00:01
Okay, so this problem is asking for which x satisfies both of these conditions, x minus 3, the absolute value of x minus 3 is less than 2, and the absolute value of x minus 5 is less than 1.
00:18
So in order to solve this, first thing we'll do is convert these two inequalities into to non -absolute value inequalities.
00:31
So we have negative 2, x minus 3, 2.
00:38
And then over here, we have negative 1 is less than x minus 5, which is less than 1.
00:46
Now once we have that, to make it simpler, we'll get rid of this internal term here by adding 3 to each thing for the first one, and adding 5 for everything.
01:01
Thing in the second one.
01:05
This would bring us to 1 is less than x, which is less than 5.
01:16
And over here we'd have 4 is less than x, which is less than 6.
01:25
So now we have two simple inequalities here.
01:31
And if we want to figure out for which x as far as both of these conditions simultaneously, we can kind of just graph this interval.
01:43
The interval here is 5 and 6.
01:51
We can graph both of these intervals and then find that area of overlap.
01:58
So now if we graph these two intervals, this one will be in red, and i'll graph this one in green.
02:11
Okay, so our first interval would fall in between this one and five fill in this area in between.
02:35
Our second interval would fall between four and six and then cover this area.
02:55
Now you can pretty clearly see the overlap between these two areas is in this area right here in between the two, from here to here.
03:13
And those values are four and five.
03:17
So also notice how these are dotted lines and unsolved lines, meaning that the ends do not count towards our solution...