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NationalTutoringObservatory
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MathEd-PII

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8
student
Ohh I see
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student
8200
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volunteer
Yes! 6800 was invested in the first account, 8200 was invested in the second account
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student
yep!
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student
I have another one that looks confusing
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student
like i dont get it
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volunteer
I think we will be able to do it :)
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student
it says "find the union or intersection of the given intervals. Write the answers in interval notation."
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volunteer
Ok! I always like to use a number line for tricky interval problems
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student
ok
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volunteer
Could you draw the first interval on the number line for me?
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student
-5 and 3?
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student
oh nvm its not negative
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student
they are already on the line
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student
5 and 3
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volunteer
I will show you what I mean
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student
this?
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volunteer
Just so we can see the interval
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student
ok
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volunteer
Now, it is a lot easier to see where these two intervals intersect!
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student
why is one open and one closed like the circle
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volunteer
We have an open circle at the 5 because, if you look at the interval, there is a parenthesis as a bracket. This means that the number 5 itself is not included in the interval
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student
Ohh ok
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volunteer
There is a closed circle at 3 because of the "hard" bracket in the interval in the question. This means that 3 is included
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student
let me write this on my paper
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student
so 3 is closed 5 is open
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volunteer
Yes
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student
ok
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student
oh bc 3 has bracket and 5 has parenthesis ok gotcha
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volunteer
Right
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student
ok now what do we do next
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volunteer
Now that we have the intervals lined up on the number line, we just need to find where they overlap
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student
at 3
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volunteer
Yes. And where does the overlap end?
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student
it ends at 5?
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volunteer
Exactly. Is 5 included in the overlap?
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student
Yes
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volunteer
It actually isn't because of the open circle on the first interval
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student
oh so 5 starts it off
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student
gotcha
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student
oh nvm
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student
no
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student
5 is overlapping 3
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volunteer
The overlap goes from 3 to 5, but 5 is not included
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student
oh
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student
why
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volunteer
As you drew on the line, the overlap only happens where the two intervals are on top of each other. We know that 3 is included, and that the numbers between 3 and 5 are included.
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volunteer
However, 5 is not included because, while the second interval includes it, the first one does not (because of the open circle). To be in the overlap, both intervals have to include 5 which the first one doesn't
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student
Ok I see
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student
so the overlap ends at 5 because that’s where the two intervals stop overlapping?
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student
this is why 5 is not included?
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volunteer
For it to be overlapping, both intervals have to include a point. They don't overlap at 5 because of the open circle (the first interval doesn't include 5)
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student
yeah so 3 starts it and it ends at 5?
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volunteer
Yes, and 3 is included and 5 is not
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student
yep got it
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student
so it should be written as [3,5)
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student
?
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volunteer
This is actually problem 87b*
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volunteer
Exactlu!
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student
i thought we were doing the first one
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student
bc im writing this on paper in order to turn in
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student
let me just erase it rq
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volunteer
I missed a detail with the symbols they gave (union vs intersection
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volunteer
The problems aren't very different, so you can keep the two interval lines on the number line
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volunteer
Sorry about that!
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student
its ok i fixed it
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student
what does the upside down U mean
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volunteer
That means the intersection between the two intervals, which is the problem that we did
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student
Ok
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student
now for A
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volunteer
Problem a wants the union between the two intervals
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student
ok
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student
lets start
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volunteer
What do you think we should do to find the union between the two intervals?
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student
draw the lines again?
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volunteer
Yep. Now the union means that we essentially combine the two intervals into one.
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volunteer
We use the word "or" to describe unions, and "and" to describe intersection
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volunteer
So the answer will be the interval of all points contained in either interval
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student
ok
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student
would it be 3,5 again or?
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student
?
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volunteer
Here, we can use the number line and want to find points that are not included in either interval.
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volunteer
Are there any points that aren't in either interval?
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student
uhhh 12
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student
67
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student
1,2,6,7
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student
4 is included
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volunteer
That would be correct for intersection. For union, we want to find points that aren't in EITHER. For example, 1 and 2 are included in the first interval. So, they are fine
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student
let me redraw my line
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student
a lot of numbers are included
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volunteer
Yes. Are there any numbers not included?
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volunteer
Let's try putting the intervals together to visualize this scenario better
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student
ok
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student
its like 1-10 are included
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volunteer
Actually, everything is included from -infinity to infinity.
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volunteer
The combined line never ends. The only interesting point is 5 because of the open circle. However, we only need one interval to include 5, which the second one does
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student
ohh yeah
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volunteer
I am glad that makes sense! Union is a very difficult concept to understand. Nice job with it!
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student
Wait so
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student
I'm still confused bc idk what the answer is like
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