Question

modeling real life you earn $10 per hour working at a grocery store and must work there at least 8 hours per week. you also teach music lessons at $15 per hour. between the two jobs, you need to earn at least $120 per week and work no more than 20 hours per week. write and graph a system that represents the situation. give one example of the number of hours you can work at each job.

Explanation:

Step1: Define variables

Let \( x = \) hours at grocery store, \( y = \) hours teaching music.

Step2: Write constraints

• Grocery hours: \( x \geq 8 \)
• Total hours: \( x + y \leq 30 \)
• Total earnings: \( 10x + 15y \geq 120 \)
• Non-negativity: \( y \geq 0 \)

Step3: Find valid example

Test \( x=8 \): \( 10(8)+15y \geq120 \Rightarrow 80+15y\geq120 \Rightarrow y\geq\frac{40}{15}\approx2.67 \). Choose \( y=3 \): total hours \( 8+3=11\leq30 \), earnings \( 80+45=125\geq120 \).

Answer:

System of inequalities: \( x \geq 8 \), \( x + y \leq 30 \), \( 10x + 15y \geq 120 \), \( y \geq 0 \); Example: 8 hours at grocery store and 3 hours teaching music.