Let's evaluate each given point to see if it satisfies the inequality [tex]\( 24x + 10y \geq 44 \)[/tex], where [tex]\( x \)[/tex] represents the number of adult tickets and [tex]\( y \)[/tex] represents the number of child tickets.
1. Point (1, 1):
- Substituting [tex]\( x = 1 \)[/tex] and [tex]\( y = 1 \)[/tex] into the inequality:
[tex]\[
24(1) + 10(1) = 24 + 10 = 34
\][/tex]
- The result [tex]\( 34 \)[/tex] is less than [tex]\( 44 \)[/tex], so the point [tex]\((1, 1)\)[/tex] is not a viable solution.
2. Point (8, 2.5):
- Substituting [tex]\( x = 8 \)[/tex] and [tex]\( y = 2.5 \)[/tex] into the inequality:
[tex]\[
24(8) + 10(2.5) = 192 + 25 = 217
\][/tex]
- The result [tex]\( 217 \)[/tex] is greater than [tex]\( 44 \)[/tex], so the point [tex]\((8, 2.5)\)[/tex] is a viable solution.
3. Point (4, -2):
- Substituting [tex]\( x = 4 \)[/tex] and [tex]\( y = -2 \)[/tex] into the inequality:
[tex]\[
24(4) + 10(-2) = 96 - 20 = 76
\][/tex]
- The result [tex]\( 76 \)[/tex] is greater than [tex]\( 44 \)[/tex], so the point [tex]\((4, -2)\)[/tex] is a viable solution. (Note: While this mathematical solution is viable, a negative number of children's tickets is not practical in a real-world context.)
4. Point (4, 6):
- Substituting [tex]\( x = 4 \)[/tex] and [tex]\( y = 6 \)[/tex] into the inequality:
[tex]\[
24(4) + 10(6) = 96 + 60 = 156
\][/tex]
- The result [tex]\( 156 \)[/tex] is greater than [tex]\( 44 \)[/tex], so the point [tex]\((4, 6)\)[/tex] is a viable solution.
Summary:
- The points [tex]\((8, 2.5)\)[/tex], [tex]\((4, -2)\)[/tex], and [tex]\((4, 6)\)[/tex] are viable solutions.
- The point [tex]\((1,1)\)[/tex] is not a viable solution.
