Sure, let's break down the given problem and constraints in detail:
1. Definitions and Variables:
- Let \( t \) represent the number of hours Nick spends tutoring each week.
- Let \( b \) represent the number of hours Nick spends working as a bag boy each week.
- Nick earns \[tex]$15 per hour tutoring and \$[/tex]8 per hour as a bag boy.
2. Constraints:
- Nick can work a maximum total of 20 hours per week.
- Nick must earn at least \$150 per week from both jobs combined.
3. Formulating the Constraints as Inequalities:
- The total hours Nick can work each week should be less than or equal to 20:
[tex]\[
t + b \leq 20
\][/tex]
- Nick must earn at least \$150 per week:
[tex]\[
15t + 8b \geq 150
\][/tex]
Given these points, let's discuss the provided system of inequalities:
1. \( t + b \leq 20 \): This correctly represents the constraint that Nick can work up to 20 hours per week.
2. \( 15t + 8b \geq 150 \): This correctly represents the constraint that Nick must earn at least \$150 per week.
The other constraints listed in the solutions are either repetitive or not relevant to the scenario described. Specifically:
- Including \( t + b \leq 20 \) multiple times does not add anything new.
- \( 15t + 8b \leq 150 \): This represents earning at most \[tex]$150, which does not align with Nick's need to earn at least \$[/tex]150.
- \( t + b \geq 20 \): This would imply that Nick must work exactly 20 hours, which is not a requirement.
- \( 15t + 8b = 150 \): This would imply that Nick must earn exactly \$150, which again is not a requirement.
Thus, the only valid and meaningful constraints that encapsulate Nick's working balance and earnings requirements are:
1. \( t + b \leq 20 \)
2. \( 15t + 8b \geq 150 \)
These two inequalities accurately represent the scenario described.
