Answer :
To solve Ben's problem of determining the number of glass and ceramic vases he bought, we need to set up a system of inequalities that reflect the given conditions.
1. Cost Condition:
- Each glass vase costs [tex]$22, and each ceramic vase costs $[/tex]14.
- The total cost of the vases must be more than $172.
- Therefore, the inequality for the cost is:
[tex]\[ 22x + 14y > 172 \][/tex]
Here, [tex]\( x \)[/tex] represents the number of glass vases, and [tex]\( y \)[/tex] represents the number of ceramic vases.
2. Quantity Condition:
- Ben bought no more than 10 vases in total.
- This gives us the inequality for the number of vases as:
[tex]\[ x + y \leq 10 \][/tex]
Now, let's match these conditions with the options provided:
- Option A:
[tex]\[ 14x + 22y > 172 \][/tex]
[tex]\[ x + y \leq 10 \][/tex]
- The cost inequality here is incorrect because it multiplies the glass vases by 14 and ceramic vases by 22.
- Option B:
[tex]\[ 14x + 22y \geq 172 \][/tex]
[tex]\[ x + y < 10 \][/tex]
- This option has the wrong cost inequality format, as well as incorrect conditions for the number of vases.
- Option C:
[tex]\[ 22x + 14y > 172 \][/tex]
[tex]\[ x + y \leq 10 \][/tex]
- This option correctly states the cost inequality with glass vases multiplied by 22 and ceramic vases by 14 and has the correct condition for the total number of vases.
- Option D:
[tex]\[ 22x + 14y \geq 172 \][/tex]
[tex]\[ x + y < 10 \][/tex]
- This option has the correct cost function but incorrect inequality signs for both conditions.
Thus, the right choice for the system of inequalities is Option C, because it correctly reflects both the total cost condition and the number of vases condition.
1. Cost Condition:
- Each glass vase costs [tex]$22, and each ceramic vase costs $[/tex]14.
- The total cost of the vases must be more than $172.
- Therefore, the inequality for the cost is:
[tex]\[ 22x + 14y > 172 \][/tex]
Here, [tex]\( x \)[/tex] represents the number of glass vases, and [tex]\( y \)[/tex] represents the number of ceramic vases.
2. Quantity Condition:
- Ben bought no more than 10 vases in total.
- This gives us the inequality for the number of vases as:
[tex]\[ x + y \leq 10 \][/tex]
Now, let's match these conditions with the options provided:
- Option A:
[tex]\[ 14x + 22y > 172 \][/tex]
[tex]\[ x + y \leq 10 \][/tex]
- The cost inequality here is incorrect because it multiplies the glass vases by 14 and ceramic vases by 22.
- Option B:
[tex]\[ 14x + 22y \geq 172 \][/tex]
[tex]\[ x + y < 10 \][/tex]
- This option has the wrong cost inequality format, as well as incorrect conditions for the number of vases.
- Option C:
[tex]\[ 22x + 14y > 172 \][/tex]
[tex]\[ x + y \leq 10 \][/tex]
- This option correctly states the cost inequality with glass vases multiplied by 22 and ceramic vases by 14 and has the correct condition for the total number of vases.
- Option D:
[tex]\[ 22x + 14y \geq 172 \][/tex]
[tex]\[ x + y < 10 \][/tex]
- This option has the correct cost function but incorrect inequality signs for both conditions.
Thus, the right choice for the system of inequalities is Option C, because it correctly reflects both the total cost condition and the number of vases condition.
