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Elam 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 most 160 books to pack and less than 30 boxes total. Let [tex]$s$[/tex] represent the number of small boxes and [tex]$l$[/tex] 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]
B. [tex]$8s - 12l \leq 160$[/tex]
C. [tex]$s + l \ \textless \ 30$[/tex]
D. [tex]$8s + 12l \leq 160$[/tex]
E. [tex]$s + l \ \textgreater \ 30$[/tex]
F. [tex]$8s + 12l \leq 160$[/tex]
G. [tex]$s + 1 \ \textless \ 30$[/tex]
H. [tex]$8s + 12l \geq 160$[/tex]

Answer :

To solve the problem of determining the correct inequalities for Elam's packing situation, we need to analyze the constraints provided:

1. Book Capacity Constraint:
- A small box holds 8 books, and a large box holds 12 books.
- Elam has at most 160 books to pack.

2. Box Limit Constraint:
- Elam has less than 30 boxes in total.

Now, let's express these constraints using inequalities:

### Book Capacity Constraint
Elam has a total of up to 160 books. If we let [tex]\( s \)[/tex] represent the number of small boxes and [tex]\( l \)[/tex] represent the number of large boxes, then the total number of books he can pack is given by [tex]\( 8s + 12l \)[/tex].

This gives us the inequality:
[tex]\[ 8s + 12l \leq 160 \][/tex]

### Box Limit Constraint
Elam has less than 30 boxes in total. This means the sum of the number of small and large boxes must be less than 30:
[tex]\[ s + l < 30 \][/tex]

These are the two inequalities that complete the system for Elam’s packing problem:

- [tex]\( 8s + 12l \leq 160 \)[/tex] (Ensures the boxes don't exceed book capacity)
- [tex]\( s + l < 30 \)[/tex] (Ensures the total number of boxes is less than 30)

Therefore, these inequalities accurately represent the situation and fulfill the given constraints.

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