To address the problem of purchasing centerpieces for up to 12 tables with a budget limit of [tex]$100, where flowers (denoted as \( f \)) cost $[/tex]4 each and candles (denoted as [tex]\( c \)[/tex]) cost [tex]$7 each, we need to formulate a system of linear inequalities.
Let's break down the constraints:
1. Number of Centerpieces Constraint:
- The total number of centerpieces (flowers plus candles) must not exceed the number of tables, which is 12. This can be written as:
\[
f + c \leq 12
\]
2. Budget Constraint:
- The total cost of the centerpieces (combined cost of flowers and candles) must not exceed $[/tex]100. With flowers costing [tex]$4 each and candles costing $[/tex]7 each, this constraint can be written as:
[tex]\[
4f + 7c \leq 100
\][/tex]
3. Non-negativity Constraints:
- The number of flowers and candles must be non-negative as it doesn't make sense to have a negative quantity of centerpieces. These constraints are:
[tex]\[
f \geq 0
\][/tex]
[tex]\[
c \geq 0
\][/tex]
Combining all these constraints, we obtain the system of linear inequalities:
[tex]\[
\begin{cases}
f + c \leq 12 \\
4f + 7c \leq 100 \\
f \geq 0 \\
c \geq 0
\end{cases}
\][/tex]
This system of inequalities correctly models the problem, allowing for the selection of flowers and/or candles while adhering to the constraints on the number of tables and the budget.
