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Answer :
To solve this problem, we need to create a system of inequalities based on the conditions given. Let's break down the scenario:
1. Understanding Box Capacity:
- A small box can hold 8 books.
- A large box can hold 12 books.
- Elam has at most 160 books to pack.
2. Limit on Total Boxes:
- The total number of boxes (small plus large) should be less than 30.
3. Conditions for Boxes:
- Let [tex]\( s \)[/tex] be the number of small boxes and [tex]\( l \)[/tex] be the number of large boxes.
- Both [tex]\( s \)[/tex] and [tex]\( l \)[/tex] must be non-negative, i.e., [tex]\( s \geq 0 \)[/tex] and [tex]\( l \geq 0 \)[/tex].
4. Modeling the Inequalities:
- Books Constraint: The total number of books that can be packed in small and large boxes combined must not exceed 160:
[tex]\[
8s + 12l \leq 160
\][/tex]
- Box Count Constraint: The total number of boxes must be less than 30:
[tex]\[
s + l < 30
\][/tex]
Thus, combining these, the system of inequalities that models the scenario is:
- [tex]\( s \geq 0 \)[/tex]
- [tex]\( l \geq 0 \)[/tex]
- [tex]\( 8s + 12l \leq 160 \)[/tex]
- [tex]\( s + l < 30 \)[/tex]
In conclusion, the complete inequalities for the system are [tex]\( s + l < 30 \)[/tex] and [tex]\( 8s + 12l \leq 160 \)[/tex], along with [tex]\( s \geq 0 \)[/tex] and [tex]\( l \geq 0 \)[/tex].
1. Understanding Box Capacity:
- A small box can hold 8 books.
- A large box can hold 12 books.
- Elam has at most 160 books to pack.
2. Limit on Total Boxes:
- The total number of boxes (small plus large) should be less than 30.
3. Conditions for Boxes:
- Let [tex]\( s \)[/tex] be the number of small boxes and [tex]\( l \)[/tex] be the number of large boxes.
- Both [tex]\( s \)[/tex] and [tex]\( l \)[/tex] must be non-negative, i.e., [tex]\( s \geq 0 \)[/tex] and [tex]\( l \geq 0 \)[/tex].
4. Modeling the Inequalities:
- Books Constraint: The total number of books that can be packed in small and large boxes combined must not exceed 160:
[tex]\[
8s + 12l \leq 160
\][/tex]
- Box Count Constraint: The total number of boxes must be less than 30:
[tex]\[
s + l < 30
\][/tex]
Thus, combining these, the system of inequalities that models the scenario is:
- [tex]\( s \geq 0 \)[/tex]
- [tex]\( l \geq 0 \)[/tex]
- [tex]\( 8s + 12l \leq 160 \)[/tex]
- [tex]\( s + l < 30 \)[/tex]
In conclusion, the complete inequalities for the system are [tex]\( s + l < 30 \)[/tex] and [tex]\( 8s + 12l \leq 160 \)[/tex], along with [tex]\( s \geq 0 \)[/tex] and [tex]\( l \geq 0 \)[/tex].
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