It takes Dimitri 9 minutes to make a simple bracelet and 20 minutes to make a deluxe bracelet. He has been making bracelets for longer than 120 minutes. If [tex] x [/tex] represents the number of simple bracelets that he has made and [tex] y [/tex] represents the number of deluxe bracelets he has made, the inequality [tex] 9x + 20y \ \textgreater \ 120 [/tex] represents the scenario.

Which is a possible combination of bracelets that Dimitri may have made?

A. 3 simple bracelets and 4 deluxe bracelets
B. 0 simple bracelets and 6 deluxe bracelets
C. 12 simple bracelets and 0 deluxe bracelets
D. 7 simple bracelets and 3 deluxe bracelets



Answer :

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.