To represent the intersection and union of the given intervals on a number line, we'll start by graphing each interval separately and then find their intersection and union.
Let's begin with the interval (-∞, 4):
On the number line, we'll mark an open circle at 4 to indicate that it is not included in the interval. Then, we'll draw an arrow pointing towards the left to indicate that the interval extends infinitely in that direction. The graph would look like this:
```
<---|---------------------o
-∞ 4
```
Next, let's move on to the interval (10, ∞):
On the number line, we'll mark an open circle at 10 to indicate that it is not included in the interval. Then, we'll draw an arrow pointing towards the right to indicate that the interval extends infinitely in that direction. The graph would look like this:
```
o---------------------|--->
10 ∞
```
Now, let's find the intersection of these two intervals:
The intersection represents the values that are common to both intervals. Since there is no overlap between (-∞, 4) and (10, ∞), their intersection would be an empty set (∅).
To represent this on the number line, we simply leave it empty:
```
<---|---------------------o
-∞ 4
o---------------------|--->
10 ∞
Intersection: ∅
```
Finally, let's find the union of these two intervals:
The union represents the combination of all values from both intervals. In this case, the union would encompass all values from negative infinity to 4, excluding 4, and all values from 10 to positive infinity. So, the union would be represented as follows:
```
<---|---------------------o o---------------------|--->
-∞ 4 10 ∞
Union: (-∞, 4) U (10, ∞)
```
I hope this helps you visualize the intersection and union of the given intervals on the number line.