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

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 2

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 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