Let's first carefully analyze the problem given and the necessary constraints for the populations of male and female gazelles.
1. Condition 1: The herd must have exactly three times more males than females.
- Representing this as an equation, we get: [tex]\( y = 3x \)[/tex].
2. Condition 2: The zoo only has room for a maximum of 12 female gazelles.
- This sets an upper limit on the number of female gazelles, [tex]\( x \)[/tex]. Thus, [tex]\( 0 < x \leq 12 \)[/tex].
3. Using Condition 1 and Condition 2:
- Given [tex]\( y = 3x \)[/tex], substituting the maximum value for [tex]\( x \)[/tex]:
- If [tex]\( x = 12 \)[/tex], then [tex]\( y = 3 \times 12 = 36 \)[/tex].
- Hence, [tex]\( y \leq 36 \)[/tex].
4. Constraints for the number of males, [tex]\( y \)[/tex]:
- Since the population of males is linked directly to the females being present, [tex]\( 0 < y \)[/tex]. Using the maximum limit for [tex]\( x \)[/tex], the maximum limit for [tex]\( y \)[/tex] becomes [tex]\( 36 \)[/tex]. Therefore, [tex]\( 0 < y \leq 36 \)[/tex].
By combining these constraints, we summarize:
- The number of female gazelles [tex]\( x \)[/tex] must be within the range [tex]\( 0 < x \leq 12 \)[/tex].
- The number of male gazelles [tex]\( y \)[/tex] must be within the range [tex]\( 0 < y \leq 36 \)[/tex].
The correct answer is:
b. [tex]\( 0
