Using a number line, find both the intersection and the union of the following intervals. (−3, +∞) and (4, +∞)
so, did you do it?
The intersection is where they overlap: (-3,4)
The union is either of them: (-3,+∞) also includes (4,+∞)
To find the intersection and union of the intervals (-3, +∞) and (4, +∞) using a number line, follow these steps:
1. Plot the intervals on the number line.
- Start by marking a point for the lower bound of the first interval (-3).
- Then, mark a point for the upper bound of the first interval (+∞). Since +∞ is not a specific number, you can just extend the line to the right to signify infinity.
- Next, mark a point for the lower bound of the second interval (4).
- Finally, mark a point for the upper bound of the second interval (+∞), just like before.
Your number line should now have two marked intervals: (-3, +∞) and (4, +∞).
2. Determine the intersection.
- The intersection of two intervals is the set of values that are common to both intervals.
- In this case, since both intervals extend to positive infinity, the overlap between them is from 4 to infinity.
- So, the intersection of (-3, +∞) and (4, +∞) is (4, +∞).
3. Determine the union.
- The union of two intervals is the set of all values contained in either interval.
- In this case, since both intervals extend to positive infinity, the union is the combination of both intervals, which is (-3, +∞).
- So, the union of (-3, +∞) and (4, +∞) is (-3, +∞).
To summarize:
- The intersection of (-3, +∞) and (4, +∞) is (4, +∞).
- The union of (-3, +∞) and (4, +∞) is (-3, +∞).
To find the intersection of two intervals, we need to identify the values that are common to both intervals. Conversely, to find the union of two intervals, we need to determine the combined set of values from both intervals.
Let's start with the intersection:
1. Draw a number line.
2. Mark the interval (-3, +∞) on the number line. Since -3 is not included, place an open circle on -3 and draw an arrow to the right to represent all numbers greater than -3.
3. Mark the interval (4, +∞) on the number line. Similarly, open circle on 4 and draw an arrow to the right.
4. Identify the overlapping section between the two intervals. In this case, there is no overlap because -3 is not greater than 4.
- Since there is no overlap, the intersection of the intervals (-3, +∞) and (4, +∞) is an empty set.
Now, let's move on to finding the union:
1. Draw a number line.
2. Mark the interval (-3, +∞) on the number line as explained in step 2 of finding the intersection.
3. Mark the interval (4, +∞) on the number line as explained in step 3 of finding the intersection.
4. To determine the union, we combine the two intervals and take all the values that are present in either interval.
- Since both intervals extend to positive infinity, the union of the intervals (-3, +∞) and (4, +∞) is simply the interval (-3, +∞), including all numbers greater than -3.
In summary:
- Intersection of (-3, +∞) and (4, +∞): Empty set (no common values).
- Union of (-3, +∞) and (4, +∞): (-3, +∞) (including all numbers greater than -3).