Systems of Linear Inequalities, Word Problems - Examples

$$\text{Thrown Through a Loop}$$

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Groundhog Day is coming up (Feb 2). Weatherman Phil is stuck in a $100$ day time loop. The first few days of the loop, he wastes a lot of time and energy, but he tries hard to improve his routine day by day.

Each day ($24$ hours), Phil spends

• $8$ hours sleeping;
• $W+R$ hours traveling by foot: $W$ hours walking at $3$ miles per hour, and $R$ hours running at $9$ miles per hour; and
• $N+S$ hours doing Good Things: $N$ hours as a normal person doing $10$ Good Things per hour, and $S$ hours as a superhero-wannabe doing $20$ Good Things per hour.

Phil needs to travel $9$ miles total (by foot), but he can only spend the rest of his time awake doing Good Things. However, Phil has a limited amount of energy: he can only spend $1$ hour running or being a superhero-wannabe, in total.