1
00:00:04,300 --> 00:00:08,120
Okay, I've got a number line here.

2
00:00:13,100 --> 00:00:17,420
How many numbers are there on that green?

3
00:00:20,480 --> 00:00:21,230
Infinitely many.

4
00:00:26,000 --> 00:00:28,580
like integers or numbers?

5
00:00:29,260 --> 00:00:30,760
Real numbers.

6
00:00:32,120 --> 00:00:36,140
There are infinitely many numbers there.

7
00:00:38,160 --> 00:00:41,260
In fact, the real number line is so dense

8
00:00:41,260 --> 00:00:45,540
that between any two numbers—no matter how close they are—there are still infinitely many numbers.

9
00:00:50,760 --> 00:00:55,860
So this is all the numbers between 2 and 5, including 2 and 5.

10
00:00:59,000 --> 00:01:00,440
This is called an interval.

11
00:01:07,800 --> 00:01:13,600
Okay, so an interval is a continuous set of numbers between two numbers, notice the endpoints.

12
00:01:15,400 --> 00:01:17,480
You can include them or exclude them

13
00:01:17,480 --> 00:01:19,340
It is still an interval of numbers.

14
00:01:21,660 --> 00:01:23,860
Or it's a set of numbers, it is an interval.

15
00:01:32,380 --> 00:01:34,040
Anytime you have an interval,

16
00:01:34,980 --> 00:01:37,500
we can use what's called interval notation

17
00:01:42,440 --> 00:01:44,200
to write down the set.

18
00:01:47,060 --> 00:01:50,260
We would not use the curly brackets here.

19
00:01:52,020 --> 00:01:55,760
Why would we not use the curly brackets here to write all the numbers?

20
00:01:58,860 --> 00:02:02,500
So, interval notation is a way to write sets.

21
00:02:03,060 --> 00:02:05,260
This is actually called roster notation.

22
00:02:18,420 --> 00:02:20,800
Why could I not use interval notation?

23
00:02:21,060 --> 00:02:23,340
Sorry, why could I not use roster notation?

24
00:02:35,500 --> 00:02:38,160
First off, I'd have to write an infinite number of things.

25
00:02:39,800 --> 00:02:41,420
I can't write down an infinite number of things.

26
00:02:42,620 --> 00:02:49,540
We can't write it as like a pattern with the ellipses because I'm missing lots of numbers.

27
00:02:49,680 --> 00:02:52,020
Like what number am I missing between 2 and 2.1?

28
00:02:55,420 --> 00:02:58,660
2.05, 2.01, 2.001.

29
00:02:59,240 --> 00:03:00,700
There's no way to write them all down.

30
00:03:00,700 --> 00:03:03,820
So we write them down a different way.

31
00:03:08,340 --> 00:03:10,260
And that's called interval notation.

32
00:03:10,800 --> 00:03:14,280
Since they have endpoints, we only write down two numbers,

33
00:03:14,920 --> 00:03:16,780
the left endpoint and the right endpoint.

34
00:03:16,940 --> 00:03:22,480
So here, the left endpoint is 2, and the right endpoint is 5.

35
00:03:24,680 --> 00:03:28,620
And then we either include those numbers or exclude them.

36
00:03:30,700 --> 00:03:33,460
Here I filled in a circle so 2 is in that set.

37
00:03:34,640 --> 00:03:37,360
If it includes the endpoint, we put a square bracket.

38
00:03:39,700 --> 00:03:42,640
Both of these are included, so I put another square bracket.

39
00:03:43,680 --> 00:03:58,740
So this is the set of numbers between 2 and 5,

40
00:03:58,740 --> 00:04:04,560
including 2 and 5.

41
00:04:04,960 --> 00:04:07,720
Let's do another example.

42
00:04:23,580 --> 00:04:28,660
This is an interval since I've shaded every number between 1 and 4.

43
00:04:29,900 --> 00:04:31,360
And what does the open circle mean?

44
00:04:32,360 --> 00:04:33,110
Not included.

45
00:04:34,860 --> 00:04:38,760
So we took out that one point on the line.

46
00:04:38,760 --> 00:04:42,620
So we still put the smaller number to the larger number.

47
00:04:44,920 --> 00:04:48,180
To exclude an endpoint in interval notation, use a round bracket.

48
00:04:48,180 --> 00:05:08,240
So this is the set of all numbers between 1

49
00:05:08,240 --> 00:05:22,240
1 and 4, not including 1 and 4, or excluding 1 and 4.

50
00:05:26,760 --> 00:05:29,240
This is called a closed set.

