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To determine the correct system of inequalities, let's analyze the problem:

1. **Cost of the vases**: 
   - Each glass vase costs $22, so the total cost for $ x $ glass vases is $ 22x $.
   - Each ceramic vase costs $14, so the total cost for $ y $ ceramic vases is $ 14y $.
   - The total cost needs to be more than $172: 
     \[
     22x + 14y > 172
     \]

2. **Total number of vases**: 
   - Ben bought no more than 10 vases in total. Thus, the inequality for the number of vases is:
     \[
     x + y \leq 10
     \]

The system of inequalities that represents this scenario is:
\[
22x + 14y > 172
\]
\[
x + y \leq 10
\]

This matches option **B**, making it the correct answer.

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Would you like more details or have any questions? 

Here are 5 related questions to expand your understanding:
1. What would change if the total cost needed to be exactly $172?
2. How would the inequalities change if Ben bought at least 5 vases?
3. If glass vases were $30 each, how would the solution change?
4. What if there was a constraint that at least 3 ceramic vases had to be bought?
5. How can this system of inequalities be graphed on a coordinate plane?

**Tip:** When dealing with inequalities, always pay attention to the direction of the inequality sign (> or <), as it determines whether the condition is "greater than" or "less than."

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

Learn how to solve a system of inequalities representing the number of glass and ceramic vases purchased by Ben, considering the total cost and quantity. Suitable for students in Grades 8-10.

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