Integrated Math 2–3 Honors — Interval Notation and Graphing Inequalities Corrected lesson transcript ## 0:00 — Intervals and Interval Notation [0:04] Okay, I've got a number line here. [0:13] How many numbers are there on that green? [0:20] Infinitely many. [0:26] like integers or numbers? [0:29] Real numbers. [0:32] There are infinitely many numbers there. [0:38] In fact, the real number line is so dense [0:41] that between any two numbers—no matter how close they are—there are still infinitely many numbers. [0:50] So this is all the numbers between 2 and 5, including 2 and 5. [0:59] This is called an interval. [1:07] Okay, so an interval is a continuous set of numbers between two numbers, notice the endpoints. [1:15] You can include them or exclude them [1:17] It is still an interval of numbers. [1:21] Or it's a set of numbers, it is an interval. [1:32] Anytime you have an interval, [1:34] we can use what's called interval notation [1:42] to write down the set. [1:47] We would not use the curly brackets here. [1:52] Why would we not use the curly brackets here to write all the numbers? [1:58] So, interval notation is a way to write sets. [2:03] This is actually called roster notation. [2:18] Why could I not use interval notation? [2:21] Sorry, why could I not use roster notation? [2:35] First off, I'd have to write an infinite number of things. [2:39] I can't write down an infinite number of things. [2:42] We can't write it as like a pattern with the ellipses because I'm missing lots of numbers. [2:49] Like what number am I missing between 2 and 2.1? [2:55] 2.05, 2.01, 2.001. [2:59] There's no way to write them all down. [3:00] So we write them down a different way. [3:08] And that's called interval notation. [3:10] Since they have endpoints, we only write down two numbers, [3:14] the left endpoint and the right endpoint. [3:16] So here, the left endpoint is 2, and the right endpoint is 5. [3:24] And then we either include those numbers or exclude them. [3:30] Here I filled in a circle so 2 is in that set. [3:34] If it includes the endpoint, we put a square bracket. [3:39] Both of these are included, so I put another square bracket. [3:43] So this is the set of numbers between 2 and 5, [3:58] including 2 and 5. ## 4:04 — Open, Closed, and Half-Open Intervals [4:04] Let's do another example. [4:23] This is an interval since I've shaded every number between 1 and 4. [4:29] And what does the open circle mean? [4:32] Not included. [4:34] So we took out that one point on the line. [4:38] So we still put the smaller number to the larger number. [4:44] To exclude an endpoint in interval notation, use a round bracket. [4:48] So this is the set of all numbers between 1 [5:08] 1 and 4, not including 1 and 4, or excluding 1 and 4. [5:26] This is called a closed set. [5:29] This is called an open set. You'll learn why later they are called open and closed. [5:45] So this is another set, sorry another interval. [5:50] We include negative 5, exclude the 10. [5:54] So in interval notation, so negative 5 to 10. [6:00] We include it, so again square bracket. [6:02] we exclude the 10 so round bracket. [6:20] This is called half-open, half-closed. [6:30] because we included the smallest number negative 5 that half is closed the right [6:38] half is open [6:56] There's another half-open, half-closed set. [7:02] What number do I write first? [7:07] Zero, the smaller the two numbers, endpoints, and then pi is it round or square around the [7:15] zero? [7:17] Round, we don't include that. [7:19] We do include the pi. ## 7:20 — Intervals with Infinity [7:20] So this is also an interval. [7:27] It doesn't have a right endpoint. [7:30] So it's all numbers between 4 all the way out here forever. [7:36] All numbers greater than or equal to 4. [7:40] So we definitely write 4 first. [7:43] And since it has no ending here, [7:44] it goes all the way to the right forever, [7:47] we put positive infinity. [7:51] We include the four, [7:55] and we never include infinity or negative infinity. [7:59] Why don't we, why would we not include that? [8:10] Well, there's really nothing to the right of infinity. [8:16] OK, the reason we don't put a square bracket is because infinity is not a number. [8:22] It's not a real number. [8:23] So