51
00:05:29,240 --> 00:05:44,900
This is called an open set. You'll learn why later they are called open and closed.

52
00:05:45,240 --> 00:05:48,200
So this is another set, sorry another interval.

53
00:05:50,800 --> 00:05:53,860
We include negative 5, exclude the 10.

54
00:05:54,720 --> 00:05:58,320
So in interval notation, so negative 5 to 10.

55
00:06:00,280 --> 00:06:02,100
We include it, so again square bracket.

56
00:06:02,100 --> 00:06:05,440
we exclude the 10 so round bracket.

57
00:06:20,460 --> 00:06:27,200
This is called half-open, half-closed.

58
00:06:30,680 --> 00:06:38,400
because we included the smallest number negative 5 that half is closed the right

59
00:06:38,400 --> 00:06:41,040
half is open

60
00:06:56,520 --> 00:06:59,280
There's another half-open, half-closed set.

61
00:07:02,520 --> 00:07:04,180
What number do I write first?

62
00:07:07,600 --> 00:07:15,700
Zero, the smaller the two numbers, endpoints, and then pi is it round or square around the

63
00:07:15,700 --> 00:07:16,450
zero?

64
00:07:17,620 --> 00:07:18,880
Round, we don't include that.

65
00:07:19,460 --> 00:07:20,420
We do include the pi.

66
00:07:20,650 --> 00:07:23,090
So this is also an interval.

67
00:07:27,310 --> 00:07:28,930
It doesn't have a right endpoint.

68
00:07:30,290 --> 00:07:35,330
So it's all numbers between 4 all the way out here forever.

69
00:07:36,150 --> 00:07:37,690
All numbers greater than or equal to 4.

70
00:07:40,070 --> 00:07:42,010
So we definitely write 4 first.

71
00:07:43,050 --> 00:07:44,870
And since it has no ending here,

72
00:07:44,890 --> 00:07:46,810
it goes all the way to the right forever,

73
00:07:47,650 --> 00:07:48,770
we put positive infinity.

74
00:07:51,870 --> 00:07:53,170
We include the four,

75
00:07:55,270 --> 00:07:59,630
and we never include infinity or negative infinity.

76
00:07:59,810 --> 00:08:01,850
Why don't we, why would we not include that?

77
00:08:10,850 --> 00:08:14,690
Well, there's really nothing to the right of infinity.

78
00:08:16,270 --> 00:08:20,310
OK, the reason we don't put a square bracket is because infinity is not a number.

79
00:08:22,450 --> 00:08:23,370
It's not a real number.

80
00:08:23,750 --> 00:08:25,190
So we don't include that.

81
00:08:25,710 --> 00:08:28,690
It means all the numbers from 4 all the way to the right.

82
00:08:30,230 --> 00:08:33,450
Okay again this is a half-open, half-closed set.

83
00:08:34,710 --> 00:08:36,970
We go the other direction.

84
00:08:37,310 --> 00:08:44,770
So here we go, all numbers less than 3. Since this doesn't end, it's from negative infinity.

85
00:08:47,670 --> 00:08:51,110
3 round bracket always with negative or positive infinity

86
00:08:51,770 --> 00:08:54,670
and then 3 do I do a square or round bracket?

87
00:08:56,490 --> 00:08:58,310
round because we're excluding 3

88
00:08:59,590 --> 00:09:01,630
that's that set of numbers

89
00:09:07,670 --> 00:09:10,810
And another example.

90
00:09:11,450 --> 00:09:12,200
Every real number.

91
00:09:14,410 --> 00:09:16,210
This is the set of real numbers.

92
00:09:16,650 --> 00:09:19,210
I think I wrote this down the other day.

93
00:09:20,710 --> 00:09:25,150
It's all numbers between negative infinity and positive infinity, and we exclude both.

94
00:09:27,050 --> 00:09:28,370
We wrote that down, right, on Friday?

95
00:09:32,610 --> 00:09:33,360
Okay.

96
00:09:33,550 --> 00:09:36,430
This is the same thing as the set of real numbers.

97
00:09:37,130 --> 00:09:39,110
Two different ways of writing the set of real numbers.

98
00:09:39,110 --> 00:09:46,670
this is interval notation this is just the symbol we use for it okay

99
00:09:46,670 --> 00:09:53,650
on this okay let me go back here for a sec

100
00:09:58,550 --> 00:10:00,550
This is all numbers between one and four correct?

101
00:10:01,350 --> 00:10:02,170
Excluding one and four?

102
00:10:04,350 --> 00:10:12,370
What if I just wrote down the very beginning class, is this an ordered pair or an interval?

