Math Problem Statement

Joey works two part-time jobs. He needs to make at least $540 a week to cover his expenses. He works as a tailor at a dress shop that pays him $14 per hour, and he also washes dishes at a restaurant that pays him $10.50 per hour after tips. Joey can work no more than 40 hours per week. A partial system of inequalities is shown below describing all ordered pairs that represent the number of hours Joey can work at the dress shop, x, and the number of hours he can work at the restaurant, y, each week and meet his conditions.

Solution

To find the missing inequality in the system, let’s examine Joey's weekly requirements.

Given Information:

  • Joey needs at least $540 per week.
  • He works two jobs:
    • Dress shop (x) at $14 per hour.
    • Restaurant (y) at $10.50 per hour.
  • Joey can work at most 40 hours per week in total.

Analysis:

  1. Hourly Limits: Joey's total working hours x+yx + y must not exceed 40 hours: x+y≤40x + y \leq 40 This inequality is already provided in the system.

  2. Weekly Income Requirement: Joey needs at least $540 from both jobs combined. His earnings from the dress shop job are 14x14x and from the restaurant job are 10.5y10.5y. Thus: 14x+10.5y≥54014x + 10.5y \geq 540 This inequality represents the minimum income requirement.

  3. Non-negative Hours: Since Joey cannot work negative hours, we also have: x≥0andy≥0x \geq 0 \quad \text{and} \quad y \geq 0

Final System of Inequalities:

The complete system should be:

x + y \leq 40 \\ 14x + 10.5y \geq 540 \\ x \geq 0 \\ y \geq 0 \\ \end{cases}$$ The missing inequality is: $$14x + 10.5y \geq 540$$ Would you like a more detailed explanation on any part of this solution? --- ### Related Questions: 1. How would the system change if Joey's minimum weekly earnings requirement increased? 2. What happens if the dress shop’s hourly rate changes? 3. How would the inequalities change if Joey’s maximum weekly hours increased to 45? 4. What if Joey wanted to save a specific amount each week, say $50, in addition to covering his expenses? 5. How could you graph this system of inequalities? ### Tip: In word problems, clearly identify constraints on both variables and determine if they affect the total quantity (like hours or earnings). This approach simplifies setting up the inequalities.

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Math Problem Analysis

Mathematical Concepts

Systems of Inequalities
Linear Inequalities
Word Problems

Formulas

x + y ≤ 40
14x + 10.5y ≥ 540

Theorems

-

Suitable Grade Level

Grades 9-10

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