Answer :
To solve the given problem, we need to determine the possible numbers of hats and scarves Alex can bring to the art fair, given the constraints provided. Let's walk through the steps:
1. Identify Variables:
- Let [tex]\( x \)[/tex] represent the number of hats Alex makes.
- Let [tex]\( y \)[/tex] represent the number of scarves Alex makes.
2. Determine Constraints:
- Alex can make no more than 40 items in total. This gives us the inequality:
[tex]\[
x + y \leq 40
\][/tex]
- Alex wants to bring at least 25 items total. This gives us another inequality:
[tex]\[
x + y \geq 25
\][/tex]
3. Formulate the System of Inequalities:
- The system of inequalities that represents Alex’s conditions is:
[tex]\[
\begin{cases}
x + y \leq 40 \\
x + y \geq 25
\end{cases}
\][/tex]
4. Analyze the Choices:
- From the options provided, we need to select the inequalities that match our system:
- Choice G: [tex]\( x + y \geq 25 \)[/tex]
- Choice H: [tex]\( x + y \leq 40 \)[/tex]
By following these steps, we conclude that Alex's possible numbers of hats and scarves are described by the inequalities [tex]\( x + y \geq 25 \)[/tex] and [tex]\( x + y \leq 40 \)[/tex]. Therefore, the correct answers are choices G and H.
1. Identify Variables:
- Let [tex]\( x \)[/tex] represent the number of hats Alex makes.
- Let [tex]\( y \)[/tex] represent the number of scarves Alex makes.
2. Determine Constraints:
- Alex can make no more than 40 items in total. This gives us the inequality:
[tex]\[
x + y \leq 40
\][/tex]
- Alex wants to bring at least 25 items total. This gives us another inequality:
[tex]\[
x + y \geq 25
\][/tex]
3. Formulate the System of Inequalities:
- The system of inequalities that represents Alex’s conditions is:
[tex]\[
\begin{cases}
x + y \leq 40 \\
x + y \geq 25
\end{cases}
\][/tex]
4. Analyze the Choices:
- From the options provided, we need to select the inequalities that match our system:
- Choice G: [tex]\( x + y \geq 25 \)[/tex]
- Choice H: [tex]\( x + y \leq 40 \)[/tex]
By following these steps, we conclude that Alex's possible numbers of hats and scarves are described by the inequalities [tex]\( x + y \geq 25 \)[/tex] and [tex]\( x + y \leq 40 \)[/tex]. Therefore, the correct answers are choices G and H.
