WEBVTT



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Okay, I've got a number line here.

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How many numbers are there on that green?

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Infinitely many.

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like integers or numbers?

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Real numbers.

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There are infinitely many numbers there.

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In fact, the real number line is so dense

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that between any two numbers—no matter how close they are—there are still infinitely many numbers.

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So this is all the numbers between 2 and 5, including 2 and 5.

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This is called an interval.

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Okay, so an interval is a continuous set of numbers between two numbers, notice the endpoints.

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You can include them or exclude them

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It is still an interval of numbers.

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Or it's a set of numbers, it is an interval.

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Anytime you have an interval,

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we can use what's called interval notation

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to write down the set.

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We would not use the curly brackets here.

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Why would we not use the curly brackets here to write all the numbers?

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So, interval notation is a way to write sets.

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This is actually called roster notation.

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Why could I not use interval notation?

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Sorry, why could I not use roster notation?

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First off, I'd have to write an infinite number of things.

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I can't write down an infinite number of things.

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We can't write it as like a pattern with the ellipses because I'm missing lots of numbers.

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Like what number am I missing between 2 and 2.1?

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2.05, 2.01, 2.001.

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There's no way to write them all down.

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So we write them down a different way.

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And that's called interval notation.

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Since they have endpoints, we only write down two numbers,

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the left endpoint and the right endpoint.

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So here, the left endpoint is 2, and the right endpoint is 5.

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And then we either include those numbers or exclude them.

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Here I filled in a circle so 2 is in that set.

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If it includes the endpoint, we put a square bracket.

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Both of these are included, so I put another square bracket.

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So this is the set of numbers between 2 and 5,

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including 2 and 5.

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Let's do another example.

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This is an interval since I've shaded every number between 1 and 4.

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And what does the open circle mean?

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Not included.

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So we took out that one point on the line.

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So we still put the smaller number to the larger number.

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To exclude an endpoint in interval notation, use a round bracket.

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So this is the set of all numbers between 1

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1 and 4, not including 1 and 4, or excluding 1 and 4.

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This is called a closed set.

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This is called an open set. You'll learn why later they are called open and closed.

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So this is another set, sorry another interval.

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We include negative 5, exclude the 10.

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So in interval notation, so negative 5 to 10.

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We include it, so again square bracket.

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we exclude the 10 so round bracket.

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This is called half-open, half-closed.

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because we included the smallest number negative 5 that half is closed the right

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half is open

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There's another half-open, half-closed set.

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What number do I write first?

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Zero, the smaller the two numbers, endpoints, and then pi is it round or square around the

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zero?

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Round, we don't include that.

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We do include the pi.

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So this is also an interval.

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It doesn't have a right endpoint.

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So it's all numbers between 4 all the way out here forever.

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All numbers greater than or equal to 4.

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So we definitely write 4 first.

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And since it has no ending here,

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it goes all the way to the right forever,

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we put positive infinity.

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We include the four,

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and we never include infinity or negative infinity.

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Why don't we, why would we not include that?

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Well, there's really nothing to the right of infinity.

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OK, the reason we don't put a square bracket is because infinity is not a number.

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It's not a real number.

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So we don't include that.

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It means all the numbers from 4 all the way to the right.

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Okay again this is a half-open, half-closed set.

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We go the other direction.

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So here we go, all numbers less than 3. Since this doesn't end, it's from negative infinity.

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3 round bracket always with negative or positive infinity

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and then 3 do I do a square or round bracket?

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round because we're excluding 3

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that's that set of numbers

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And another example.

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Every real number.

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This is the set of real numbers.

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I think I wrote this down the other day.

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It's all numbers between negative infinity and positive infinity, and we exclude both.

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We wrote that down, right, on Friday?

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Okay.

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This is the same thing as the set of real numbers.

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Two different ways of writing the set of real numbers.

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this is interval notation this is just the symbol we use for it okay

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on this okay let me go back here for a sec

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This is all numbers between one and four correct?

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Excluding one and four?

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What if I just wrote down the very beginning class, is this an ordered pair or an interval?

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Interval.

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Right, if you see this now,

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is this an ordered pair or an interval?

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Interval.

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Interval, why is it obvious?

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Square bracket.

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It's a square bracket.

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Okay.

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When they're both open,

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it looks exactly the same as an ordered pair.

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Okay, the only way you know the difference

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is we're talking about intervals and numbers.

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this would be an interval.

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If you have no context, we don't know if that's an interval

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or an ordered pair.

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If I said graph the ordered pair,

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then we would definitely know it's one comma four.

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Okay, they look exactly the same, so be careful with that.

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Make sense?

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Okay.

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What do we call this thing that I've written on the board?

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An inequality.

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Is it true, false, or open?

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Open.

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Are the solutions numbers or are they ordered pairs?

