1
00:00:04,350 --> 00:00:06,310
There are two things written on the board, one on

2
00:00:06,310 --> 00:00:07,130
the left, one on the right.

3
00:00:08,730 --> 00:00:11,630
What do we call the thing on the left?

4
00:00:13,350 --> 00:00:15,690
An expression. And what do we call the thing on

5
00:00:15,690 --> 00:00:18,590
the right? What

6
00:00:18,590 --> 00:00:30,930
is

7
00:00:30,930 --> 00:00:35,430
an equation? Anybody know what an equation or what's the

8
00:00:35,430 --> 00:00:37,110
definition of an equation is?

9
00:00:37,910 --> 00:00:38,650
Take a wild guess.

10
00:00:39,450 --> 00:00:43,050
Yeah. Like two expressions brought together by an equal sign.

11
00:00:43,050 --> 00:00:47,510
Two expressions brought together by an equal sign So

12
00:00:47,510 --> 00:00:51,050
we definitely have one expression another expression then like an

13
00:00:51,050 --> 00:00:55,310
equal sign in between okay, we're close Would this be

14
00:00:55,310 --> 00:00:59,810
an equation definitely not Will this be an

15
00:00:59,810 --> 00:01:02,490
equation? Definitely not. We definitely need an equal sign.

16
00:01:02,990 --> 00:01:04,810
And we need an expression on the left and an

17
00:01:04,810 --> 00:01:05,460
expression on the right.

18
00:01:06,050 --> 00:01:07,390
Here's the actual definition.

19
00:01:08,190 --> 00:01:12,550
An equation is a mathematical statement that

20
00:01:12,550 --> 00:01:14,590
two expressions are equal.

21
00:01:15,390 --> 00:01:18,710
So an equation is a mathematical statement

22
00:01:18,710 --> 00:01:22,150
that can be written in many different forms.

23
00:01:22,950 --> 00:01:24,410
So why is this an equation?

24
00:01:25,210 --> 00:01:26,190
Whoever said it was.

25
00:01:26,530 --> 00:01:27,370
Why is it an equation?

26
00:01:27,870 --> 00:01:30,190
Because it's saying that 2x minus 3 is equal to

27
00:01:30,190 --> 00:01:33,910
5. Yeah, it's a statement that this expression is equal

28
00:01:33,910 --> 00:01:37,150
to this expression. Is this an equation?

29
00:01:38,310 --> 00:01:39,430
Yes. Yes.

30
00:01:40,550 --> 00:01:44,790
this an equation? see everybody's thumb thumbs up yes thumbs

31
00:01:44,790 --> 00:01:47,150
down no

32
00:01:47,150 --> 00:01:55,450
they're

33
00:01:55,450 --> 00:01:59,350
definitely not equal we agree yeah is

34
00:01:59,350 --> 00:02:03,330
this a

35
00:02:03,330 --> 00:02:04,770
mathematical statement

36
00:02:04,770 --> 00:02:11,730
that

37
00:02:11,730 --> 00:02:16,310
two Mr. Wittman is 19 years old is

38
00:02:16,310 --> 00:02:17,090
that a statement?

39
00:02:19,130 --> 00:02:23,250
That true statement No definitely

40
00:02:23,250 --> 00:02:27,790
not Okay, so is this a

41
00:02:27,790 --> 00:02:32,150
statement that? One expression is equal to another expression.

42
00:02:32,450 --> 00:02:36,470
Yes. Yes, this is an equation Okay

43
00:02:37,270 --> 00:02:41,770
this is an equation and is the statement true

44
00:02:41,770 --> 00:02:44,730
yeah yeah this is called a true equation

45
00:02:44,730 --> 00:02:50,850
this

46
00:02:50,850 --> 00:02:54,870
is called a false equation Is this equation red?

47
00:02:55,070 --> 00:02:59,370
Is it true, false, or neither?

