To solve the compound inequality [tex]\(x \leq -4\)[/tex] or [tex]\(x \geq 5\)[/tex] and express it in interval notation, we proceed as follows:
1. Identify the individual inequalities:
- [tex]\(x \leq -4\)[/tex]: This inequality includes all numbers less than or equal to [tex]\(-4\)[/tex].
- [tex]\(x \geq 5\)[/tex]: This inequality includes all numbers greater than or equal to [tex]\(5\)[/tex].
2. Express each inequality as an interval:
- For [tex]\(x \leq -4\)[/tex], we use [tex]\(-\infty\)[/tex] for the lowest possible value because there is no lower bound, and we go up to [tex]\(-4\)[/tex], including [tex]\(-4\)[/tex]. So, the interval is [tex]\((-\infty, -4]\)[/tex].
- For [tex]\(x \geq 5\)[/tex], we start from [tex]\(5\)[/tex] (including [tex]\(5\)[/tex]) and go up to [tex]\(\infty\)[/tex] because there is no upper bound. Thus, the interval is [tex]\([5, \infty)\)[/tex].
3. Combine the intervals:
- Since the compound inequality allows for [tex]\(x\)[/tex] to be in either one of these intervals (it’s an "or"), we combine them using the union symbol [tex]\(U\)[/tex]. Therefore, the combined interval notation is:
[tex]\[
(-\infty, -4] \cup [5, \infty)
\][/tex]
So the correct interval notation for the compound inequality [tex]\( x \leq -4 \text { or } x \geq 5 \)[/tex] is:
[tex]\[
(-\infty, -4] \cup [5, \infty)
\][/tex]
