To find the union of the sets [tex]\( A = [-5, 3) \)[/tex] and [tex]\( B = [-8, 8] \)[/tex], we consider the overall coverage of these intervals on the number line.
1. Identifying endpoints of the intervals:
- Interval [tex]\( A = [-5, 3) \)[/tex] starts at [tex]\(-5\)[/tex] and goes up to, but does not include, [tex]\(3\)[/tex].
- Interval [tex]\( B = [-8, 8] \)[/tex] starts at [tex]\(-8\)[/tex] and goes up to [tex]\(8\)[/tex], including both endpoints.
2. Finding the union of the intervals:
- The union of two sets combines all the elements that are in either set or in both sets.
- The smallest value considered is [tex]\(-8\)[/tex] from interval [tex]\( B \)[/tex].
- The largest value considered is [tex]\(8\)[/tex] from interval [tex]\( B \)[/tex].
- Thus, we encompass all values starting from [tex]\(-8\)[/tex] up to [tex]\(8\)[/tex], including both endpoints since interval [tex]\( B \)[/tex] itself already includes these endpoints.
3. Simplifying the union:
- When we take the union of [tex]\( [-5, 3) \)[/tex] and [tex]\( [-8, 8] \)[/tex], we essentially span an uninterrupted section of the number line from [tex]\(-8\)[/tex] to [tex]\(8\)[/tex].
Therefore, the simplest version of the resulting set in interval notation is:
[tex]\[ A \cup B = [-8, 8] \][/tex]
