00:01
So in this problem, we're asked to find the real numbers x that satisfy this equation.
00:10
So we're here equalling the ceiling function to the floor function.
00:17
So since these two functions have a very different definition, it's not a very clear.
00:27
We don't have an explicit formula to plug in here.
00:31
And then sulfur x, we're going to have to talk through this one.
00:36
So this is the definition, or it would be the definition of the floor and ceiling functions.
00:43
So for every real number, any real number is between two integers, right? so in this case, we're calling it and it doesn't matter.
00:59
We know it's an integer, and if n is between n and n plus 1, it can be equal to 1, then the floor function is going to, the output will be n.
01:16
And if it, so that's for the floor function, for the ceiling function, if x is between n minus 1 and n, including n, then we have that the ceiling function.
01:29
Will out of n.
01:33
So what we're doing here is if n, if x is the integer n plus some decimal, it's going to go back to n.
01:44
And here, if n is any, we're rounding up here.
01:53
Yeah, so that's the definition for x, for floor and ceiling function.
01:58
And we want to find not just any x, all the xes that, such that these two are equal.
02:08
So what we're going to do, since this is based a lot on integers, we are going to say that, let's pick another color.
02:23
Okay, so if x, actually let's call m.
02:34
To be some number that's between zero and one.
02:44
So that way we can see, we want to see what the ceiling and floor functions of x are when x is some some integer n minus m, when x is exactly an integer, when x is exactly an integer, and when it is a little bit more than this integer.
03:24
So since we are not assigning any specific value to this, here we're covering all the real numbers.
03:34
Yeah, okay, so that's my approach to this problem.
03:43
So when n is a little bit, on x is a little bit less than n, so it's not n minus 1, but it's also not 1.
03:51
Not n.
03:53
That means it's somewhere around here, which means that the ceiling function is n, because we're rounding up.
04:07
I want to use like here.
04:08
Okay.
04:10
This is n.
04:12
Now, as for the floor function, remember, if it's between two numbers, two integers, we round down.
04:20
So we're taking away whatever.
04:24
If it's n plus some decimal, we're rounding down to n for the floor function.
04:31
So in this case, it's a little bit less than n.
04:35
So we round down to n minus 1.
04:40
Okay, so when it's n, so here for the floor function, it's equal to an integer.
04:51
Floor function is going to be that same integer.
04:57
Same thing with this one.
04:58
If it's equal to one integer, the floor, i'm sorry, ceiling function is going to be that same integer.
05:07
So here we have that when x is an integer, both the floor and ceiling functions are equal.
05:20
So next, for the last case, if it's it's n plus a little bit more.
05:26
So this is n plus some decimal part.
05:31
For the flower function, we round down.
05:34
So it will be something like this.
05:39
It's between n and n plus one.
05:42
We round down to n...