we don't include that. [8:25] It means all the numbers from 4 all the way to the right. [8:30] Okay again this is a half-open, half-closed set. [8:34] We go the other direction. [8:37] So here we go, all numbers less than 3. Since this doesn't end, it's from negative infinity. [8:47] 3 round bracket always with negative or positive infinity [8:51] and then 3 do I do a square or round bracket? [8:56] round because we're excluding 3 [8:59] that's that set of numbers [9:07] And another example. [9:11] Every real number. [9:14] This is the set of real numbers. [9:16] I think I wrote this down the other day. [9:20] It's all numbers between negative infinity and positive infinity, and we exclude both. [9:27] We wrote that down, right, on Friday? [9:32] Okay. [9:33] This is the same thing as the set of real numbers. [9:37] Two different ways of writing the set of real numbers. [9:39] this is interval notation this is just the symbol we use for it okay ## 9:46 — Intervals Versus Ordered Pairs [9:46] on this okay let me go back here for a sec [9:58] This is all numbers between one and four correct? [10:01] Excluding one and four? [10:04] What if I just wrote down the very beginning class, is this an ordered pair or an interval? [10:12] Interval. [10:18] Right, if you see this now, [10:20] is this an ordered pair or an interval? [10:22] Interval. [10:23] Interval, why is it obvious? [10:25] Square bracket. [10:26] It's a square bracket. [10:28] Okay. [10:30] When they're both open, [10:32] it looks exactly the same as an ordered pair. [10:36] Okay, the only way you know the difference [10:39] is we're talking about intervals and numbers. [10:40] this would be an interval. [10:42] If you have no context, we don't know if that's an interval [10:45] or an ordered pair. [10:47] If I said graph the ordered pair, [10:50] then we would definitely know it's one comma four. [10:53] Okay, they look exactly the same, so be careful with that. [10:56] Make sense? [10:58] Okay. ## 10:58 — Solving a One-Variable Inequality [10:58] What do we call this thing that I've written on the board? [11:04] An inequality. [11:06] Is it true, false, or open? [11:11] Open. [11:12] Are the solutions numbers or are they ordered pairs? [11:16] Numbers. [11:17] Numbers. [11:19] One. [11:20] It only has one variable. [11:21] Okay. [11:21] Okay, we're gonna solve this and what's it mean to solve? [11:30] the solution set and what's a solution set? [11:34] The set of all solutions, what is a solution? [11:42] When the number is replaced for the variable it makes the inequality true. [11:48] Okay, we're gonna solve this. [11:50] You saw these last year, I think. [11:54] What, subtract 5? [11:55] And then when we divide by negative 2, what do we have to do? [11:58] Flip. [12:04] We get x is greater than or equal to 8. [12:16] And is this my solution? [12:20] No. [12:21] What is this? [12:24] Another open inequality. [12:26] The solutions are numbers. [12:33] The graph of the solution set. [12:42] We need a number line. [12:48] And it's all numbers greater than or equal to 8. [12:57] And is the solution set an interval? [13:03] It is an interval, so we can write the solution set in interval notation. [13:10] It is all numbers between 8 and positive infinity. [13:19] Square bracket or round bracket around the 8? Square. We include it. Square around the [13:26] infinity. Why round? It's not a number. Okay. This is the only way to write down the, well, [13:35] I shouldn't say that. [13:36] This is the solution set in interval notation. [13:40] This is the graph of the solution set. ## 13:43 — Graphing a Two-Variable Inequality [13:43] Ordered pairs. Why? [13:45] Two variables. [13:47] Two variables. [13:49] Two [13:52] Uh, the two variables. [14:03] Before we graph the inequality, let me write down its boundary equation. [14:14] How many solutions does this have? [14:24] We went over this one. [14:25] Infinitely many, right? We graphed the solution set. [14:43] Its y-intercept is 2 and its slope is 3. [14:51] Earlier, we graphed the solution set of this equation. [14:55] Everything in purple here is a solution. [14:58] Everything not in purple is not a solution. [15:01] We agree? [15:01] Yeah. [15:02] Okay. [15:03] Are all these ordered pairs, these