48
00:03:00,170 --> 00:03:02,270
It depends on what the value of x is.

49
00:03:02,490 --> 00:03:05,170
If x is 3, it would be a true equation.

50
00:03:05,210 --> 00:03:07,170
If x is any other number, it would be a

51
00:03:07,170 --> 00:03:10,450
false equation. So this is neither true nor false.

52
00:03:10,450 --> 00:03:12,770
this is called an open equation.

53
00:03:19,830 --> 00:03:23,490
Basically, anytime you see a variable inside an equation it's

54
00:03:23,490 --> 00:03:27,890
an open equation. All equations are

55
00:03:27,890 --> 00:03:31,690
either true, false,

56
00:03:34,050 --> 00:03:34,700
or open.

57
00:03:36,430 --> 00:03:37,850
This is very important.

58
00:03:38,650 --> 00:03:40,290
Is this an equation?

59
00:03:41,090 --> 00:03:44,430
Yes, yes X is an expression X is an expression

60
00:03:44,430 --> 00:03:47,950
is a statement the two expressions are equal

61
00:03:49,030 --> 00:03:51,410
This contains a variable however.

62
00:03:51,410 --> 00:03:55,370
This is the rare case that this is not an

63
00:03:55,370 --> 00:03:57,190
open equation. This would be a what?

64
00:03:58,210 --> 00:04:00,750
So but most any other time

65
00:04:02,190 --> 00:04:06,750
like this equation is no longer true always it's

66
00:04:06,750 --> 00:04:11,290
open okay but almost every time except for special

67
00:04:11,290 --> 00:04:15,670
cases if an equation contains a

68
00:04:15,670 --> 00:04:19,610
variable it's an open equation Can

69
00:04:21,950 --> 00:04:24,950
Can we solve these?

70
00:04:25,750 --> 00:04:27,510
7th grade? 8th grade?

71
00:04:28,150 --> 00:04:29,190
Something like that.

72
00:04:30,170 --> 00:04:31,590
What would be the first step?

73
00:04:32,390 --> 00:04:34,830
Subtract 5 both sides or add negative 5 both sides.

74
00:04:35,630 --> 00:04:38,950
Which property allows me to do that?

75
00:04:39,750 --> 00:04:43,750
Either the addition property or the subtraction property.

76
00:04:43,750 --> 00:04:46,490
If you add a negative I just call everything the

77
00:04:46,490 --> 00:04:47,650
addition property of equality.

78
00:04:49,690 --> 00:04:50,910
And then what's the next step?

79
00:04:51,890 --> 00:04:55,270
Divide by 2. Okay,

80
00:04:55,270 --> 00:04:56,670
let

81
00:04:56,670 --> 00:05:01,070
me come back to

82
00:05:01,070 --> 00:05:05,510
this. So if the directions were

83
00:05:05,510 --> 00:05:06,160
solved,

84
00:05:07,390 --> 00:05:10,330
what does solve mean?

85
00:05:11,130 --> 00:05:13,370
Make the statement true.

86
00:05:14,170 --> 00:05:17,390
the statement true. We'll come back to that in a

87
00:05:17,390 --> 00:05:18,090
second.

88
00:05:19,610 --> 00:05:20,730
Some people said find the answer.

89
00:05:21,290 --> 00:05:21,940
What else did we say?

90
00:05:22,310 --> 00:05:23,670
Find the unknown value.

91
00:05:24,630 --> 00:05:25,970
Okay, it's a possibility.

92
00:05:26,770 --> 00:05:28,410
So let's write down the definition.

93
00:05:28,410 --> 00:05:29,890
to solve

94
00:05:29,890 --> 00:05:36,750
means

95
00:05:36,750 --> 00:05:38,550
to find

96
00:05:38,550 --> 00:05:41,650
the

97
00:05:41,650 --> 00:05:44,250
solution

98
00:05:44,250 --> 00:05:48,470
set We'll come back to what

99
00:05:48,470 --> 00:05:50,210
solutions that means in a second.

