IDNLearn.com provides a platform for sharing and gaining valuable knowledge. Join our knowledgeable community and get detailed, reliable answers to all your questions.
Sagot :
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]
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]
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Your questions deserve accurate answers. Thank you for visiting IDNLearn.com, and see you again for more solutions.