Integrated Math 2–3 Honors — Compound Inequalities
Corrected lesson transcript


## 0:00 — Compound Inequalities


## 0:04 — Lesson Overview

[0:04] Okay, we're talking about something called compound inequalities, which you should have
[0:12] learned last year, but we're going to kind of review a little bit.

## 0:16 — AND vs. OR

[0:16] Let me explain a little more of this.
[0:18] Compound inequalities are two or more.
[0:20] We can have more than two, but typically it's just two inequalities.
[0:27] And then there's two different types.
[0:29] One's called an OR, one's called an AND compound inequality.
[0:35] For the AND compound inequality, a number is a solution if it's a solution to both inequalities.
[0:45] That's the AND.
[0:47] And the OR inequality is a number is a solution to the compound if it's a solution to one
[0:53] or the other inequality.
[0:56] So let me give you an example.
[0:57] So we have AND and OR inequalities.

## 1:08 — AND: Use the Overlap

[1:08] They will either say the words
[1:11] They will either use the word AND or the word OR. There is another way to write an AND inequality where you will not see the word AND.
[1:24] solution to the compound inequality the number must be a solution to this and
[1:30] and also a solution to this.
[1:36] So these are already solved for x.
[1:49] So all the solutions to x greater than or equal to 4
[1:58] Are all of these numbers correct?
[2:06] And solutions to this, x is less than 10, are all these solutions?
[2:20] So solutions to this compound inequality
[2:24] are just the numbers that are solutions to this and this.
[2:31] So what on the graph is going to be a solution?
[2:34] 5.
[2:36] Whatever is in both green and blue.
[2:40] OK.
[2:41] So the numbers are in both green and blue would be is four in both colors?
[2:53] Yes. Yes and then everything here what about ten is ten in both colors? No. No
[3:02] because of the open circle there. Okay so this is the solution set to that
[3:10] compound and inequality.
[3:13] And so the solution set, of course that's the graph of the solution set.
[3:20] The actual solution set in interval notation is all the numbers between 4 and 10,
[3:26] including 4 but not including 10: [4, 10).

## 3:29 — Write AND as a Chain

[3:29] If this compound inequality could be written in one line,
[3:37] Okay. If, uh, that's not the best way of writing it. Let's do it this way.
[3:58] That's this guy right here, right?
[4:00] And if I flip this guy around, same thing as 4 is less than
[4:05] or equal to x.
[4:07] So we can just put this guy right here.
[4:15] So if you ever see something like this,
[4:19] that's a compound inequality.
[4:22] 4 is less than or equal to x, which is less than 10.
[4:25] Was it less than 10?
[4:27] OK.
[4:28] This is two inequalities in one, a two for one.
[4:33] Okay, if you ever see two inequalities in one,
[4:36] it's a compound and inequality.
[4:39] Has to be a solution to this and a solution to this.

## 4:42 — OR: Use the Union

[4:42] Where or inequalities have to be written separately,
[4:46] you have to use the word or in between.
[4:47] So x is greater than or equal to 5, or x is less than 0. You always see the word OR in between.
[5:01] word or in between
[5:34] OK, so in orange is the solution to the x is less than zero, green is solution to x
[5:40] greater than or equal to 5.
[5:43] So the solutions to the compound inequality
[5:47] is the ones that are in green or orange.
[5:52] We draw these.
[5:58] And then to write this in interval notation,
[6:03] we have to write two separate intervals.
[6:06] So, from negative infinity to zero, and five to infinity, which are two different sets.
[6:21] And how do we join them into one set?
[6:25] Did I teach you guys this?
[6:28] It's a U.
[6:30] Yeah.
[6:30] So we put the this big U looking thing
[6:35] stands for union
[6:37] all that means is
[6:43] These are two separate sets. To combine them into one set, put a union sign between them.
[6:49] Now this is one big set even though it's two separate intervals

## 6:52 — Solve an OR Inequality

[6:52] So is this an or or and?
[6:55] How about an and?
[6:56] Definitely, we see the word or.
[6:59] So the solutions are all the solutions to this, or all the solutions to this. For each inequality, we have to do a little work.
[7:06] of these we got to do a tiny bit of work.
[7:07] So we do divide by three.
[7:14] Subtract six.
[7:20] Fourteen.
[7:52] So solutions to the compound inequalities is everything we see in red.
[7:59] So in interval notation: negative infinity to 4, union 14 to infinity.

## 8:09 — Solve an AND Inequality

[8:09] Is this an and or an or?
[8:12] And.
[8:13] And how do we know?
[8:19] You know we got this inequality
[8:23] We have this inequality a
[8:27] Solution it's a solution to the compound inequality if it's a solution to the left inequality, and it's a solution to the right
[8:34] Okay, the beauty about these is we don't have to write two separate ones. We could solve it all at the same time
[8:40] So first is we subtract 6 from all three sides, which is really from each side of the two
[8:49] separate inequalities.
[8:53] So we get negative 10 is less than or equal to 2x, which is less than 4.
[9:04] Then we use the multiplication property of inequalities—or division—and divide all three sides by 2.
[9:11] by two, divide by two, divide by two.
[9:39] So this one, x is less than 2.
[9:50] That.
[9:54] This one, which same thing as x is greater than or equal to negative five.
[10:10] So the solution set has to be both in green and red here.
[10:28] So, that set of numbers, which is only one interval, so that's all the numbers between
[10:36] negative five and two excluding two, including negative five.
[10:55] Questions on this so far?

## 10:56 — OR Can Give All Real Numbers

[10:56] So is the solution set where they overlap?
[11:00] Yes, but also when they overlap.
[11:02] Okay, it's not just where they overlap
[11:06] it's all the solutions to this which is everything in blue or
[11:13] Everything in purple whatever color that is
[11:21] Which is
[11:23] All real numbers
[11:25] We're gonna be careful just because it overlaps the solution set is not where it overlaps. It's very important
[11:31] And it's a solution to this, or it's a solution to this.
[11:34] So the solution set is all real numbers.
[11:41] This would be the graph of the solution set.
[11:45] In interval notation: negative infinity to positive infinity.
[11:49] So be real careful whether it's an or or an and.
[11:53] Just because it overlaps.
[11:55] That does not mean the solution set is only where they overlap.
[12:01] Okay.

## 12:01 — AND Can Give the Empty Set

[12:01] So is the solution set those two intervals?
[12:05] No.
[12:05] No.
[12:07] It has to be both in purple and in blue and there are no numbers in purple and blue.
[12:15] So what is the solution set?
[12:19] There are no solutions but what is the solution set?
[12:25] The empty set.
[12:26] Either that or we write it with curly brackets.
[12:34] Okay, be real careful.
[12:37] The AND and the OR make a big difference.
[12:42] There are no numbers, there are both solutions to this and solutions to this.
[12:47] So there are no solutions; the solution set is empty.
[12:50] Let me pass out the homework and stop here.