Math Problem Statement

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 for $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 a system that represents the situation. Give one example of the number of hours you can work at each job.

Solution

The problem states:

  • You earn $10 per hour at a grocery store, with a minimum of 8 hours per week.
  • You teach music lessons for $15 per hour.
  • You need to make at least $120 per week and work no more than 20 hours per week.

Let's define:

  • xx: hours worked at the grocery store.
  • yy: hours spent teaching music lessons.

Step 1: Set Up the System of Inequalities

  1. Earnings Requirement: You need to earn at least $120. 10x+15y≥12010x + 15y \geq 120

  2. Hours Limit: You cannot work more than 20 hours per week. x+y≤20x + y \leq 20

  3. Minimum Hours at the Grocery Store: You must work at least 8 hours at the grocery store. x≥8x \geq 8

  4. Non-Negative Hours: You can't work negative hours. x≥0,y≥0x \geq 0, \quad y \geq 0

Final System of Inequalities

The system that represents the situation is:

10x + 15y \geq 120 \\ x + y \leq 20 \\ x \geq 8 \\ x \geq 0, \, y \geq 0 \end{cases}$$ ### Example Solution To satisfy the requirements, let's try an example where: - $$x = 8$$ hours at the grocery store. - Substitute into the total hours constraint: $$8 + y \leq 20 \Rightarrow y \leq 12$$. - Suppose $$y = 4$$. Then: - Earnings: $$10(8) + 15(4) = 80 + 60 = 140$$, which meets the $120 minimum. - Total hours: $$8 + 4 = 12$$, within the 20-hour limit. This solution satisfies all constraints. --- Would you like further explanation on this, or have questions? --- 1. How do you determine feasible solutions within these inequalities? 2. How can graphing help visualize the solution set? 3. How would changing the pay rate affect the inequalities? 4. Can you find other possible combinations of hours that satisfy all conditions? 5. How does setting limits on hours influence potential earnings? **Tip:** To check if solutions satisfy the inequalities, substitute them back into each inequality and verify.

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Math Problem Analysis

Mathematical Concepts

Linear Inequalities
Systems of Inequalities
Linear Programming

Formulas

10x + 15y ≥ 120 (Earnings Requirement)
x + y ≤ 20 (Hours Limit)
x ≥ 8 (Minimum Hours Requirement)

Theorems

Feasibility of Solutions in Systems of Inequalities

Suitable Grade Level

Grades 9-11

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