To determine which combination of bracelets Dimitri may have made, we need to check each option against the inequality [tex]\(9x + 20y > 120\)[/tex]:
1. For 3 simple bracelets and 4 deluxe bracelets:
- [tex]\( x = 3 \)[/tex]
- [tex]\( y = 4 \)[/tex]
- Calculate [tex]\(9x + 20y\)[/tex]:
[tex]\[
9(3) + 20(4) = 27 + 80 = 107
\][/tex]
- Since [tex]\( 107 \leq 120 \)[/tex], this combination does not satisfy the inequality.
2. For 0 simple bracelets and 6 deluxe bracelets:
- [tex]\( x = 0 \)[/tex]
- [tex]\( y = 6 \)[/tex]
- Calculate [tex]\(9x + 20y\)[/tex]:
[tex]\[
9(0) + 20(6) = 0 + 120 = 120
\][/tex]
- Since [tex]\( 120 \leq 120 \)[/tex], this combination does not satisfy the inequality.
3. For 12 simple bracelets and 0 deluxe bracelets:
- [tex]\( x = 12 \)[/tex]
- [tex]\( y = 0 \)[/tex]
- Calculate [tex]\(9x + 20y\)[/tex]:
[tex]\[
9(12) + 20(0) = 108 + 0 = 108
\][/tex]
- Since [tex]\( 108 \leq 120 \)[/tex], this combination does not satisfy the inequality.
4. For 7 simple bracelets and 3 deluxe bracelets:
- [tex]\( x = 7 \)[/tex]
- [tex]\( y = 3 \)[/tex]
- Calculate [tex]\(9x + 20y\)[/tex]:
[tex]\[
9(7) + 20(3) = 63 + 60 = 123
\][/tex]
- Since [tex]\( 123 > 120 \)[/tex], this combination does satisfy the inequality.
From these calculations, we can see that the possible combination of bracelets that Dimitri may have made, satisfying the inequality [tex]\(9x + 20y > 120\)[/tex], is 7 simple bracelets and 3 deluxe bracelets.
