Question

1. you have at most 8 hours to spend at a park and a planetarium. you want to spend more than 3 hours at the park and at least 3 hours at the planetarium. let x be the number of hours at the park and let y be the number of hours at the planetarium. write a system to represent the situation.

Explanation:

Step1: Analyze total time constraint

The total time at park (\(x\)) and planetarium (\(y\)) is at most 8 hours, so \(x + y\leqslant8\).

Step2: Analyze park time constraint

Spend more than 3 hours at park, so \(x > 3\).

Step3: Analyze planetarium time constraint

Spend at least 3 hours at planetarium, so \(y\geqslant3\).
Also, since time can't be negative, \(x\geqslant0\) and \(y\geqslant0\), but the main constraints from the problem are the three above.

Answer:

The system of inequalities is \(

$$\begin{cases}x + y\leqslant8\\x > 3\\y\geqslant3\end{cases}$$

\)