https://file.aiquickdraw.com/mathbot/file/17274506223526ns086im.jpg

Let's analyze and solve the problem based on the provided image.

### **Problem Summary:**
Myesha is buying cupcakes and cookies:
- Cupcakes cost $4.25 each.
- Cookies cost $0.75 each.
- She has $30 to spend.
- She needs to buy at least 16 items (cupcakes and cookies combined).

Let $ x $ represent the number of cupcakes and $ y $ represent the number of cookies.

### **Step 1: Setting Up the Inequalities**

#### **Inequality 1: Budget Constraint**
The total cost must be less than or equal to $30.
\[
4.25x + 0.75y \leq 30
\]

#### **Inequality 2: Minimum Quantity Constraint**
The total number of items (cupcakes and cookies) must be at least 16.
\[
x + y \geq 16
\]

### **Step 2: Solve the System of Inequalities**

We have the system:
\[
4.25x + 0.75y \leq 30
\]
\[
x + y \geq 16
\]

Let's solve and find the intercepts to understand these inequalities graphically.

#### **For the first inequality: $ 4.25x + 0.75y \leq 30 $**

1. **Find x-intercept (where $ y = 0 $):**
   \[
   4.25x = 30 \implies x = \frac{30}{4.25} \approx 7.06
   \]

2. **Find y-intercept (where $ x = 0 $):**
   \[
   0.75y = 30 \implies y = \frac{30}{0.75} = 40
   \]

Thus, the intercepts are $ (7.06, 0) $ and $ (0, 40) $.

#### **For the second inequality: $ x + y \geq 16 $**

1. **Find x-intercept (where $ y = 0 $):**
   \[
   x = 16
   \]

2. **Find y-intercept (where $ x = 0 $):**
   \[
   y = 16
   \]

The intercepts for this are $ (16, 0) $ and $ (0, 16) $.

### **Step 3: Determine One Possible Solution**

Graphically, these inequalities form a region of possible solutions. A potential solution needs to satisfy both inequalities. Let's find one feasible solution:

#### **Substitute $ x = 5 $ and $ y = 12 $ into the inequalities:**

1. Check $ 4.25x + 0.75y \leq 30 $:
   \[
   4.25(5) + 0.75(12) = 21.25 + 9 = 30.25 \quad \text{(not valid as it exceeds 30)}
   \]

Try $ x = 5 $ and $ y = 10 $:

1. Check $ 4.25(5) + 0.75(10) = 21.25 + 7.5 = 28.75 \quad \text{(valid)}
2. Check \( x + y \geq 16 $: $ 5 + 10 = 15 \quad \text{(not valid)}

Let's try \( x = 4 $ and $ y = 12 $:

1. Check $ 4.25(4) + 0.75(12) = 17 + 9 = 26 \quad \text{(valid)}
2. Check \( x + y \geq 16 $: $ 4 + 12 = 16 \quad \text{(valid)}

Hence, \( x = 4 $ and $ y = 12 $ is a valid solution.

### **Answer**

A possible solution is purchasing **4 cupcakes** and **12 cookies**.

Would you like a more detailed explanation of any part, or do you have any questions?

### **5 Related Questions:**
1. What is the graphical representation of the system of inequalities?
2. How do you find the feasible region for a system of inequalities?
3. Why is $ x = 4 $ and $ y = 12 $ a solution to both inequalities?
4. What does it mean when two inequalities intersect on a graph?
5. How would the solution change if the budget increased?

### **Tip:** 
Always verify a potential solution by substituting it back into all inequalities to ensure it satisfies the constraints.

Myesha and her children went into a bakery that sells cupcakes for $4.25 each and cookies for $0.75 each. Myesha has $30 to spend and must buy at least 16 cupcakes and cookies altogether. If x represents the number of cupcakes purchased and y represents the number of cookies purchased, write and solve a system of inequalities graphically and determine one possible solution.

Learn how to graph and solve a system of inequalities based on a bakery budget where cupcakes cost $4.25 and cookies cost $0.75. Myesha has $30 to spend and must purchase at least 16 items. Find the solution graphically and determine the number of cupcakes and cookies she can buy.

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