To solve the problem, let's identify the constraints given in the problem and form the relevant inequalities accordingly:
1. The first constraint is about the total cost of the vases. Ben bought glass vases that cost [tex]$22 each and ceramic vases that cost $[/tex]14 each. The total cost came to more than $172.
This translates to the inequality:
[tex]\[
22x + 14y > 172
\][/tex]
where \( x \) is the number of glass vases and \( y \) is the number of ceramic vases.
2. The second constraint is about the total number of vases. Ben bought no more than 10 vases in all.
This translates to the inequality:
[tex]\[
x + y \leq 10
\][/tex]
Combining these two inequalities, we get the system of inequalities:
[tex]\[
22x + 14y > 172
\][/tex]
[tex]\[
x + y \leq 10
\][/tex]
Now, let's match this system of inequalities with the given options:
A. \(22x + 14y > 172\)
[tex]\[
x + y \leq 10
\][/tex]
B. \(14x + 22y \geq 172\)
[tex]\[
x + y < 10
\][/tex]
C. \(14x + 22y > 172\)
[tex]\[
x + y \leq 10
\][/tex]
D. \(22x + 14y \geq 172\)
[tex]\[
x + y < 10
\][/tex]
Comparing the derived inequalities with the given options, we find that Option A matches the system of inequalities:
[tex]\[
22x + 14y > 172
\][/tex]
[tex]\[
x + y \leq 10
\][/tex]
Therefore, the correct answer is:
A. \(22 x + 14 y > 172\)
[tex]\[
x + y \leq 10\][/tex]
