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Answer :
To solve this problem, we need to determine the inequalities that reflect Elam's constraints for packing his books. Let's break it down step by step using the information provided:
1. Variables and their meanings:
- [tex]\( s \)[/tex]: the number of small boxes.
- [tex]\( l \)[/tex]: the number of large boxes.
2. Understand each type of box's capacity:
- A small box can hold 8 books.
- A large box can hold 12 books.
3. Examine the constraints:
- Total book constraint:
Elam can pack at most 160 books. The equation for this is:
[tex]\[
8s + 12l \leq 160
\][/tex]
This inequality ensures that the total number of books packed in both small and large boxes does not exceed 160.
- Total box constraint:
Elam can use less than 30 boxes in total. The equation for this is:
[tex]\[
s + l < 30
\][/tex]
This inequality assures that the total number of boxes (small and large combined) is less than 30.
4. Non-negativity constraints:
- Since Elam cannot have a negative number of boxes, we inherently have [tex]\( s \geq 0 \)[/tex] and [tex]\( l \geq 0 \)[/tex].
Combining all these, the system of inequalities that models this scenario is:
- [tex]\( s + l < 30 \)[/tex]
- [tex]\( 8s + 12l \leq 160 \)[/tex]
- [tex]\( s \geq 0 \)[/tex]
- [tex]\( l \geq 0 \)[/tex]
By using these inequalities, Elam can determine how many of each type of box he needs to use without exceeding the total number of books or boxes allowed.
1. Variables and their meanings:
- [tex]\( s \)[/tex]: the number of small boxes.
- [tex]\( l \)[/tex]: the number of large boxes.
2. Understand each type of box's capacity:
- A small box can hold 8 books.
- A large box can hold 12 books.
3. Examine the constraints:
- Total book constraint:
Elam can pack at most 160 books. The equation for this is:
[tex]\[
8s + 12l \leq 160
\][/tex]
This inequality ensures that the total number of books packed in both small and large boxes does not exceed 160.
- Total box constraint:
Elam can use less than 30 boxes in total. The equation for this is:
[tex]\[
s + l < 30
\][/tex]
This inequality assures that the total number of boxes (small and large combined) is less than 30.
4. Non-negativity constraints:
- Since Elam cannot have a negative number of boxes, we inherently have [tex]\( s \geq 0 \)[/tex] and [tex]\( l \geq 0 \)[/tex].
Combining all these, the system of inequalities that models this scenario is:
- [tex]\( s + l < 30 \)[/tex]
- [tex]\( 8s + 12l \leq 160 \)[/tex]
- [tex]\( s \geq 0 \)[/tex]
- [tex]\( l \geq 0 \)[/tex]
By using these inequalities, Elam can determine how many of each type of box he needs to use without exceeding the total number of books or boxes allowed.
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