Answer:
Step-by-step explanation:
The correct system of inequalities that represents Liam's situation is:
1. \(1.5x + 3.25y > 300\)
2. \(x + y \geq 120\)
Where:
- \(x\) represents the number of dozens of cookies Liam sells.
- \(y\) represents the number of dozens of brownies Liam sells.
Step-by-step explanation:
1. The first inequality \(1.5x + 3.25y > 300\) represents Liam's goal to make more than $300 in total sales. Here, \(1.5x\) represents the total amount he earns from selling cookies, and \(3.25y\) represents the total amount he earns from selling brownies. The sum of these amounts must be greater than $300.
2. The second inequality \(x + y \geq 120\) ensures that Liam sells at least 10 dozen baked goods (which is equivalent to 120 individual items when considering dozens). This inequality represents Liam's goal to sell at least 10 dozen baked goods.
Given this system of inequalities, if Liam sells 3 dozen cookies (\(x = 3\)), we can substitute this value into the inequalities to find the range of possible values for \(y\), the number of dozens of brownies he needs to sell:
1. \(1.5(3) + 3.25y > 300\)
\(4.5 + 3.25y > 300\)
\(3.25y > 295.5\)
\(y > \frac{295.5}{3.25}\)
\(y > 90.923\)
2. \(3 + y \geq 120\)
\(y \geq 117\)
So, Liam needs to sell at least 91 dozen brownies (or 117 individual brownies) to meet his personal goal of making more than $300 in total sales while selling 3 dozen cookies.