100
00:05:50,790 --> 00:05:53,370
What uh, what is the solution to this equation?

101
00:05:54,910 --> 00:05:58,990
X equals 3. Uh, it is not x equals 3.

102
00:05:59,570 --> 00:06:03,150
It is 3. 3

103
00:06:03,150 --> 00:06:07,490
is the solution.

104
00:06:09,170 --> 00:06:10,710
So what's the mean to be a solution?

105
00:06:10,710 --> 00:06:12,030
OK,

106
00:06:14,830 --> 00:06:15,890
this is very important.

107
00:06:16,510 --> 00:06:19,530
Here's the definition. a solution

108
00:06:19,530 --> 00:06:27,250
is

109
00:06:27,250 --> 00:06:28,490
a number

110
00:06:28,490 --> 00:06:32,790
that

111
00:06:32,790 --> 00:06:35,790
when replaced

112
00:06:35,790 --> 00:06:39,610
for the variable

113
00:06:40,410 --> 00:06:43,210
it makes

114
00:06:43,210 --> 00:06:46,950
the equation

115
00:06:46,950 --> 00:06:50,890
true

116
00:06:51,690 --> 00:06:55,130
OK, so why is the number 3 a solution?

117
00:06:55,910 --> 00:06:58,350
Because when we replace it with a variable,

118
00:07:01,590 --> 00:07:05,430
we no longer have an open equation, we have a

119
00:07:05,430 --> 00:07:06,130
true equation.

120
00:07:07,370 --> 00:07:11,410
OK, so a solution is a number that when replaced

121
00:07:11,410 --> 00:07:13,730
with a variable, it makes the equation true.

122
00:07:14,530 --> 00:07:19,110
Why is the number 7 not a solution?

123
00:07:19,910 --> 00:07:22,410
Because when it replaces the variable, it does not make

124
00:07:22,410 --> 00:07:25,330
the equation true. Okay.

125
00:07:26,130 --> 00:07:26,780
Okay.

126
00:07:28,070 --> 00:07:29,670
Solution set,

127
00:07:30,890 --> 00:07:32,330
the definition is

128
00:07:33,910 --> 00:07:36,690
a solution set

129
00:07:37,490 --> 00:07:42,070
is the set of

130
00:07:42,070 --> 00:07:46,630
all solutions. So

131
00:07:46,630 --> 00:07:49,830
we still haven't solved yet because we haven't found the

132
00:07:49,830 --> 00:07:52,390
solution set. Well, we kind of did, but

133
00:07:52,390 --> 00:07:56,850
our final answer has to be a solution

134
00:07:56,850 --> 00:07:57,500
set.

135
00:07:59,420 --> 00:08:01,670
Is x equals 3 the solution?

136
00:08:03,670 --> 00:08:05,870
Why is x equals 3 not the solution?

137
00:08:06,670 --> 00:08:11,050
It's still an open equation a solution is

138
00:08:11,050 --> 00:08:14,690
a number not an equation, okay?

139
00:08:15,490 --> 00:08:17,630
So the solution is

140
00:08:20,050 --> 00:08:24,210
is three and the

141
00:08:24,210 --> 00:08:25,410
solution set

142
00:08:25,410 --> 00:08:29,790
you

143
00:08:29,790 --> 00:08:33,110
simply take all the solutions there's only one of them

144
00:08:33,110 --> 00:08:37,610
here and we put it in these curly brackets

145
00:08:37,610 --> 00:08:42,180
there's several ways to

146
00:08:42,180 --> 00:08:43,940
that denotes sets in mathematics.

147
00:08:45,160 --> 00:08:47,700
If we could list them all, we simply use curly

148
00:08:47,700 --> 00:08:50,700
brackets and put them all in there separated by comma.

149
00:08:50,900 --> 00:08:52,060
There's only one here.

