tommy is saving for college and works less than 22 hours per week between two part-time jobs. he earns $8 an…

Explanation:

Step1: Define total hours inequality

Total weekly hours < 22. $x + y < 22$

Step2: Define earnings inequality

Total earnings ≥ $253. $8x + 21y \geq 253$

Step3: Add non-negativity constraints

Hours cannot be negative. $x \geq 0, y \geq 0$

Step4: Find intersection of bounds

First, rewrite $x + y < 22$ as $x < 22 - y$. Substitute into earnings inequality: $8(22 - y) + 21y \geq 253$ $176 - 8y + 21y \geq 253$ $13y \geq 77$ $y \geq \frac{77}{13} \approx 5.92$, so $y \geq 6$ (since hours are whole numbers) Substitute $y=6$ into $x < 22 - y$: $x < 16$ Also, from $8x + 21y \geq 253$, when $x=0$: $y \geq \frac{253}{21} \approx 12.05$, so $y \geq 13$ when $x=0$ When $y=0$: $8x \geq 253 \implies x \geq 31.625$, which conflicts with $x < 22$, so $y$ cannot be 0.

Answer:

The system of inequalities is: $$ \begin{cases} x + y < 22 \ 8x + 21y \geq 253 \ x \geq 0 \ y \geq 0 \end{cases} $$ The solution set consists of all integer pairs $(x,y)$ where $y \geq 6$, $x < 22 - y$, and $8x + 21y \geq 253$. Graphically, this is the dark shaded region in the provided plot, representing valid combinations of babysitting and tutoring hours that meet both the weekly hour limit and earnings goal.