Write the set of points from -4 to -2 but excluding -3 and -2 as a union of intervals
The correct answer and explanation is :
The given set of points is from −4-4 to −2-2, but we are excluding −3-3 and −2-2. To express this set as a union of intervals, we need to break it into parts where the excluded points create gaps.
- The original interval before exclusions: [−4,−2][-4, -2]
- Excluding −3-3 means breaking the interval into two parts: [−4,−3)∪(−3,−2][-4, -3) \cup (-3, -2]
- Excluding −2-2 further means removing it from (−3,−2](-3, -2], leaving just (−3,−2)(-3, -2) instead.
Thus, the correct union of intervals representation is: [−4,−3)∪(−3,−2)[-4, -3) \cup (-3, -2)
Explanation (300 words)
To correctly express a set of numbers as a union of intervals, we need to determine the continuous sections of the set and use interval notation appropriately. Intervals come in four main types:
- Closed interval [a,b][a, b] includes both endpoints aa and bb.
- Open interval (a,b)(a, b) excludes both endpoints.
- Half-open (or half-closed) intervals [a,b)[a, b) or (a,b](a, b] include one endpoint but not the other.
For our given problem, the set of points is initially from −4-4 to −2-2, meaning the complete closed interval is [−4,−2][-4, -2]. However, we must exclude the points −3-3 and −2-2, which introduces breaks in the interval.
- Step 1: Exclude −3-3
Removing −3-3 means splitting the interval [−4,−2][-4, -2] into two parts: [−4,−3)[-4, -3) and (−3,−2](-3, -2]. The notation [−4,−3)[-4, -3) means −4-4 is included, but −3-3 is not. - Step 2: Exclude −2-2
Since −2-2 was included in the second interval (−3,−2](-3, -2], removing it turns that into (−3,−2)(-3, -2).
Now, the final expression is: [−4,−3)∪(−3,−2)[-4, -3) \cup (-3, -2)
This means we include all numbers from −4-4 to just before −3-3, and from just after −3-3 up to just before −2-2, effectively excluding −3-3 and −2-2.