Marisol is making a rectangular wooden frame. She wants the length of the frame to be no more than 12 inches. She has less than 30 inches of wood to use. Which system of inequalities represents the possible length, [tex]l[/tex], and the possible width, [tex]w[/tex], that her frame could have?
[tex]
\begin{aligned}
l & \leq 12 \\
2 l+2 w & \ \textless \ 30
\end{aligned}
[/tex]
[tex]
\begin{aligned}
l & \ \textgreater \ 12 \\
2 l+2 w & \ \textless \ 30
\end{aligned}
[/tex]
[tex]
\begin{aligned}
l & \leq 12 \\
1+w & \ \textless \ 30
\end{aligned}
[/tex]
[tex]
\begin{aligned}
l & \ \textgreater \ 12 \\
1+w & \ \textless \ 30
\end{aligned}
[/tex]
Answer (2)
The length l is no more than 12 inches: l ≤ 12 .
The perimeter 2 l + 2 w is less than 30 inches: 2 l + 2 w < 30 .
The system of inequalities is: l ≤ 12 and 2 l + 2 w < 30 .
The correct answer is the system: l ≤ 12 ; 2 l + 2 w < 30 .
Explanation
Problem Analysis Let's analyze the problem. Marisol is building a rectangular frame with length l and width w . We have two constraints:
The length l must be no more than 12 inches, which means l l e q 12 .
The total wood used, which is the perimeter of the rectangle, must be less than 30 inches. The perimeter is given by 2 l + 2 w , so 2 l + 2 w < 30 .
System of Inequalities Now, let's write down the system of inequalities that represents these constraints:
l ≤ 12 2 l + 2 w < 30
Matching the Solution Comparing this system with the given options, we see that the first option matches our system:
l ≤ 12 2 l + 2 w < 30
Examples
Imagine you're designing a garden bed and have a limited amount of fencing. You want the length to be no more than 12 feet to fit the space, and you have less than 30 feet of fencing in total. This problem helps you determine the possible dimensions (length and width) of the garden bed so that you don't exceed your fencing limit and the length constraint. Understanding these constraints ensures you design a garden bed that fits your space and uses your resources efficiently.
The system of inequalities representing Marisol's frame constraints is l ≤ 12 and 2 l + 2 w < 30 . This means the length must not exceed 12 inches, and the total perimeter must be less than 30 inches of wood. The correct answer is l ≤ 12 ; 2 l + 2 w < 30 .
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