### Video Transcript

Given that 𝑋 is equal to the
closed interval from negative seven to negative six and 𝑌 is equal to the
left-closed right-open interval from three to ∞, find 𝑋 union 𝑌.

In this question, we are given two
sets, 𝑋 and 𝑌, in interval notation. And we need to find the union of
these two sets. To do this, we can start by
recalling that a union of sets means that we take all of the elements in both
sets. So, we say that 𝑎 is a member of
the union of 𝑋 and 𝑌 if 𝑎 is a member of 𝑋 or if 𝑎 is a member of 𝑌.

To find the union of these two
intervals, we should start by determining exactly which numbers are members of each
set. First, we note that 𝑋 is a closed
interval. So, the endpoints of the interval
are included. This means we want to include all
real numbers between negative seven and negative six including the endpoints. We can follow a similar process for
𝑌. We note that 𝑌 is closed at three
and unbounded above. So, the set 𝑌 includes all real
values greater than or equal to three.

At this point, we could find the
union of 𝑋 and 𝑌 using set builder notation. However, when working with
intervals, it is often easier to visualize the sets using a number line first. First, 𝑋 is the set of real
numbers between negative seven and negative six, including the endpoints. We can represent this on a number
line using solid dots at negative seven and negative six to show that they are
included in the set.

We can apply a similar process to
𝑌. We want to include all of the
values greater than or equal to three in the set. So, we sketch a solid dot at three
and a line to the right of three. We include an arrow on the right to
show that the set is not bounded above.

There are a few different ways of
finding an expression for the union of these sets. One way is to find a set that
includes both 𝑋 and 𝑌. For instance, we can see that the
set of all real values greater than or equal to negative seven encompasses both 𝑋
and 𝑌.

However, we can see that this set
includes all of the real values between negative six and three, which are not
elements of 𝑋 or 𝑌. So, we need to remove these values
from our set to find the union of 𝑋 and 𝑌. This means that we want to remove
the open interval from negative six to three from our set to find the union of 𝑋
and 𝑌.

Finally, we can recall that we can
remove the elements of a set from another set by using the set minus operation. This means that we can write the
union of 𝑋 and 𝑌 as the left-closed, right-open interval from negative seven to ∞
minus the open interval from negative six to three. It is also worth noting that we can
use parentheses or reversed brackets to show that an interval does not include the
endpoints. Both notations are common, and it
is personal preference which one to use.