150
00:08:52,860 --> 00:08:55,660
Are these three the same equation?

151
00:08:56,440 --> 00:08:58,380
They're definitely not the same.

152
00:09:00,980 --> 00:09:05,580
This is a statement that this expression is equal

153
00:09:05,580 --> 00:09:07,960
to this expression, which this expression is way different than

154
00:09:07,960 --> 00:09:10,780
this expression, and this expression is way different.

155
00:09:12,260 --> 00:09:13,640
And this is way different than that one.

156
00:09:14,000 --> 00:09:15,200
They're all different equations.

157
00:09:15,980 --> 00:09:17,720
However they all have something in common.

158
00:09:17,920 --> 00:09:18,620
What do they have in common?

159
00:09:19,420 --> 00:09:20,580
The same exact solution set.

160
00:09:21,020 --> 00:09:23,400
The same exact solution set.

161
00:09:24,500 --> 00:09:28,620
Okay. Like the addition property of equality says

162
00:09:28,620 --> 00:09:32,800
you can add the same number to both sides of

163
00:09:32,800 --> 00:09:36,940
the equation and the new equation is

164
00:09:36,940 --> 00:09:38,240
equivalent.

165
00:09:39,780 --> 00:09:41,480
Not the same equations, equivalent.

166
00:09:41,980 --> 00:09:44,280
So these are all called equivalent equations.

167
00:09:45,080 --> 00:09:47,880
Equivalent equations

168
00:09:47,880 --> 00:09:52,160
are

169
00:09:52,160 --> 00:09:55,820
equations. that have

170
00:09:55,820 --> 00:09:59,100
the same

171
00:09:59,100 --> 00:10:02,560
solution

172
00:10:02,560 --> 00:10:04,840
set.

173
00:10:06,640 --> 00:10:10,700
Another definition. Okay, so we use the addition property of

174
00:10:10,700 --> 00:10:11,440
equality to go to here.

175
00:10:12,120 --> 00:10:15,160
Here we use the division or multiplication property of equality

176
00:10:15,160 --> 00:10:19,620
which says you can divide both

177
00:10:19,620 --> 00:10:23,480
numbers by the both sides of the equation by the

178
00:10:23,480 --> 00:10:27,140
same number and the new equation, it one's not zero

179
00:10:27,140 --> 00:10:30,060
is equivalent which means that is the same solution set

180
00:10:30,060 --> 00:10:34,480
okay and once we're here we know the obvious

181
00:10:34,480 --> 00:10:37,220
solution is three can we plug in three equals three

182
00:10:37,220 --> 00:10:39,280
okay

183
00:10:39,280 --> 00:10:43,420
let's try another

184
00:10:43,420 --> 00:10:45,960
one The directions say solve.

185
00:10:47,280 --> 00:10:48,360
What does it mean to solve?

186
00:10:50,860 --> 00:10:52,360
Find the solution set.

187
00:10:52,960 --> 00:10:54,940
And what is the solution set?

188
00:10:55,920 --> 00:10:57,580
The set of all solutions.

189
00:10:57,960 --> 00:10:58,780
The set of all solutions.

190
00:10:58,980 --> 00:11:00,640
You take them all and you put them in curly

191
00:11:00,640 --> 00:11:01,290
brackets.

192
00:11:02,200 --> 00:11:05,800
Okay. And what is the solution?

193
00:11:07,180 --> 00:11:09,960
A number that when replacing the variable makes the statement

194
00:11:09,960 --> 00:11:11,920
true. Makes the equation true.

195
00:11:12,320 --> 00:11:15,780
A number that when replaced for the variable makes the

196
00:11:15,780 --> 00:11:20,140
equation true. A number that when replaced

197
00:11:20,140 --> 00:11:22,940
for the variable makes the equation true.

