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Ben is decorating his home with vases of flowers. He bought glass vases that cost [tex]\[tex]$22[/tex] each and ceramic vases that cost [tex]\$[/tex]14[/tex] each. The total cost of the vases came to more than [tex]\$172[/tex]. Also, Ben bought no more than 10 vases in all.

Which system of inequalities can be used to determine the number of glass vases, [tex]x[/tex], and the number of ceramic vases, [tex]y[/tex], that Ben could have bought?

A. [tex]22x + 14y \ \textgreater \ 172 \\ x + y \leq 10[/tex]

B. [tex]14x + 22y \geq 172 \\ x + y \ \textless \ 10[/tex]

C. [tex]14x + 22y \ \textgreater \ 172 \\ x + y \leq 10[/tex]

D. [tex]\begin{aligned} 22x + 14y &\geq 172 \\ x + y &\ \textless \ 10 \end{aligned}[/tex]



Answer :

GinnyAnswer
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]

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