Answer :
To solve the problem of determining which set of inequalities correctly represents Caitlyn's situation, let's break down what we know:
1. Total Number of Toys Sold: Caitlyn needs to sell at least 10 toys in total. This means the total number of xylophones ([tex]\(x\)[/tex]) and yo-yos ([tex]\(y\)[/tex]) sold should satisfy the inequality:
[tex]\[
x + y \geq 10
\][/tex]
2. Sales Revenue Requirement: Caitlyn sells xylophones for [tex]$2.00 each and yo-yos for $[/tex]1.00 each. She needs to earn at least $16.00 to make a profit. Therefore, the revenue from selling [tex]\(x\)[/tex] xylophones and [tex]\(y\)[/tex] yo-yos should satisfy the inequality:
[tex]\[
2.00x + 1.00y \geq 16.00
\][/tex]
Now, we need to find the correct set of inequalities that matches the described conditions from the options given:
- [tex]\(\left\{\begin{array}{l}x+y \geq 16.00 \\ 1.00 x+2.00 y \geq 10\end{array}\right.\)[/tex]
- [tex]\(\left\{\begin{array}{l}x+y \geq 10 \\ 1.00 x+2.00 y \geq 16.00\end{array}\right.\)[/tex]
- [tex]\(\left\{\begin{array}{l}x+y \geq 16.00 \\ 2.00 x+1.00 y \geq 10\end{array}\right.\)[/tex]
- [tex]\(\left\{\begin{array}{l}x+y \geq 10 \\ 2.00 x+1.00 y \geq 16.00\end{array}\right.\)[/tex]
Considering the conditions:
- The condition [tex]\(x + y \geq 10\)[/tex] is for the total number of toys.
- The condition [tex]\(2.00x + 1.00y \geq 16.00\)[/tex] is for the total sales amount.
By comparing these with each set of options, we see that the correct choice is the fourth set of inequalities:
[tex]\[
\left\{\begin{array}{l}
x + y \geq 10 \\
2.00x + 1.00y \geq 16.00
\end{array}\right.
\][/tex]
This matches Caitlyn's requirements both for the total number of toys and total sales revenue.
1. Total Number of Toys Sold: Caitlyn needs to sell at least 10 toys in total. This means the total number of xylophones ([tex]\(x\)[/tex]) and yo-yos ([tex]\(y\)[/tex]) sold should satisfy the inequality:
[tex]\[
x + y \geq 10
\][/tex]
2. Sales Revenue Requirement: Caitlyn sells xylophones for [tex]$2.00 each and yo-yos for $[/tex]1.00 each. She needs to earn at least $16.00 to make a profit. Therefore, the revenue from selling [tex]\(x\)[/tex] xylophones and [tex]\(y\)[/tex] yo-yos should satisfy the inequality:
[tex]\[
2.00x + 1.00y \geq 16.00
\][/tex]
Now, we need to find the correct set of inequalities that matches the described conditions from the options given:
- [tex]\(\left\{\begin{array}{l}x+y \geq 16.00 \\ 1.00 x+2.00 y \geq 10\end{array}\right.\)[/tex]
- [tex]\(\left\{\begin{array}{l}x+y \geq 10 \\ 1.00 x+2.00 y \geq 16.00\end{array}\right.\)[/tex]
- [tex]\(\left\{\begin{array}{l}x+y \geq 16.00 \\ 2.00 x+1.00 y \geq 10\end{array}\right.\)[/tex]
- [tex]\(\left\{\begin{array}{l}x+y \geq 10 \\ 2.00 x+1.00 y \geq 16.00\end{array}\right.\)[/tex]
Considering the conditions:
- The condition [tex]\(x + y \geq 10\)[/tex] is for the total number of toys.
- The condition [tex]\(2.00x + 1.00y \geq 16.00\)[/tex] is for the total sales amount.
By comparing these with each set of options, we see that the correct choice is the fourth set of inequalities:
[tex]\[
\left\{\begin{array}{l}
x + y \geq 10 \\
2.00x + 1.00y \geq 16.00
\end{array}\right.
\][/tex]
This matches Caitlyn's requirements both for the total number of toys and total sales revenue.