198
00:11:22,940 --> 00:11:26,920
So solutions, okay first step here is

199
00:11:28,200 --> 00:11:32,620
Add eight to both sides Next step

200
00:11:34,580 --> 00:11:37,900
Divide by five or multiply by one fifth same thing

201
00:11:38,480 --> 00:11:40,320
Is this my solution

202
00:11:42,200 --> 00:11:44,860
No, we're getting better, I like this.

203
00:11:45,580 --> 00:11:50,140
What's my solution? Twelve fifths, or two point

204
00:11:50,140 --> 00:11:51,180
four, something like that.

205
00:11:52,020 --> 00:11:53,480
And what is the solution set?

206
00:11:55,060 --> 00:11:59,120
Twelve fifths. So the way we say it, we say

207
00:11:59,120 --> 00:12:03,440
the set containing

208
00:12:03,440 --> 00:12:06,780
12

209
00:12:06,780 --> 00:12:11,200
fifths.

210
00:12:11,200 --> 00:12:14,000
We write down. It's just curly brackets and put the

211
00:12:14,000 --> 00:12:17,680
number inside And why is 12 fifth the solution?

212
00:12:18,480 --> 00:12:21,540
Because when replace for the variable or plug into the

213
00:12:21,540 --> 00:12:24,380
variable it makes the equation true Why is the number

214
00:12:24,380 --> 00:12:25,600
zero not a solution?

215
00:12:26,360 --> 00:12:30,860
Because when you replace a

216
00:12:30,860 --> 00:12:32,560
variable it does not make the equation true.

217
00:12:33,620 --> 00:12:34,270
Okay.

218
00:12:36,060 --> 00:12:37,700
Let's do another one.

219
00:12:38,500 --> 00:12:41,260
Okay, here we got a very well on both sides.

220
00:12:42,320 --> 00:12:43,820
So what's our first step?

221
00:12:43,820 --> 00:12:47,220
extract 3x both

222
00:12:47,220 --> 00:12:51,340
sides And

223
00:12:51,340 --> 00:12:53,380
we have four equals seven.

224
00:12:54,180 --> 00:12:55,760
What does that tell us?

225
00:12:55,960 --> 00:12:57,140
It's a false equation.

226
00:12:58,000 --> 00:12:58,650
Good.

227
00:13:00,000 --> 00:13:03,520
So we use the addition property of equality.

228
00:13:03,760 --> 00:13:06,680
We end up with a false equation, always false equation.

229
00:13:08,240 --> 00:13:10,640
So what is the solutions?

230
00:13:11,440 --> 00:13:12,500
There's none.

231
00:13:27,880 --> 00:13:29,780
So is there a solution set?

232
00:13:30,220 --> 00:13:30,870
No.

233
00:13:31,940 --> 00:13:33,580
I'll leave it. there is.

234
00:13:34,380 --> 00:13:36,200
Okay, it's simply

235
00:13:38,320 --> 00:13:39,780
this set.

236
00:13:41,080 --> 00:13:43,320
The set contains all of its solutions.

237
00:13:44,120 --> 00:13:48,040
There are no solutions to this equation, but there is

238
00:13:48,040 --> 00:13:52,220
a solution set. So can we even solve this

239
00:13:52,220 --> 00:13:54,860
equation? What does it mean to solve?

240
00:13:55,660 --> 00:13:57,400
Did we find the solution set?

241
00:13:58,080 --> 00:14:00,020
Yes, we can solve this equation.

242
00:14:02,100 --> 00:14:05,080
We can't find any solutions but we can solve it.

243
00:14:06,100 --> 00:14:10,220
Ok, because by the definition we found the solution set.

244
00:14:11,020 --> 00:14:13,800
In fact this is a very important set.

