Use graphs to find the set. $(-\infty, 0) \cup[-2,4)$
Select the correct choice below and fill in any answer boxes within your choice.
A. The set is $\square$ - (Type your answer in interval notation.)
B. The answer is the empty set.
Answer (1)
The problem asks to find the union of the sets ( − ∞ , 0 ) and [ − 2 , 4 ) .
The set ( − ∞ , 0 ) includes all numbers less than 0, and [ − 2 , 4 ) includes numbers from -2 (inclusive) to 4 (exclusive).
Combining these sets, the union extends from negative infinity up to 4 (exclusive).
The union of the two sets is ( − ∞ , 4 ) .
Explanation
Understanding the Problem We are asked to find the union of two sets: ( − ∞ , 0 ) and [ − 2 , 4 ) . The union of two sets contains all elements that are in either set.
Analyzing the Sets The first set, ( − ∞ , 0 ) , includes all real numbers less than 0, but not including 0. The second set, [ − 2 , 4 ) , includes all real numbers greater than or equal to -2 and less than 4.
Finding the Union To find the union, we combine these two sets. The interval ( − ∞ , 0 ) extends to negative infinity. The interval [ − 2 , 4 ) starts at -2 (inclusive) and goes up to 4 (exclusive). Since -2 is greater than negative infinity, the union will start at negative infinity. The interval extends to 4, not including 4.
Determining the Result Therefore, the union of the two sets is ( − ∞ , 4 ) .
Examples
Understanding set unions is crucial in many areas, such as scheduling and resource allocation. For example, if one group of students is available to work on a project during the time interval ( − ∞ , 0 ) (representing times before noon) and another group is available during the interval [ − 2 , 4 ) (representing times from 10 AM to 4 PM), then the combined availability of both groups is ( − ∞ , 4 ) , meaning someone is available at any time before 4 PM. This helps in coordinating schedules and ensuring coverage.
