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Answer :
To solve the problem of determining the inequalities that model Elam's packing scenario, let's break down the conditions and requirements step-by-step:
1. Variables Defined:
- [tex]\( s \)[/tex] represents the number of small boxes.
- [tex]\( l \)[/tex] represents the number of large boxes.
2. Box Constraints:
- A small box can hold 8 books.
- A large box can hold 12 books.
3. Given Conditions:
- He has at most 160 books to pack.
- He can use less than 30 boxes in total.
4. Inequality Constraints:
- For the total number of books not exceeding 160, the inequality is:
[tex]\[
8s + 12l \leq 160
\][/tex]
This inequality ensures that the total number of books packed does not exceed 160, considering that each small box holds 8 books and each large box holds 12 books.
- For the total number of boxes being less than 30, the inequality is:
[tex]\[
s + l < 30
\][/tex]
This inequality limits the number of boxes (small plus large) to be less than 30, adhering to the constraint of using fewer than 30 boxes.
5. Non-Negative Constraints:
- Additionally, as given, [tex]\( s \geq 0 \)[/tex] and [tex]\( l \geq 0 \)[/tex] are included to ensure that the number of boxes can't be negative.
To summarize, the inequalities that complete the system for Elam's scenario are:
- [tex]\( s + l < 30 \)[/tex]
- [tex]\( 8s + 12l \leq 160 \)[/tex]
These represent the combined requirements based on the problem's conditions.
1. Variables Defined:
- [tex]\( s \)[/tex] represents the number of small boxes.
- [tex]\( l \)[/tex] represents the number of large boxes.
2. Box Constraints:
- A small box can hold 8 books.
- A large box can hold 12 books.
3. Given Conditions:
- He has at most 160 books to pack.
- He can use less than 30 boxes in total.
4. Inequality Constraints:
- For the total number of books not exceeding 160, the inequality is:
[tex]\[
8s + 12l \leq 160
\][/tex]
This inequality ensures that the total number of books packed does not exceed 160, considering that each small box holds 8 books and each large box holds 12 books.
- For the total number of boxes being less than 30, the inequality is:
[tex]\[
s + l < 30
\][/tex]
This inequality limits the number of boxes (small plus large) to be less than 30, adhering to the constraint of using fewer than 30 boxes.
5. Non-Negative Constraints:
- Additionally, as given, [tex]\( s \geq 0 \)[/tex] and [tex]\( l \geq 0 \)[/tex] are included to ensure that the number of boxes can't be negative.
To summarize, the inequalities that complete the system for Elam's scenario are:
- [tex]\( s + l < 30 \)[/tex]
- [tex]\( 8s + 12l \leq 160 \)[/tex]
These represent the combined requirements based on the problem's conditions.
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