245
00:14:14,760 --> 00:14:18,620
Its own name is called the empty set And in

246
00:14:18,620 --> 00:14:22,100
fact, it's so important we give it its own symbol

247
00:14:23,540 --> 00:14:24,190
Okay

248
00:14:25,620 --> 00:14:29,180
It's a circle with a slightly diagonal line through it

249
00:14:31,320 --> 00:14:33,300
This is also the empty set

250
00:14:35,720 --> 00:14:39,820
You could either write this as your solution set or

251
00:14:39,820 --> 00:14:42,400
you could write this as your solution set.

252
00:14:43,200 --> 00:14:45,060
Simply an empty set.

253
00:14:45,300 --> 00:14:47,300
In fact, it's one of the most important sets in

254
00:14:47,300 --> 00:14:51,200
all of mathematics. Is this the empty set?

255
00:14:52,740 --> 00:14:53,880
Definitely not, this is not empty.

256
00:14:54,520 --> 00:14:56,160
It contains another set.

257
00:14:57,220 --> 00:14:59,660
The set containing the empty set is definitely not the

258
00:14:59,660 --> 00:15:01,480
empty set. So don't ever write this.

259
00:15:02,580 --> 00:15:03,230
Terrible.

260
00:15:04,480 --> 00:15:07,060
You can either write this or write this.

261
00:15:07,720 --> 00:15:09,360
Don't put one inside the other.

262
00:15:10,020 --> 00:15:10,670
Okay?

263
00:15:13,000 --> 00:15:15,240
Okay. Let's try a new equation.

264
00:15:16,040 --> 00:15:19,440
This has lots of variables, what do we get first?

265
00:15:19,440 --> 00:15:23,690
I like my terms like that

266
00:15:24,240 --> 00:15:28,120
And now what You could either stop here

267
00:15:29,100 --> 00:15:33,420
This is always equal to this If you want to

268
00:15:33,420 --> 00:15:37,620
we could subtract 4x we get this equation

269
00:15:38,540 --> 00:15:41,200
and it doesn't matter if we do that or not

270
00:15:42,000 --> 00:15:44,520
and what do we end up with?

271
00:15:46,180 --> 00:15:50,520
an always true equation so what are the solutions?

272
00:15:51,320 --> 00:15:54,320
Every single number we know is a solution.

273
00:15:55,820 --> 00:15:58,040
And what is the biggest set of numbers we know?

274
00:16:01,240 --> 00:16:05,580
All real numbers. Why is every real number a

275
00:16:05,580 --> 00:16:08,180
solution? Because when you plug it in for the variable

276
00:16:08,180 --> 00:16:12,280
it makes the variables in this case makes the equation

277
00:16:12,280 --> 00:16:16,840
true. Okay, what is the solution

278
00:16:16,840 --> 00:16:21,000
set? Okay, the solution set the set of real

279
00:16:21,000 --> 00:16:23,060
numbers. Okay.

280
00:16:24,580 --> 00:16:27,860
And there's lots of different ways we denote the set

281
00:16:27,860 --> 00:16:28,780
of real numbers

282
00:16:30,220 --> 00:16:32,420
One way is we write it out in words like

283
00:16:32,420 --> 00:16:33,070
that

284
00:16:34,980 --> 00:16:39,580
Another way is It looks like this have

285
00:16:39,580 --> 00:16:40,300
we seen that before

286
00:16:41,320 --> 00:16:41,970
Okay

287
00:16:43,220 --> 00:16:46,900
That is simply the letter R.

288
00:16:46,900 --> 00:16:48,500
But in boldface

289
00:16:49,500 --> 00:16:53,900
That's not IR it's bold face

290
00:16:53,900 --> 00:16:57,580
are Right when you're in Microsoft Word or whatever how

291
00:16:57,580 --> 00:16:58,550
do you make something bold?

292
00:16:59,180 --> 00:17:01,960
We highlight it and press bold

293
00:17:03,500 --> 00:17:06,760
Okay Before computers you couldn't do that.

