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Answer :
To solve this problem, we need to set up a system of inequalities that models the packing of books into boxes.
Let's break it down step-by-step:
1. Variables Defined:
- Let [tex]\( s \)[/tex] represent the number of small boxes.
- Let [tex]\( l \)[/tex] represent the number of large boxes.
2. Constraints Based on the Problem:
- A small box can hold 8 books, while a large box can hold 12 books.
- Elam has at most 160 books to pack.
- There are fewer than 30 boxes in total.
3. Inequalities:
- Since you can't have a negative number of boxes, the first two inequalities are:
[tex]\[
s \geq 0
\][/tex]
[tex]\[
l \geq 0
\][/tex]
- The total number of boxes (small and large combined) is less than 30:
[tex]\[
s + l < 30
\][/tex]
- The total number of books packed, which includes books from both small and large boxes, cannot exceed 160:
[tex]\[
8s + 12l \leq 160
\][/tex]
These inequalities form a complete system that describes Elam's constraints for packing his books:
- [tex]\[ s \geq 0 \][/tex]
- [tex]\[ l \geq 0 \][/tex]
- [tex]\[ s + l < 30 \][/tex]
- [tex]\[ 8s + 12l \leq 160 \][/tex]
These inequalities account for the nonnegative number of boxes, the limit on the total number of books, and the total number of boxes available.
Let's break it down step-by-step:
1. Variables Defined:
- Let [tex]\( s \)[/tex] represent the number of small boxes.
- Let [tex]\( l \)[/tex] represent the number of large boxes.
2. Constraints Based on the Problem:
- A small box can hold 8 books, while a large box can hold 12 books.
- Elam has at most 160 books to pack.
- There are fewer than 30 boxes in total.
3. Inequalities:
- Since you can't have a negative number of boxes, the first two inequalities are:
[tex]\[
s \geq 0
\][/tex]
[tex]\[
l \geq 0
\][/tex]
- The total number of boxes (small and large combined) is less than 30:
[tex]\[
s + l < 30
\][/tex]
- The total number of books packed, which includes books from both small and large boxes, cannot exceed 160:
[tex]\[
8s + 12l \leq 160
\][/tex]
These inequalities form a complete system that describes Elam's constraints for packing his books:
- [tex]\[ s \geq 0 \][/tex]
- [tex]\[ l \geq 0 \][/tex]
- [tex]\[ s + l < 30 \][/tex]
- [tex]\[ 8s + 12l \leq 160 \][/tex]
These inequalities account for the nonnegative number of boxes, the limit on the total number of books, and the total number of boxes available.
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