Liam is packing his room to move into a new house. A small box can hold 8 books without breaking, while a large box can hold 12 books without breaking. He has at least 160 books to pack and less than 30 boxes total. Let s represent the number of small boxes and l represent the number of large boxes.
The inequalities [tex]$s \geq 0$[/tex] and [tex]$l \geq 0$[/tex] are part of the system that models this scenario.
Which inequalities complete the system?
A. [tex]$s+l\ \textless \ 30$[/tex] and [tex]$8s+12l \leq 160$[/tex]
B. [tex]$s-l\ \textless \ 30$[/tex] and [tex]$8s-12l \leq 160$[/tex]
C. [tex]$s+l\ \textgreater \ 30$[/tex] and [tex]$8s+12l \leq 160$[/tex]
D. [tex]$s+l\ \textless \ 30$[/tex] and [tex]$8s+12l \geq 160$[/tex]
Answer (2)
The total number of books packed in small and large boxes must be greater than or equal to 160: 8 s + 12 l ≥ 160 .
The total number of boxes must be less than 30: s + l < 30 .
Combine these inequalities with s ≥ 0 and l ≥ 0 to define the complete system.
The inequalities that complete the system are s + l < 30 and 8 s + 12 l ≥ 160 .
Explanation
Setting up the inequalities We are given that s represents the number of small boxes and l represents the number of large boxes. We know that a small box can hold 8 books and a large box can hold 12 books. Liam has at least 160 books to pack, so the total number of books packed in small and large boxes must be greater than or equal to 160. This can be represented by the inequality:
Books Inequality = 160"> 8 s + 12 l " >= 160
Boxes Inequality We are also given that Liam has less than 30 boxes total. This means the sum of the number of small boxes and the number of large boxes must be less than 30. This can be represented by the inequality:
Boxes Inequality s + l < 30
Final Answer Therefore, the inequalities that complete the system are s + l < 30 and = 160"> 8 s + 12 l " >= 160 .
Examples
Imagine you're organizing a school event and need to transport materials using boxes. Small boxes hold 8 items each, and large boxes hold 12 items each. You need to transport at least 160 items, and you can only use fewer than 30 boxes in total. This problem helps you determine how many small and large boxes you need to meet these constraints, ensuring you have enough space for all the items while staying within the box limit. Understanding these constraints helps in efficient planning and resource management.
The inequalities that complete the system for Liam's packing scenario are: s + l < 30 and 8 s + 12 l ≥ 160 . This confirms that he has less than 30 boxes in total and that the total number of books must be at least 160. The answer is option D.
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