To find the domain of the relationship, we need to determine the range of possible values that [tex]\( x \)[/tex] (the number of youth tickets) can take, given the total amount spent on tickets is \[tex]$90 and the costs of the youth and adult tickets are \$[/tex]5 and \[tex]$9, respectively.
1. Formulating the equation:
We are given:
- Youth ticket cost: \$[/tex]5
- Adult ticket cost: \[tex]$9
- Total amount spent: \$[/tex]90
We can set up the following equation based on the information provided:
[tex]\[ 5x + 9y = 90 \][/tex]
2. Solving for [tex]\( x \)[/tex]:
To find the domain of [tex]\( x \)[/tex], let's consider the extreme cases:
- Maximum value of [tex]\( x \)[/tex]: This happens when the number of adult tickets [tex]\( y \)[/tex] is minimized. The minimum value for [tex]\( y \)[/tex] is 0. If [tex]\( y = 0 \)[/tex], the equation simplifies to:
[tex]\[ 5x = 90 \Rightarrow x = \frac{90}{5} = 18 \][/tex]
So, the maximum possible value for [tex]\( x \)[/tex] is 18.
3. Minimum value of [tex]\( x \)[/tex]: [tex]\( x \)[/tex] can be a minimum of 0 since the number of tickets cannot be negative. Now the equation would be:
[tex]\[ 5(0) + 9y = 90 \Rightarrow y = \frac{90}{9} = 10 \][/tex]
Therefore, the minimum value for [tex]\( x \)[/tex] is 0.
4. Combining these constraints:
[tex]\( x \)[/tex] can range from 0 to 18, inclusive.
Therefore, the domain of the relationship [tex]\( x \)[/tex] is:
[tex]\[ 0 \leq x \leq 18 \][/tex]
Given the multiple-choice options, the correct answer is:
[tex]\[ \boxed{0 \leq x \leq 18} \][/tex]
