Sure, let's analyze the given multiple-choice options to choose the solution for the equation [tex]\( y_1 = y_2 \)[/tex]:
1. Option A: [tex]\((-2, 5)\)[/tex]
- This option represents a pair of points, specifically the point [tex]\((-2, 5)\)[/tex]. This is not a set of solution points, and thus it is not appropriate for denoting the solutions to an equation.
2. Option B: [tex]\(\varnothing\)[/tex]
- The symbol [tex]\(\varnothing\)[/tex] (phi) represents an empty set, which means there are no solutions. This choice indicates that there are no points where [tex]\( y_1 \)[/tex] equals [tex]\( y_2 \)[/tex]. This can be the correct answer only if there is no intersection between the two functions.
3. Option C: [tex]\(\{-2,5\}\)[/tex]
- This option represents a set containing the points [tex]\(-2\)[/tex] and [tex]\(5\)[/tex]. This indicates that the equation [tex]\( y_1 = y_2 \)[/tex] has solutions at [tex]\( x = -2 \)[/tex] and [tex]\( x = 5 \)[/tex].
4. Option D: [tex]\((-\infty, -2) \cup (5, \infty)\)[/tex]
- This option represents an interval on the number line, namely all real numbers less than [tex]\(-2\)[/tex] together with all real numbers greater than [tex]\(5\)[/tex]. This is suggesting a range of solutions, excluding points [tex]\(-2\)[/tex] and [tex]\(5\)[/tex].
Upon carefully analyzing these options, the most accurate representation of the solution set for [tex]\( y_1 = y_2 \)[/tex] where there are specific points that satisfy the equation is:
Option C: [tex]\(\{-2, 5\}\)[/tex]
Thus, the solution for [tex]\( y_1 = y_2 \)[/tex] is:
C. [tex]\(\{-2, 5\}\)[/tex]