103
00:10:12,370 --> 00:10:13,120
Interval.

104
00:10:18,090 --> 00:10:19,670
Right, if you see this now,

105
00:10:20,590 --> 00:10:22,090
is this an ordered pair or an interval?

106
00:10:22,650 --> 00:10:23,400
Interval.

107
00:10:23,290 --> 00:10:24,890
Interval, why is it obvious?

108
00:10:25,790 --> 00:10:26,540
Square bracket.

109
00:10:26,370 --> 00:10:27,120
It's a square bracket.

110
00:10:28,170 --> 00:10:28,920
Okay.

111
00:10:30,570 --> 00:10:32,050
When they're both open,

112
00:10:32,190 --> 00:10:34,650
it looks exactly the same as an ordered pair.

113
00:10:36,250 --> 00:10:38,450
Okay, the only way you know the difference

114
00:10:39,070 --> 00:10:40,730
is we're talking about intervals and numbers.

115
00:10:40,730 --> 00:10:42,150
this would be an interval.

116
00:10:42,850 --> 00:10:45,550
If you have no context, we don't know if that's an interval

117
00:10:45,550 --> 00:10:46,370
or an ordered pair.

118
00:10:47,570 --> 00:10:50,410
If I said graph the ordered pair,

119
00:10:50,570 --> 00:10:52,370
then we would definitely know it's one comma four.

120
00:10:53,290 --> 00:10:56,070
Okay, they look exactly the same, so be careful with that.

121
00:10:56,890 --> 00:10:57,640
Make sense?

122
00:10:58,070 --> 00:10:58,820
Okay.

123
00:10:58,560 --> 00:11:00,640
What do we call this thing that I've written on the board?

124
00:11:04,620 --> 00:11:05,700
An inequality.

125
00:11:06,920 --> 00:11:08,100
Is it true, false, or open?

126
00:11:11,030 --> 00:11:11,780
Open.

127
00:11:12,430 --> 00:11:15,470
Are the solutions numbers or are they ordered pairs?

128
00:11:16,070 --> 00:11:16,820
Numbers.

129
00:11:17,710 --> 00:11:18,460
Numbers.

130
00:11:19,510 --> 00:11:20,260
One.

131
00:11:20,250 --> 00:11:21,190
It only has one variable.

132
00:11:21,830 --> 00:11:22,580
Okay.

133
00:11:21,970 --> 00:11:26,350
Okay, we're gonna solve this and what's it mean to solve?

134
00:11:30,210 --> 00:11:32,610
the solution set and what's a solution set?

135
00:11:34,310 --> 00:11:38,010
The set of all solutions, what is a solution?

136
00:11:42,730 --> 00:11:47,790
When the number is replaced for the variable it makes the inequality true.

137
00:11:48,790 --> 00:11:49,690
Okay, we're gonna solve this.

138
00:11:50,210 --> 00:11:52,370
You saw these last year, I think.

139
00:11:54,050 --> 00:11:55,270
What, subtract 5?

140
00:11:55,530 --> 00:11:58,810
And then when we divide by negative 2, what do we have to do?

141
00:11:58,810 --> 00:11:59,560
Flip.

142
00:12:04,010 --> 00:12:09,510
We get x is greater than or equal to 8.

143
00:12:16,430 --> 00:12:19,250
And is this my solution?

144
00:12:20,250 --> 00:12:21,000
No.

145
00:12:21,070 --> 00:12:22,170
What is this?

146
00:12:24,950 --> 00:12:26,730
Another open inequality.

147
00:12:26,730 --> 00:12:29,010
The solutions are numbers.

148
00:12:33,570 --> 00:12:39,190
The graph of the solution set.

149
00:12:42,270 --> 00:12:43,730
We need a number line.

150
00:12:48,770 --> 00:12:51,370
And it's all numbers greater than or equal to 8.

151
00:12:57,770 --> 00:13:00,010
And is the solution set an interval?

152
00:13:03,950 --> 00:13:10,630
It is an interval, so we can write the solution set in interval notation.

153
00:13:10,630 --> 00:13:16,810
It is all numbers between 8 and positive infinity.

154
00:13:19,350 --> 00:13:25,290
Square bracket or round bracket around the 8? Square. We include it. Square around the

155
00:13:26,070 --> 00:13:35,010
infinity. Why round? It's not a number. Okay. This is the only way to write down the, well,

156
00:13:35,010 --> 00:13:36,090
I shouldn't say that.

157
00:13:36,830 --> 00:13:39,490
This is the solution set in interval notation.

