Math Problem Statement
Jeremy has a babysitting job after school that pays $7.25 per hour. He earns twice as much per hour working as a library assistant. He wants to make more than $100 a week, but work at most 20 hours per week. Which system of linear inequalities represents all possible combinations of the number of hours babysitting, x, and the number of hours as a library assistant, y, Jeremy can work?
Solution
To solve this problem, let's break down each part of the question.
Step 1: Understanding the Variables
Let:
- = hours Jeremy works babysitting (at $7.25 per hour).
- = hours Jeremy works as a library assistant (at $14.50 per hour, which is twice as much as babysitting).
Step 2: Analyzing the Conditions
-
Total Weekly Hours: Jeremy can work a maximum of 20 hours per week. This gives the inequality:
-
Income Goal: Jeremy wants to earn more than $100 per week. The total earnings are given by:
Step 3: Identifying the Correct Inequalities
The correct system of inequalities that represents these conditions is:
x + y \leq 20 \\ 7.25x + 14.50y > 100 \end{cases}$$ ### Step 4: Matching with the Options From the options given: - **Option D** correctly represents the system of inequalities: $$x + y \leq 20 \quad \text{and} \quad 7.25x + 14.50y > 100$$ ### Final Answer The correct answer is **D**. Would you like further details or explanations on any part of this solution? --- Here are five related questions to deepen understanding: 1. What would the inequalities look like if Jeremy could work up to 25 hours instead of 20? 2. How would the inequalities change if the library assistant job paid $10 per hour instead? 3. If Jeremy wanted to earn at least $150 per week, how would the inequality change? 4. How can this system of inequalities be graphed on a coordinate plane? 5. How many hours should Jeremy work in each job to maximize his income under these constraints? **Tip:** When setting up inequalities, carefully analyze each condition separately before combining them into a system.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Inequalities
Systems of Inequalities
Formulas
Total Hours Inequality: x + y ≤ 20
Income Inequality: 7.25x + 14.50y > 100
Theorems
-
Suitable Grade Level
Grades 8-10