**Applications of Systems of Inequalities**

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 

In this problem, we are tasked with formulating a system of inequalities based on Jeremy's budget and work hour constraints for his two employees, Erica and Megan. Let’s define the variables and write the system of inequalities step by step.

### **Step 1: Define the Variables**

- Let $ x $ be the number of hours Erica works per week.
- Let $ y $ be the number of hours Megan works per week.

### **Step 2: Write the Inequality Equations**

From the problem, we know two things:
1. Jeremy has at most $470 to pay his employees weekly.
2. There are at most 30 hours of work available in total.

#### **Budget Constraint**
Erica gets paid $18 per hour, and Megan gets paid $10 per hour. The total wage paid cannot exceed $470. This gives us the inequality:

\[
18x + 10y \leq 470
\]

#### **Work Hours Constraint**
The total number of hours worked by Erica and Megan cannot exceed 30 hours. This gives us the inequality:

\[
x + y \leq 30
\]

#### **Non-negative Constraints**
Since neither Erica nor Megan can work negative hours, we also have:

\[
x \geq 0 \quad \text{and} \quad y \geq 0
\]

### **Step 3: Use Intercepts to Graph**

To graph these inequalities, it’s helpful to find the intercepts of the lines represented by the inequalities.

#### **Intercepts for $ 18x + 10y \leq 470 $**

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

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

#### **Intercepts for $ 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) $.

### **Graphing the Inequalities**

To graph the system of inequalities:
1. Plot the line for $ 18x + 10y = 470 $ using the intercepts $ (26.11, 0) $ and $ (0, 47) $.
2. Plot the line for $ x + y = 30 $ using the intercepts $ (30, 0) $ and $ (0, 30) $.
3. Shade the region below both lines (since the inequalities are $ \leq $) and within the first quadrant, as $ x \geq 0 $ and $ y \geq 0 $.

The feasible region where both inequalities are satisfied represents all possible combinations of hours worked by Erica and Megan within the constraints of Jeremy's budget and total available hours.

Would you like a graphical representation of this system, or further explanation?

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**Related Questions:**
1. What happens to the system if Erica’s pay is increased to $20 per hour?
2. How would the system change if the budget was reduced to $400?
3. How many hours can Megan work if Erica works 20 hours?
4. What are the implications if Megan is paid more than Erica?
5. How does the system change if there are no constraints on total hours?

**Tip:** Always verify the intercepts by plugging them into the inequality to check if they satisfy the conditions.

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.

Learn how to apply systems of inequalities to budget employee hours and wages. Jeremy has a weekly budget of $470 and a maximum of 30 work hours to distribute between two employees, Erica and Megan. This problem explores linear inequalities, intercepts, and graphing. Suitable for Grades 9-11.

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