
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.
 

### Step 1: Define the Variables
Let:
- $ x $ represent the number of hours Erica works per week.
- $ y $ represent the number of hours Megan works per week.

### Step 2: Write the Inequality Equations

1. **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:
   \[
   18x + 10y \leq 470
   \]

2. **Hours Constraint:**
   The total number of hours that both employees work should not exceed 30 hours per week. This gives the inequality:
   \[
   x + y \leq 30
   \]

Additionally, since the number of hours worked cannot be negative:
   \[
   x \geq 0 \quad \text{and} \quad y \geq 0
   \]

### Step 3: Use Intercepts to Graph the Inequalities

To graph these inequalities, we first find the intercepts for the two main inequalities.

#### Inequality 1: $ 18x + 10y \leq 470 $

- **x-intercept** (set $ y = 0 $):
  \[
  18x = 470 \implies x = \frac{470}{18} \approx 26.11
  \]
  So, the x-intercept is approximately $ (26.11, 0) $.

- **y-intercept** (set $ x = 0 $):
  \[
  10y = 470 \implies y = \frac{470}{10} = 47
  \]
  So, the y-intercept is $ (0, 47) $.

#### Inequality 2: $ x + y \leq 30 $

- **x-intercept** (set $ y = 0 $):
  \[
  x = 30
  \]
  So, the x-intercept is $ (30, 0) $.

- **y-intercept** (set $ x = 0 $):
  \[
  y = 30
  \]
  So, the y-intercept is $ (0, 30) $.

### Step 4: Graph the Inequalities
To graph the system:
- Plot the points $ (26.11, 0) $ and $ (0, 47) $ for the inequality $ 18x + 10y = 470 $.
- Plot the points $ (30, 0) $ and $ (0, 30) $ for the inequality $ x + y = 30 $.
- The solution region will be the area below both lines, including the region where they overlap, and bounded by the non-negative axes (since $ x \geq 0 $ and $ y \geq 0 $).

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:
1. How can you find the point where the two lines intersect?
2. What is the maximum number of hours Erica can work if Megan works 15 hours?
3. If Jeremy increased his budget to $500, how would the inequality change?
4. How would the graph change if Megan's wage increased to $12 per hour?
5. 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).

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.

Explore a real-world application of linear inequalities by solving how Jeremy can allocate up to 30 work hours between two employees while staying within a weekly wage budget of $470. Understand the graphical solution using intercepts and system constraints. Suitable for Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation