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Answer :
To solve the problem of packing books for Elam, we need to identify which inequalities complete the system based on the given conditions:
1. Understand the Box Capacity:
- A small box can hold 8 books.
- A large box can hold 12 books.
2. Total Books Constraint:
- Elam has at most 160 books. So, the total number of books packed in the small and large boxes must be at most 160.
- This gives us the inequality: [tex]\(8s + 12l \leq 160\)[/tex].
3. Total Box Constraint:
- Elam can use less than 30 boxes in total. This means that the total of small and large boxes used must be less than 30.
- This condition is represented by the inequality: [tex]\(s + l < 30\)[/tex].
4. Non-Negativity Constraints:
- Since Elam cannot use a negative number of boxes, both small and large box counts must be non-negative.
- These constraints are given as: [tex]\(s \geq 0\)[/tex] and [tex]\(l \geq 0\)[/tex].
Putting it all together, the system of inequalities that models this situation is:
- [tex]\(8s + 12l \leq 160\)[/tex]: This ensures that the total number of books does not exceed 160.
- [tex]\(s + l < 30\)[/tex]: This ensures that the total number of boxes used is less than 30.
- [tex]\(s \geq 0\)[/tex] and [tex]\(l \geq 0\)[/tex]: These ensure that the numbers of small and large boxes are non-negative.
These inequalities comprehensively describe the constraints Elam faces when packing his books into boxes.
1. Understand the Box Capacity:
- A small box can hold 8 books.
- A large box can hold 12 books.
2. Total Books Constraint:
- Elam has at most 160 books. So, the total number of books packed in the small and large boxes must be at most 160.
- This gives us the inequality: [tex]\(8s + 12l \leq 160\)[/tex].
3. Total Box Constraint:
- Elam can use less than 30 boxes in total. This means that the total of small and large boxes used must be less than 30.
- This condition is represented by the inequality: [tex]\(s + l < 30\)[/tex].
4. Non-Negativity Constraints:
- Since Elam cannot use a negative number of boxes, both small and large box counts must be non-negative.
- These constraints are given as: [tex]\(s \geq 0\)[/tex] and [tex]\(l \geq 0\)[/tex].
Putting it all together, the system of inequalities that models this situation is:
- [tex]\(8s + 12l \leq 160\)[/tex]: This ensures that the total number of books does not exceed 160.
- [tex]\(s + l < 30\)[/tex]: This ensures that the total number of boxes used is less than 30.
- [tex]\(s \geq 0\)[/tex] and [tex]\(l \geq 0\)[/tex]: These ensure that the numbers of small and large boxes are non-negative.
These inequalities comprehensively describe the constraints Elam faces when packing his books into boxes.
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