158
00:13:40,370 --> 00:13:42,410
This is the graph of the solution set.

159
00:13:43,570 --> 00:13:45,330
Ordered pairs. Why?

160
00:13:45,610 --> 00:13:46,360
Two variables.

161
00:13:47,330 --> 00:13:48,130
Two variables.

162
00:13:49,750 --> 00:13:50,500
Two

163
00:13:52,610 --> 00:13:57,570
Uh, the two variables.

164
00:14:03,290 --> 00:14:08,210
Before we graph the inequality, let me write down its boundary equation.

165
00:14:14,590 --> 00:14:16,850
How many solutions does this have?

166
00:14:24,550 --> 00:14:25,810
We went over this one.

167
00:14:25,810 --> 00:14:30,750
Infinitely many, right? We graphed the solution set.

168
00:14:43,450 --> 00:14:46,490
Its y-intercept is 2 and its slope is 3.

169
00:14:51,200 --> 00:14:55,700
Earlier, we graphed the solution set of this equation.

170
00:14:55,900 --> 00:14:57,860
Everything in purple here is a solution.

171
00:14:58,060 --> 00:14:59,820
Everything not in purple is not a solution.

172
00:15:01,140 --> 00:15:01,890
We agree?

173
00:15:01,960 --> 00:15:02,710
Yeah.

174
00:15:02,380 --> 00:15:03,130
Okay.

175
00:15:03,920 --> 00:15:06,440
Are all these ordered pairs, these are a bunch of ordered pairs.

176
00:15:07,360 --> 00:15:11,840
Are all these ordered pairs solutions to this inequality?

177
00:15:11,840 --> 00:15:12,590
Why

178
00:15:12,220 --> 00:15:13,920
Somebody give me an idea.

179
00:15:16,320 --> 00:15:17,920
What you think?

180
00:15:22,580 --> 00:15:23,330
Like

181
00:15:24,120 --> 00:15:25,080
So like

182
00:15:25,700 --> 00:15:29,920
The graph of the equation gives solutions to the inequality,

183
00:15:30,620 --> 00:15:33,400
but it is not all of the inequality's solutions.

184
00:15:33,400 --> 00:15:34,920
Those points all make the inequality true.

185
00:15:40,480 --> 00:15:41,880
Okay, that's correct

186
00:15:41,880 --> 00:15:44,440
So this is (1, 5), right?

187
00:15:46,360 --> 00:15:47,660
It's a solution here

188
00:15:48,460 --> 00:15:54,800
Let's check to see if it's a solution here is 5 less than or equal to 3 times 1 plus 2

189
00:15:55,480 --> 00:15:57,580
It's 5 less than or equal to 5

190
00:15:58,640 --> 00:15:59,390
Yes

191
00:16:00,000 --> 00:16:02,700
You take any ordered pair on that purple line

192
00:16:02,700 --> 00:16:07,440
Plug it in you'll get this number exactly equals this number

193
00:16:08,060 --> 00:16:09,800
because of the “or equal to” part.

194
00:16:11,380 --> 00:16:14,280
For sure it's going to be a solution to this too.

195
00:16:16,000 --> 00:16:16,750
Does this make sense?

196
00:16:18,140 --> 00:16:27,480
Okay so everything in purple here is a solution to this and there's way way way more.

197
00:16:31,740 --> 00:16:36,940
Okay and the way we do that, I think you did this last year, we do like a test point.

198
00:16:39,060 --> 00:16:41,700
You take any ordered pair not on that line.

199
00:16:42,980 --> 00:16:45,980
I always pick 0, 0 unless the line goes through the origin.

200
00:16:46,940 --> 00:16:48,120
And we test it.

201
00:16:49,200 --> 00:16:53,140
We go 0 less than or equal to 3 times 0 plus 2.

202
00:16:54,560 --> 00:16:55,310
Is this true?

203
00:16:57,500 --> 00:16:59,580
Zero is less than or equal to 2.

204
00:17:00,640 --> 00:17:01,390
It's less than.

205
00:17:03,360 --> 00:17:06,780
Okay. So this right here is a solution.

206
00:17:11,400 --> 00:17:17,840
Okay, and therefore everything on this side of the line is going to be a solution.

207
00:17:18,960 --> 00:17:21,320
Every ordered pair on the other side will not be a solution.

208
00:17:22,700 --> 00:17:30,020
If we did a test point and it was false, we would not shade this side, we'd shade the

209
00:17:30,020 --> 00:17:31,280
opposite side of the test point.

