Using a number line, find both the intersection and the union of the following interval:
(− ∞ , 7) and [−5, +∞)
Ah, the intersection and union of intervals, huh? Let's clown around with it, shall we?
First, let's find the intersection. Imagine the number line, and here we have (-∞, 7) and [-5, +∞). So, from minus infinity to 7, we take the overlapping part with -5 onwards.
And what do we get? Well, the overlapping portion is [-5, 7). Think of it as the VIP section at a party. It's exclusive and includes -5, but not quite as cool as 7.
Now, the union is like the best of both worlds, where everyone's invited! So, we take all the numbers from -∞ to 7 and add to it anything from -5 onwards. The result is (-∞, +∞).
It's like a never-ending party on the number line! No need to worry about any bouncers kicking you out. So, enjoy the infinite fun at (-∞, +∞)!
To find the intersection and the union of the two intervals, we can use a number line.
First, let's plot the interval (-∞, 7) on the number line. Since it starts at negative infinity and ends at 7, we can represent it by drawing an open circle at 7 and extending an arrow to the left.
```
-∞─────(7)─────>
```
Next, let's plot the interval [-5, +∞) on the same number line. Since it starts at -5 and extends to positive infinity, we can represent it by drawing a closed circle at -5 and extending an arrow to the right.
```
<─────[-5]────────∞─────>
```
To find the intersection, we need to identify the shared region between the two intervals. Looking at the number line, we can see that the only shared region is the interval from -5 to 7. Therefore, the intersection of (-∞, 7) and [-5, +∞) is [-5, 7), where the square bracket at -5 indicates that it is included in the interval, and the open parenthesis at 7 indicates that it is not included.
To find the union, we need to identify the combined region of both intervals. Looking at the number line, we can see that the union includes all values from negative infinity to positive infinity, except for the value 7. Therefore, the union of (-∞, 7) and [-5, +∞) is (-∞, 7) U (-∞, -5] U (7, +∞), where the parentheses indicate that the values are excluded except for -5 and 7, and the square bracket at -5 indicates that it is included.