Let's break down the problem and formulate the necessary inequalities step-by-step.
1. Time Constraints:
- Greg can work at most 18 hours a day.
- It takes him 30 minutes (or 0.5 hours) to design one T-shirt and 45 minutes (or 0.75 hours) to design one pair of shorts.
The total working time constraint can be expressed as:
[tex]\[
0.5t + 0.75s \leq 18
\][/tex]
2. Minimum and Maximum Number of Items:
- Greg must design at least 12 items (T-shirts + shorts) each day.
- He cannot design more than 28 items each day.
These constraints can be expressed as:
[tex]\[
t + s \geq 12
\][/tex]
and
[tex]\[
t + s \leq 28
\][/tex]
By isolating [tex]\( s \)[/tex]:
[tex]\[
s \geq 12 - t
\][/tex]
and
[tex]\[
s \leq 28 - t
\][/tex]
3. Non-negative Quantities:
- Greg cannot design a negative number of items, so:
[tex]\[
s \geq 0
\][/tex]
[tex]\[
t \geq 0
\][/tex]
By combining these constraints, we obtain the complete set of inequalities:
[tex]\[
\begin{aligned}
&s \geq 12 - t, \\
&s \leq 28 - t, \\
&0.5t + 0.75s \leq 18, \\
&s \geq 0, \\
&t \geq 0.
\end{aligned}
\][/tex]
Reviewing the given answer options:
- Option A: [tex]\(s \geq 12 - t, s \geq 28 - t, s \leq 24 - 0.66t, s \geq 0, t \geq 0\)[/tex]
- Option B: [tex]\(s \geq 12 - t, s \leq 28 - t, 0.5s \geq 18 - 0.66t, s \geq 0, t \geq 0\)[/tex]
- Option C: [tex]\(s \geq 12 - t, s \leq 28 - t, 0.5t + 0.66s \leq 18, s \geq 0, t \geq 0\)[/tex]
- Option D: [tex]\(s \geq 12 + t, s \leq 28 + t, s \leq 24 - 0.66t, s \geq 0, t \geq 0\)[/tex]
The correct set of inequalities is reflected in Option C:
[tex]\[
s \geq 12 - t, s \leq 28 - t, 0.5t + 0.75s \leq 18, s \geq 0, t \geq 0
\][/tex]
