Math Problem Statement
Jeremy has two part-time employees, Erica has more experience and gets paid $18 per hour and Megan gets paid $10 per hour. Jeremy budgets $470 per week for their wages and has at most 30 hours of work available.
Define the variables: Write the inequality equations:
Use the intercept to help graph.
Solution
Step 1: Define the Variables
Let:
- represent the number of hours Erica works per week.
- represent the number of hours Megan works per week.
Step 2: Write the Inequality Equations
-
Budget Inequality: The total wages paid to both Erica and Megan should not exceed Jeremy's budget of $470 per week. The wage per hour for Erica is $18, and for Megan, it's $10. Thus, the inequality representing this constraint is:
-
Hours Constraint: The total number of hours that both employees work should not exceed 30 hours per week. This gives the inequality:
Additionally, since the number of hours worked cannot be negative:
Step 3: Use Intercepts to Graph the Inequalities
To graph these inequalities, we first find the intercepts for the two main inequalities.
Inequality 1:
-
x-intercept (set ): So, the x-intercept is approximately .
-
y-intercept (set ): So, the y-intercept is .
Inequality 2:
-
x-intercept (set ): So, the x-intercept is .
-
y-intercept (set ): So, the y-intercept is .
Step 4: Graph the Inequalities
To graph the system:
- Plot the points and for the inequality .
- Plot the points and for the inequality .
- The solution region will be the area below both lines, including the region where they overlap, and bounded by the non-negative axes (since and ).
Would you like me to provide a graph to help visualize this? Let me know if you'd like more details on any step or if you have questions.
Here are five related questions:
- How can you find the point where the two lines intersect?
- What is the maximum number of hours Erica can work if Megan works 15 hours?
- If Jeremy increased his budget to $500, how would the inequality change?
- How would the graph change if Megan's wage increased to $12 per hour?
- What combination of hours minimizes the total wage paid to both employees?
Tip: When solving linear inequalities, always consider the feasibility of values based on the context (e.g., time or money can't be negative).
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Math Problem Analysis
Mathematical Concepts
Linear Inequalities
Systems of Equations
Budgeting Constraints
Formulas
18x + 10y ≤ 470 (Budget Constraint)
x + y ≤ 30 (Total Hours Constraint)
x ≥ 0 and y ≥ 0 (Non-negative Hours)
Theorems
Linear Inequalities
Graphing Intercepts
Suitable Grade Level
Grades 9-11
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