To solve for [tex]\( y_1 < y_2 \)[/tex], we need to determine the intervals or specific points in the domain where [tex]\( y_1 \)[/tex] is strictly less than [tex]\( y_2 \)[/tex]. Let's examine each given option carefully.
A. [tex]\( (-2, 5) \)[/tex]
- This option represents the open interval between -2 and 5. Within this interval, we cannot guarantee that [tex]\( y_1 \)[/tex] is always less than [tex]\( y_2 \)[/tex] for all possible values of [tex]\( y_1 \)[/tex] and [tex]\( y_2 \)[/tex].
B. [tex]\( \{ -2, 5 \} \)[/tex]
- This option represents the set containing the points -2 and 5. At these specific points, we still cannot determine a consistent inequality relationship between [tex]\( y_1 \)[/tex] and [tex]\( y_2 \)[/tex].
C. [tex]\( \varnothing \)[/tex]
- This option represents the empty set, implying there are no values where [tex]\( y_1 \)[/tex] is less than [tex]\( y_2 \)[/tex]. However, it is unlikely that there are no such values altogether, unless [tex]\( y_1 \)[/tex] and [tex]\( y_2 \)[/tex] are identical functions, which is not specified here.
D. [tex]\( (-\infty, -2) \cup (5, \infty) \)[/tex]
- This option represents the union of two intervals: one extending from negative infinity to -2 (not including -2), and the other extending from 5 to positive infinity (not including 5). In these intervals, it can be inferred that [tex]\( y_1 \)[/tex] is consistently less than [tex]\( y_2 \)[/tex].
Given our choices and the need for [tex]\( y_1 \)[/tex] to be consistently less than [tex]\( y_2 \)[/tex], option D, [tex]\((-\infty, -2) \cup (5, \infty)\)[/tex], is the correct interval where [tex]\( y_1 < y_2 \)[/tex].
Thus, the correct answer for the inequality [tex]\( y_1 < y_2 \)[/tex] is:
[tex]\[ \boxed{4} \][/tex]
Which corresponds to:
[tex]\[ \boxed{(-\infty, -2) \cup (5, \infty)} \][/tex]
