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Answer :
To solve this problem, we need to determine the system of inequalities that reflects the constraints given:
1. Cost Constraint: Each glass vase costs [tex]$22, and each ceramic vase costs $[/tex]14. Ben's total spending on vases is more than [tex]$172.
2. Quantity Constraint: Ben bought no more than 10 vases in total.
We'll use the following variables:
- \( x \) represents the number of glass vases.
- \( y \) represents the number of ceramic vases.
Step 1: Formulate the Cost Inequality
The total cost of the vases can be calculated using the price of each type of vase and the quantities purchased:
- The cost for glass vases is \( 22x \).
- The cost for ceramic vases is \( 14y \).
The total cost of these vases is more than $[/tex]172, so we have the inequality:
[tex]\[ 22x + 14y > 172 \][/tex]
Step 2: Formulate the Quantity Inequality
The problem states that the total number of vases is no more than 10. This gives us the inequality:
[tex]\[ x + y \leq 10 \][/tex]
Step 3: Combine the Two Inequalities
The system of inequalities that represents the problem is:
1. [tex]\( 22x + 14y > 172 \)[/tex]
2. [tex]\( x + y \leq 10 \)[/tex]
Thus, the correct choice from the given options is:
D.
[tex]\[ 22x + 14y > 172 \][/tex]
[tex]\[ x + y \leq 10 \][/tex]
This system respects the conditions given — that Ben's total cost for vases is more than $172 and that he bought no more than 10 vases in total.
1. Cost Constraint: Each glass vase costs [tex]$22, and each ceramic vase costs $[/tex]14. Ben's total spending on vases is more than [tex]$172.
2. Quantity Constraint: Ben bought no more than 10 vases in total.
We'll use the following variables:
- \( x \) represents the number of glass vases.
- \( y \) represents the number of ceramic vases.
Step 1: Formulate the Cost Inequality
The total cost of the vases can be calculated using the price of each type of vase and the quantities purchased:
- The cost for glass vases is \( 22x \).
- The cost for ceramic vases is \( 14y \).
The total cost of these vases is more than $[/tex]172, so we have the inequality:
[tex]\[ 22x + 14y > 172 \][/tex]
Step 2: Formulate the Quantity Inequality
The problem states that the total number of vases is no more than 10. This gives us the inequality:
[tex]\[ x + y \leq 10 \][/tex]
Step 3: Combine the Two Inequalities
The system of inequalities that represents the problem is:
1. [tex]\( 22x + 14y > 172 \)[/tex]
2. [tex]\( x + y \leq 10 \)[/tex]
Thus, the correct choice from the given options is:
D.
[tex]\[ 22x + 14y > 172 \][/tex]
[tex]\[ x + y \leq 10 \][/tex]
This system respects the conditions given — that Ben's total cost for vases is more than $172 and that he bought no more than 10 vases in total.
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