][/tex]
For this system, for [tex]\( x + 3y \geq 0.5 \)[/tex] to be true, [tex]\( x + 3y \)[/tex] must be at least 0.5. Additionally, for [tex]\( x + 3y \geq 2.5 \)[/tex] to be true, [tex]\( x + 3y \)[/tex] must also be at least 2.5. The stronger condition [tex]\( x + 3y \geq 2.5 \)[/tex] encompasses the weaker condition [tex]\( x + 3y \geq 0.5 \)[/tex]. Thus, the solutions for the second inequality alone also satisfy the first inequality, ensuring that every solution to the second inequality is also a solution to the first inequality. Therefore, this system has a solution.
System 3:
[tex]\[
\begin{aligned}
x + 3y \leq 0.5 \\
x + 3y \geq 2.5
\end{aligned}
\][/tex]
For this system, [tex]\( x + 3y \)[/tex] must be less than or equal to 0.5 for the first inequality and greater than or equal to 2.5 for the second inequality. These two conditions are contradictory because a number cannot be simultaneously less than or equal to 0.5 and greater than or equal to 2.5. Therefore, this system has no solution.
Conclusion:
The system of inequalities with no solution is:
[tex]\[
\begin{aligned}
x + 3y \leq 0.5 \\
x + 3y \geq 2.5
\end{aligned}
\][/tex]
Thus, the provided answer is correct, and the system of inequalities that has no solution corresponds to the third range in the original list.
