Answer :
To solve the problem of determining which system of inequalities can be used to determine the number of glass vases (represented by [tex]\( x \)[/tex]) and ceramic vases (represented by [tex]\( y \)[/tex]) that Ben could have bought, we need to consider the conditions provided:
1. Cost Condition: Each glass vase costs [tex]$22 and each ceramic vase costs $[/tex]14. The total cost of these vases is more than [tex]$172.
2. Quantity Condition: Ben bought no more than 10 vases in total.
Step-by-step Solution:
1. Express the Cost Condition Mathematically:
- The total cost of the vases can be expressed as \( 22x + 14y \).
- The problem states that the total cost is more than $[/tex]172, which translates to the inequality:
[tex]\[
22x + 14y > 172
\][/tex]
2. Express the Quantity Condition Mathematically:
- The total number of vases Ben bought is [tex]\( x + y \)[/tex].
- Ben bought no more than 10 vases, which means:
[tex]\[
x + y \leq 10
\][/tex]
3. System of Inequalities:
- Combine the inequalities:
- The cost inequality: [tex]\( 22x + 14y > 172 \)[/tex]
- The quantity inequality: [tex]\( x + y \leq 10 \)[/tex]
Based on these steps, the correct system of inequalities that satisfies both the cost and quantity conditions is:
- Option B:
[tex]\[
\begin{align*}
22x + 14y & > 172 \\
x + y & \leq 10
\end{align*}
\][/tex]
This system accurately reflects the conditions provided by the problem: the total cost is more than $172, and no more than 10 vases were bought.
1. Cost Condition: Each glass vase costs [tex]$22 and each ceramic vase costs $[/tex]14. The total cost of these vases is more than [tex]$172.
2. Quantity Condition: Ben bought no more than 10 vases in total.
Step-by-step Solution:
1. Express the Cost Condition Mathematically:
- The total cost of the vases can be expressed as \( 22x + 14y \).
- The problem states that the total cost is more than $[/tex]172, which translates to the inequality:
[tex]\[
22x + 14y > 172
\][/tex]
2. Express the Quantity Condition Mathematically:
- The total number of vases Ben bought is [tex]\( x + y \)[/tex].
- Ben bought no more than 10 vases, which means:
[tex]\[
x + y \leq 10
\][/tex]
3. System of Inequalities:
- Combine the inequalities:
- The cost inequality: [tex]\( 22x + 14y > 172 \)[/tex]
- The quantity inequality: [tex]\( x + y \leq 10 \)[/tex]
Based on these steps, the correct system of inequalities that satisfies both the cost and quantity conditions is:
- Option B:
[tex]\[
\begin{align*}
22x + 14y & > 172 \\
x + y & \leq 10
\end{align*}
\][/tex]
This system accurately reflects the conditions provided by the problem: the total cost is more than $172, and no more than 10 vases were bought.