are a bunch of ordered pairs. [15:07] Are all these ordered pairs solutions to this inequality? [15:11] Why [15:12] Somebody give me an idea. [15:16] What you think? [15:22] Like [15:24] So like [15:25] The graph of the equation gives solutions to the inequality, [15:30] but it is not all of the inequality's solutions. [15:33] Those points all make the inequality true. [15:40] Okay, that's correct [15:41] So this is (1, 5), right? [15:46] It's a solution here [15:48] Let's check to see if it's a solution here is 5 less than or equal to 3 times 1 plus 2 [15:55] It's 5 less than or equal to 5 [15:58] Yes [16:00] You take any ordered pair on that purple line [16:02] Plug it in you'll get this number exactly equals this number [16:08] because of the “or equal to” part. [16:11] For sure it's going to be a solution to this too. [16:16] Does this make sense? [16:18] Okay so everything in purple here is a solution to this and there's way way way more. [16:31] Okay and the way we do that, I think you did this last year, we do like a test point. [16:39] You take any ordered pair not on that line. [16:42] I always pick 0, 0 unless the line goes through the origin. [16:46] And we test it. [16:49] We go 0 less than or equal to 3 times 0 plus 2. [16:54] Is this true? [16:57] Zero is less than or equal to 2. [17:00] It's less than. [17:03] Okay. So this right here is a solution. [17:11] Okay, and therefore everything on this side of the line is going to be a solution. [17:18] Every ordered pair on the other side will not be a solution. [17:22] If we did a test point and it was false, we would not shade this side, we'd shade the [17:30] opposite side of the test point. [17:33] Okay, so now we just got to put dots on every single ordered pair. ## 17:37 — Half-Planes and Dashed Boundaries [17:37] There are so many solutions to this inequality that they fill an entire half-plane. [17:46] We sometimes call the solution set a half plane. [18:02] OK. [18:06] Can I write the solution set in interval notation? [18:17] Can I... oops. [18:20] Can I write this entire shaded region in interval notation? [18:31] Like I did right here? [18:35] The answer is no. [18:38] Intervals are sets of numbers, not sets of ordered pairs. [18:45] OK. [18:46] Only inequalities in one variable can have [18:50] solution sets written in interval notation. [18:54] The way to show a two-variable solution set [18:58] is to graph it. [19:03] What would change if it was strictly less than? [19:17] Okay, if it was strictly less than, everything that's a... oops... a solution here would [19:27] not be a solution. [19:30] Okay, and instead of like we did up here, putting... where did we put it? [19:35] An open circle. [19:37] We basically put open circles on this entire line. [19:44] But we don't really put open circles, we just put a dashed line. [19:49] We would put a dotted line. [19:52] That means exclude that line, all of our pairs in that line. [19:59] Okay, let's try another one. ## 20:04 — A False Test Point and Opposite-Side Shading [20:05] We got two variables. [20:10] Sorry, I am going to graph on a coordinate plane. [20:17] Sorry, let me erase this. [20:18] I want strictly less than. [20:23] We start by graphing this line. [20:26] Am I going to put a dotted line or solid line? [20:32] Dotted. [20:33] So all these solutions to this are not solutions here. [20:36] So we exclude the boundary. [20:51] And then what do we do? [20:54] A test point. [20:59] And let's test (0, 0). [21:02] 0 less than negative 3 times 0 minus 2. [21:07] Is 0 less than negative 2? [21:13] 0 is not less than negative 2. [21:15] This is false. [21:21] So this test point right here is not a solution. [21:25] It failed. [21:26] so that means everything on the other side of the line is a solution so we just start [21:32] shading [21:37] this side of the line. [21:39] Again, for an inequality in two variables you almost always end up with a half a plane of solutions. [21:46] There are a lot of them. [21:50] Okay what questions do we have on this? [21:53] You guys did this last year or no? [21:56] Yes. [21:56] This is a review but very important [22:00] We have to understand [22:03] interval notation and [22:06] Know when interval notation cannot be used: with two variables.