Express the answer in interval notation.
$(-\infty,-9] \cup[-5, \infty)$
A. [-9,-5]
B. (-9,5)
C. [-5, \infty)
Answer (1)
The problem asks to express the union of intervals ( − ∞ , − 9 ] ∪ [ − 5 , ∞ ) in interval notation.
The interval ( − ∞ , − 9 ] includes all real numbers less than or equal to -9.
The interval [ − 5 , ∞ ) includes all real numbers greater than or equal to -5.
The union of these two intervals is ( − ∞ , − 9 ] ∪ [ − 5 , ∞ ) . Since there is a gap between -9 and -5, the interval cannot be simplified further. The final answer is ( − ∞ , − 9 ] ∪ [ − 5 , ∞ ) .
Explanation
Understanding the problem We are asked to express the union of two intervals in interval notation. The given intervals are ( − ∞ , − 9 ] and [ − 5 , ∞ ) . The union of these intervals includes all the numbers that are in either of the intervals.
Analyzing the intervals The interval ( − ∞ , − 9 ] represents all real numbers less than or equal to -9. The interval [ − 5 , ∞ ) represents all real numbers greater than or equal to -5.
Finding the union The union of the intervals ( − ∞ , − 9 ] and [ − 5 , ∞ ) is written as ( − ∞ , − 9 ] ∪ [ − 5 , ∞ ) . Since there is a gap between -9 and -5, the expression cannot be simplified further.
Final Answer Therefore, the answer is ( − ∞ , − 9 ] ∪ [ − 5 , ∞ ) .
Examples
Interval notation is used in various fields, such as statistics, calculus, and computer science, to represent ranges of values. For example, in statistics, confidence intervals are expressed using interval notation to indicate the range within which a population parameter is likely to fall. In calculus, intervals are used to define the domain and range of functions. In computer science, intervals can represent ranges of valid inputs or outputs for algorithms.
