Question

Jace is buying cookies and brownies for a party. He needs at least 8 cookies and at least 4 brownies. A cookie costs $3 and a brownie costs $5. The most he can spend is $60. A. Create a system of linear inequalities in two variables from the description. Define your variables.

Answer 1

B. Consider each of the following potential solutions. For each potential solution, determine whether it is a viable solution, a nonviable solution, or not a solution.(3 points) viable solution nonviable solution not a solution 8.5 cookies and 4 brownies 10 cookies and 5 brownies 20 cookies and 0 brownies

Answer 2

Let's define our variables:

- Let \( c \) represent the number of cookies Jace buys.
- Let \( b \) represent the number of brownies Jace buys.

From the problem description, we can derive the following inequalities:

1. **Cookies Requirement**: Jace needs at least 8 cookies.
\[
c \geq 8
\]

2. **Brownies Requirement**: Jace needs at least 4 brownies.
\[
b \geq 4
\]

3. **Budget Constraint**: Each cookie costs $3 and each brownie costs $5, and Jace's total spending should not exceed $60.
\[
3c + 5b \leq 60
\]

Therefore, the system of linear inequalities that represents Jace's situation is:

\[
\begin{align*}
c & \geq 8 \quad \text{(1)} \\
b & \geq 4 \quad \text{(2)} \\
3c + 5b & \leq 60 \quad \text{(3)} \\
\end{align*}
\]

In summary, the inequalities are:

1. \( c \geq 8 \)
2. \( b \geq 4 \)
3. \( 3c + 5b \leq 60 \)

Answer 3

Let's evaluate each potential solution against the system of inequalities:

1. **Potential Solution: 8.5 cookies and 4 brownies**
- \( c = 8.5 \)
- \( b = 4 \)
- Check against inequalities:
- **1.** \( c \geq 8 \): \( 8.5 \geq 8 \) (True)
- **2.** \( b \geq 4 \): \( 4 \geq 4 \) (True)
- **3.** \( 3c + 5b \leq 60 \): \( 3(8.5) + 5(4) = 25.5 + 20 = 45.5 \leq 60 \) (True)
- **Conclusion**: This is a **viable solution**.

2. **Potential Solution: 10 cookies and 5 brownies**
- \( c = 10 \)
- \( b = 5 \)
- Check against inequalities:
- **1.** \( c \geq 8 \): \( 10 \geq 8 \) (True)
- **2.** \( b \geq 4 \): \( 5 \geq 4 \) (True)
- **3.** \( 3c + 5b \leq 60 \): \( 3(10) + 5(5) = 30 + 25 = 55 \leq 60 \) (True)
- **Conclusion**: This is a **viable solution**.

3. **Potential Solution: 20 cookies and 0 brownies**
- \( c = 20 \)
- \( b = 0 \)
- Check against inequalities:
- **1.** \( c \geq 8 \): \( 20 \geq 8 \) (True)
- **2.** \( b \geq 4 \): \( 0 \geq 4 \) (False)
- **3.** \( 3c + 5b \leq 60 \): \( 3(20) + 5(0) = 60 + 0 = 60 \leq 60 \) (True)
- **Conclusion**: This is a **nonviable solution** because it does not meet the requirement for the number of brownies.

### Summary of Conclusions:
- 8.5 cookies and 4 brownies: **Viable solution**
- 10 cookies and 5 brownies: **Viable solution**
- 20 cookies and 0 brownies: **Nonviable solution**