To solve this problem, let's break down the constraints and requirements one by one.
1. Food Quantity Requirement:
- We need at least three bratwursts or hamburgers per guest.
- There are 50 guests.
- Therefore, we need at least [tex]\(3 \times 50 = 150\)[/tex] items in total.
- This translates to the inequality [tex]\( x + y \geq 150 \)[/tex], where [tex]\(x\)[/tex] is the number of bratwursts and [tex]\(y\)[/tex] is the number of hamburgers.
2. Budget Constraint:
- Bratwursts cost \[tex]$1.25 each and hamburgers cost \$[/tex]0.95 each.
- We have a total budget of \[tex]$200.
- Therefore, the total cost of purchasing \(x\) bratwursts and \(y\) hamburgers should not exceed \$[/tex]200.
- This translates to the inequality [tex]\( 1.25x + 0.95y \leq 200 \)[/tex].
Putting these constraints together, we get the system of inequalities:
[tex]\[
\begin{cases}
x + y \geq 150 \\
1.25x + 0.95y \leq 200
\end{cases}
\][/tex]
This matches one of the provided options. Therefore, the correct system of inequalities representing this situation is:
[tex]\[
\begin{array}{ll}
x+y \geq 150 & 1.25x + 0.95y \leq 200
\end{array}
\][/tex]