294
00:17:06,780 --> 00:17:10,480
You had to make it bold by hand and so

295
00:17:10,480 --> 00:17:11,680
back who knows how long

296
00:17:12,660 --> 00:17:16,980
Okay, they made it bold by making the left side

297
00:17:18,400 --> 00:17:22,720
Put a line there, okay, that's simply bold face

298
00:17:22,720 --> 00:17:26,760
are It's just that is the one way we denote

299
00:17:26,760 --> 00:17:27,720
the set of real numbers.

300
00:17:27,860 --> 00:17:29,400
We're gonna learn another way.

301
00:17:29,400 --> 00:17:30,940
Some of you might know this

302
00:17:31,960 --> 00:17:36,520
we do this all the numbers between negative infinity and

303
00:17:36,520 --> 00:17:41,060
positive infinity okay we'll typically do this way this is

304
00:17:41,060 --> 00:17:45,500
called interval notation how many of you guys learned interval

305
00:17:45,500 --> 00:17:46,880
notation last year

306
00:17:47,980 --> 00:17:52,400
and not too many we'll learn it uh beginning next

307
00:17:52,400 --> 00:17:53,090
week i think

308
00:17:55,390 --> 00:17:59,650
Okay, so some equations have one solution,

309
00:17:59,810 --> 00:18:03,790
some equations have no solutions,

310
00:18:05,270 --> 00:18:07,450
and very similar equation all of a sudden we have

311
00:18:07,450 --> 00:18:09,150
an infinite number of solutions.

312
00:18:09,950 --> 00:18:14,370
Alright, either way, any equation we can

313
00:18:14,370 --> 00:18:16,690
solve even if there's no solutions.

314
00:18:17,490 --> 00:18:18,850
Is this an equation?

315
00:18:19,650 --> 00:18:20,650
Why is it an equation?

316
00:18:21,450 --> 00:18:24,110
It's a statement that two expressions are equal.

317
00:18:24,310 --> 00:18:25,930
As long as you see an equal sign and some

318
00:18:25,930 --> 00:18:27,490
expression on the left and some expression on the right,

319
00:18:27,590 --> 00:18:31,070
it's an equation. Anybody know a solution to

320
00:18:31,070 --> 00:18:34,970
this? Just like say

321
00:18:34,970 --> 00:18:38,030
like x can go y and y can go like

322
00:18:38,030 --> 00:18:38,680
5.

323
00:18:40,810 --> 00:18:41,460
Close.

324
00:18:42,750 --> 00:18:45,710
So this is fundamentally different because there's two variables.

325
00:18:47,630 --> 00:18:52,130
So we actually have to alter our definition.

326
00:18:54,670 --> 00:18:56,430
Where's our definition of equation?

327
00:18:57,650 --> 00:19:00,250
Here it is. An equation is a mathematical statement that...

328
00:19:01,490 --> 00:19:02,140
Sorry.

329
00:19:03,110 --> 00:19:04,110
Definition of a solution.

330
00:19:04,770 --> 00:19:08,670
A solution is a number that when replaced for the

331
00:19:08,670 --> 00:19:09,320
variable,

332
00:19:10,190 --> 00:19:13,770
this is really the only definition for an equation in

333
00:19:13,770 --> 00:19:17,850
one variable. Okay, when we have an equation in two

334
00:19:17,850 --> 00:19:22,090
variables, it's a different

335
00:19:22,090 --> 00:19:23,490
definition. Let's write down

336
00:19:25,730 --> 00:19:27,130
definition.

337
00:19:31,150 --> 00:19:33,350
A solution,

338
00:19:34,710 --> 00:19:37,650
equation,

339
00:19:40,710 --> 00:19:43,010
and two variables.

340
00:19:49,210 --> 00:19:51,470
It's not a number.

341
00:19:52,210 --> 00:19:54,950
We need two different numbers

342
00:19:54,950 --> 00:19:59,210
okay and

343
00:19:59,210 --> 00:20:03,730
right for example one comma five

344
00:20:03,730 --> 00:20:07,470
somebody said or one for the X five for the

345
00:20:07,470 --> 00:20:10,790
Y is the solution so we had to know which

346
00:20:10,790 --> 00:20:13,750
number goes to which variable and the way we denote

347
00:20:13,750 --> 00:20:16,750
that is an ordered pair.