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Numbers.

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Numbers.

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One.

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It only has one variable.

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Okay.

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Okay, we're gonna solve this and what's it mean to solve?

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the solution set and what's a solution set?

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The set of all solutions, what is a solution?

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When the number is replaced for the variable it makes the inequality true.

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Okay, we're gonna solve this.

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You saw these last year, I think.

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What, subtract 5?

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And then when we divide by negative 2, what do we have to do?

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Flip.

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We get x is greater than or equal to 8.

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And is this my solution?

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No.

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What is this?

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Another open inequality.

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The solutions are numbers.

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The graph of the solution set.

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We need a number line.

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And it's all numbers greater than or equal to 8.

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And is the solution set an interval?

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It is an interval, so we can write the solution set in interval notation.

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It is all numbers between 8 and positive infinity.

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Square bracket or round bracket around the 8? Square. We include it. Square around the

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infinity. Why round? It's not a number. Okay. This is the only way to write down the, well,

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I shouldn't say that.

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This is the solution set in interval notation.

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This is the graph of the solution set.

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Ordered pairs. Why?

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Two variables.

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Two variables.

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Two

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Uh, the two variables.

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Before we graph the inequality, let me write down its boundary equation.

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How many solutions does this have?

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We went over this one.

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Infinitely many, right? We graphed the solution set.

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Its y-intercept is 2 and its slope is 3.

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Earlier, we graphed the solution set of this equation.

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Everything in purple here is a solution.

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Everything not in purple is not a solution.

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We agree?

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Yeah.

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Okay.

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Are all these ordered pairs, these are a bunch of ordered pairs.

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Are all these ordered pairs solutions to this inequality?

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Why

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Somebody give me an idea.

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What you think?

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Like

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So like

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The graph of the equation gives solutions to the inequality,

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but it is not all of the inequality's solutions.

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Those points all make the inequality true.

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Okay, that's correct

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So this is (1, 5), right?

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It's a solution here

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Let's check to see if it's a solution here is 5 less than or equal to 3 times 1 plus 2

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It's 5 less than or equal to 5

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Yes

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You take any ordered pair on that purple line

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Plug it in you'll get this number exactly equals this number

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because of the “or equal to” part.

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For sure it's going to be a solution to this too.

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Does this make sense?

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Okay so everything in purple here is a solution to this and there's way way way more.

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Okay and the way we do that, I think you did this last year, we do like a test point.

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You take any ordered pair not on that line.

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I always pick 0, 0 unless the line goes through the origin.

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And we test it.

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We go 0 less than or equal to 3 times 0 plus 2.

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Is this true?

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Zero is less than or equal to 2.

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It's less than.

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Okay. So this right here is a solution.

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Okay, and therefore everything on this side of the line is going to be a solution.

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Every ordered pair on the other side will not be a solution.

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If we did a test point and it was false, we would not shade this side, we'd shade the

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opposite side of the test point.

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Okay, so now we just got to put dots on every single ordered pair.

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There are so many solutions to this inequality that they fill an entire half-plane.

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We sometimes call the solution set a half plane.

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OK.

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Can I write the solution set in interval notation?

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Can I... oops.

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Can I write this entire shaded region in interval notation?

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Like I did right here?

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The answer is no.

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Intervals are sets of numbers, not sets of ordered pairs.

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OK.

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Only inequalities in one variable can have

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solution sets written in interval notation.

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The way to show a two-variable solution set

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is to graph it.

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What would change if it was strictly less than?

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Okay, if it was strictly less than, everything that's a... oops... a solution here would

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not be a solution.

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Okay, and instead of like we did up here, putting... where did we put it?

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An open circle.

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We basically put open circles on this entire line.

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But we don't really put open circles, we just put a dashed line.

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We would put a dotted line.

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That means exclude that line, all of our pairs in that line.

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Okay, let's try another one.

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We got two variables.

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Sorry, I am going to graph on a coordinate plane.

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Sorry, let me erase this.

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I want strictly less than.

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We start by graphing this line.

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Am I going to put a dotted line or solid line?

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Dotted.

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So all these solutions to this are not solutions here.

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So we exclude the boundary.

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And then what do we do?

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A test point.

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And let's test (0, 0).

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0 less than negative 3 times 0 minus 2.

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Is 0 less than negative 2?

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0 is not less than negative 2.

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This is false.

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So this test point right here is not a solution.

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It failed.

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so that means everything on the other side of the line is a solution so we just start

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shading

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this side of the line.

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Again, for an inequality in two variables you almost always end up with a half a plane of solutions.

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There are a lot of them.

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Okay what questions do we have on this?

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You guys did this last year or no?

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Yes.

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This is a review but very important

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We have to understand

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interval notation and

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Know when interval notation cannot be used: with two variables.