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Answer :
To solve this problem, let's start by identifying the constraints given by the situation and translate them into mathematical inequalities.
1. Book Constraint:
- A small box can hold 8 books, and a large box can hold 12 books.
- Elam has at most 160 books to pack. Therefore, the total number of books packed in both small and large boxes should be less than or equal to 160. This gives us the inequality:
[tex]\[
8s + 12l \leq 160
\][/tex]
2. Box Constraint:
- The total number of boxes used (both small and large) should be less than 30. This gives us another inequality:
[tex]\[
s + l < 30
\][/tex]
3. Non-negative Constraint:
- Since you can't have a negative number of boxes, we also have:
[tex]\[
s \geq 0 \quad \text{and} \quad l \geq 0
\][/tex]
Combining these, the complete 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]
These inequalities ensure that Elam does not exceed the number of books the boxes can hold and does not use more boxes than available, all while accounting for non-negative numbers of small and large boxes.
1. Book Constraint:
- A small box can hold 8 books, and a large box can hold 12 books.
- Elam has at most 160 books to pack. Therefore, the total number of books packed in both small and large boxes should be less than or equal to 160. This gives us the inequality:
[tex]\[
8s + 12l \leq 160
\][/tex]
2. Box Constraint:
- The total number of boxes used (both small and large) should be less than 30. This gives us another inequality:
[tex]\[
s + l < 30
\][/tex]
3. Non-negative Constraint:
- Since you can't have a negative number of boxes, we also have:
[tex]\[
s \geq 0 \quad \text{and} \quad l \geq 0
\][/tex]
Combining these, the complete 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]
These inequalities ensure that Elam does not exceed the number of books the boxes can hold and does not use more boxes than available, all while accounting for non-negative numbers of small and large boxes.
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