To solve the problem of finding the union of the given intervals [tex]\(<-\infty,-2\rangle \cup[-1,3] \cup<4,+\infty\rangle\)[/tex], let's analyze each interval separately and then combine them to form the union.
1. First Interval: [tex]\(<-\infty, -2\rangle\)[/tex]
- This interval starts from [tex]\(-\infty\)[/tex] and goes up to [tex]\(-2\)[/tex], but it does not include [tex]\(-2\)[/tex] itself. It can be written as:
[tex]\[(-\infty, -2)\][/tex]
2. Second Interval: [tex]\([-1, 3]\)[/tex]
- This interval includes all the numbers from [tex]\(-1\)[/tex] to [tex]\(3\)[/tex], including both [tex]\(-1\)[/tex] and [tex]\(3\)[/tex]. Hence, it is a closed interval:
[tex]\([-1, 3]\)[/tex]
3. Third Interval: [tex]\(<4, +\infty\rangle\)[/tex]
- This interval starts just after [tex]\(4\)[/tex] and extends infinitely towards the positive direction, but it does not include [tex]\(4\)[/tex]. It is written as:
[tex]\((4, +\infty)\)[/tex]
Next, let's combine these intervals to form the union.
- The first interval [tex]\((-\infty, -2)\)[/tex] does not overlap with any other intervals.
- The second interval [tex]\([-1, 3]\)[/tex] stands alone, too; no other interval intersects with it.
- The third interval [tex]\((4, +\infty)\)[/tex] also remains separate from the others.
Combining these results, the union of the intervals [tex]\(<-\infty, -2\rangle \cup [-1, 3] \cup <4, +\infty\rangle\)[/tex] is:
[tex]\[
[(-\infty, -2), [-1, 3], (4, +\infty)]
\][/tex]