210
00:17:33,760 --> 00:17:37,090
Okay, so now we just got to put dots on every single ordered pair.

211
00:17:37,180 --> 00:17:42,760
There are so many solutions to this inequality that they fill an entire half-plane.

212
00:17:46,240 --> 00:17:54,400
We sometimes call the solution set a half plane.

213
00:18:02,420 --> 00:18:03,560
OK.

214
00:18:06,900 --> 00:18:10,800
Can I write the solution set in interval notation?

215
00:18:17,280 --> 00:18:18,360
Can I... oops.

216
00:18:20,440 --> 00:18:24,480
Can I write this entire shaded region in interval notation?

217
00:18:31,980 --> 00:18:33,180
Like I did right here?

218
00:18:35,600 --> 00:18:36,560
The answer is no.

219
00:18:38,660 --> 00:18:42,940
Intervals are sets of numbers, not sets of ordered pairs.

220
00:18:45,400 --> 00:18:46,150
OK.

221
00:18:46,960 --> 00:18:50,040
Only inequalities in one variable can have

222
00:18:50,040 --> 00:18:52,800
solution sets written in interval notation.

223
00:18:54,620 --> 00:18:58,760
The way to show a two-variable solution set

224
00:18:58,760 --> 00:19:00,340
is to graph it.

225
00:19:03,700 --> 00:19:09,560
What would change if it was strictly less than?

226
00:19:17,300 --> 00:19:27,420
Okay, if it was strictly less than, everything that's a... oops... a solution here would

227
00:19:27,420 --> 00:19:28,600
not be a solution.

228
00:19:30,280 --> 00:19:34,900
Okay, and instead of like we did up here, putting... where did we put it?

229
00:19:35,060 --> 00:19:36,140
An open circle.

230
00:19:37,060 --> 00:19:41,620
We basically put open circles on this entire line.

231
00:19:44,940 --> 00:19:47,620
But we don't really put open circles, we just put a dashed line.

232
00:19:49,700 --> 00:19:51,740
We would put a dotted line.

233
00:19:52,420 --> 00:19:58,120
That means exclude that line, all of our pairs in that line.

234
00:19:59,480 --> 00:20:04,990
Okay, let's try another one.

235
00:20:05,220 --> 00:20:07,020
We got two variables.

236
00:20:10,440 --> 00:20:14,420
Sorry, I am going to graph on a coordinate plane.

237
00:20:17,720 --> 00:20:18,800
Sorry, let me erase this.

238
00:20:18,880 --> 00:20:19,740
I want strictly less than.

239
00:20:23,840 --> 00:20:26,100
We start by graphing this line.

240
00:20:26,580 --> 00:20:29,820
Am I going to put a dotted line or solid line?

241
00:20:32,360 --> 00:20:33,110
Dotted.

242
00:20:33,260 --> 00:20:36,680
So all these solutions to this are not solutions here.

243
00:20:36,680 --> 00:20:41,980
So we exclude the boundary.

244
00:20:51,840 --> 00:20:52,940
And then what do we do?

245
00:20:54,680 --> 00:20:55,660
A test point.

246
00:20:59,940 --> 00:21:01,780
And let's test (0, 0).

247
00:21:02,600 --> 00:21:06,920
0 less than negative 3 times 0 minus 2.

248
00:21:07,820 --> 00:21:09,660
Is 0 less than negative 2?

249
00:21:13,620 --> 00:21:15,240
0 is not less than negative 2.

250
00:21:15,480 --> 00:21:16,230
This is false.

251
00:21:21,100 --> 00:21:24,160
So this test point right here is not a solution.

252
00:21:25,340 --> 00:21:26,090
It failed.

253
00:21:26,960 --> 00:21:32,360
so that means everything on the other side of the line is a solution so we just start

254
00:21:32,360 --> 00:21:33,240
shading

255
00:21:37,240 --> 00:21:38,990
this side of the line.

256
00:21:39,200 --> 00:21:45,380
Again, for an inequality in two variables you almost always end up with a half a plane of solutions.

257
00:21:46,800 --> 00:21:48,160
There are a lot of them.

258
00:21:50,820 --> 00:21:52,760
Okay what questions do we have on this?

259
00:21:53,300 --> 00:21:55,120
You guys did this last year or no?

260
00:21:56,440 --> 00:21:57,190
Yes.

261
00:21:56,720 --> 00:21:59,700
This is a review but very important

262
00:22:00,940 --> 00:22:02,680
We have to understand

263
00:22:03,240 --> 00:22:05,240
interval notation and

264
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Know when interval notation cannot be used: with two variables.