To convert the inequality [tex]\(\{x \mid 5 \leq x\}\)[/tex] into interval notation, let's go through the steps in detail:
1. Identify the Inequality: The given inequality is [tex]\(5 \leq x\)[/tex], which means [tex]\(x\)[/tex] is any number that is greater than or equal to 5.
2. Determine the Lower Bound: The inequality specifies that [tex]\(x\)[/tex] can be equal to 5, which means 5 is included in the set. In interval notation, numbers that are included in the set are represented with a square bracket "[", indicating that the boundary is included.
3. Determine the Upper Bound: There is no upper limit provided for [tex]\(x\)[/tex]. This means [tex]\(x\)[/tex] can be any number greater than or equal to 5, and there is no restriction on how large [tex]\(x\)[/tex] can be. In interval notation, this is represented by the symbol [tex]\(\infty\)[/tex] (infinity) to indicate that there is no upper bound.
4. Use Proper Symbols: Since infinity is not an actual number but rather a concept that represents an unbounded quantity, it is never included in the interval. Thus, we use a parenthesis ")", not a square bracket, to denote that infinity is not included.
5. Combine the Bounds: Putting together the lower bound of 5 (which is included) and the upper bound of infinity (which cannot be included), the interval notation is written as [tex]\([5, \infty)\)[/tex].
Thus, the inequality [tex]\(\{x \mid 5 \leq x\}\)[/tex] in interval notation is:
[tex]\[
[5, \infty)
\][/tex]
This indicates that [tex]\(x\)[/tex] can take any value from 5 upwards, including 5 itself, and without any upper limit.