348
00:20:18,110 --> 00:20:21,610
So a solution to the equation in two variables is

349
00:20:21,610 --> 00:20:24,410
an ordered pair

350
00:20:24,410 --> 00:20:28,790
such

351
00:20:28,790 --> 00:20:29,470
that

352
00:20:29,470 --> 00:20:33,650
when

353
00:20:33,650 --> 00:20:34,650
the numbers

354
00:20:34,650 --> 00:20:39,370
are

355
00:20:39,370 --> 00:20:43,650
replaced for

356
00:20:43,650 --> 00:20:46,510
their respective

357
00:20:46,510 --> 00:20:51,410
variables

358
00:20:51,410 --> 00:20:55,810
it

359
00:20:55,810 --> 00:20:57,250
makes the equation true.

360
00:21:05,870 --> 00:21:08,570
So it's not a number, it's two numbers.

361
00:21:10,050 --> 00:21:11,450
We should call it ordered pair.

362
00:21:12,930 --> 00:21:15,790
Such that when the numbers are replaced for the respective

363
00:21:15,790 --> 00:21:18,110
variables, the x for the x my life or why

364
00:21:18,110 --> 00:21:19,850
it makes the equation true.

365
00:21:27,670 --> 00:21:30,450
So 1 comma 5 is a solution.

366
00:21:30,970 --> 00:21:31,890
Somebody give me another solution

367
00:21:31,890 --> 00:21:38,150
zero

368
00:21:38,150 --> 00:21:42,570
comma three well okay

369
00:21:42,570 --> 00:21:45,090
interesting I like that one half four

370
00:21:45,090 --> 00:21:48,850
three comma

371
00:21:48,850 --> 00:21:49,950
nine

372
00:21:49,950 --> 00:21:54,770
100

373
00:21:54,770 --> 00:21:55,420
comma

374
00:21:57,190 --> 00:22:00,270
203

375
00:22:00,270 --> 00:22:04,810
negative 10 comma

376
00:22:04,810 --> 00:22:08,010
negative 17

377
00:22:08,010 --> 00:22:13,170
and

378
00:22:13,170 --> 00:22:17,470
we could keep going how many solutions are there to

379
00:22:17,470 --> 00:22:19,470
this equation? Infinite.

380
00:22:23,570 --> 00:22:28,010
Okay there's no way we're gonna

381
00:22:28,010 --> 00:22:31,750
write down all the solutions so we're not going to

382
00:22:31,750 --> 00:22:36,250
do a curly bracket so we can't write

383
00:22:36,250 --> 00:22:36,900
them all plan.

384
00:22:40,630 --> 00:22:43,950
So how are we going to write down the

385
00:22:43,950 --> 00:22:47,470
solution set? Any guesses?

386
00:22:51,930 --> 00:22:52,580
Okay,

387
00:22:53,230 --> 00:22:53,880
so I have a question.

388
00:22:54,590 --> 00:22:57,190
So what would I write?

389
00:22:57,990 --> 00:23:00,930
So all real numbers?

390
00:23:01,450 --> 00:23:04,110
And the range is?

391
00:23:05,430 --> 00:23:06,290
All real numbers.

392
00:23:08,190 --> 00:23:11,450
So does that mean every ordered pair is a solution?

393
00:23:12,210 --> 00:23:14,570
No. It's definitely not.

394
00:23:16,090 --> 00:23:17,830
0, 10 is not a solution.

395
00:23:19,890 --> 00:23:21,930
So we can't just write down the domain range.

396
00:23:22,010 --> 00:23:23,990
We'll talk about what domain range means in a minute.

397
00:23:24,790 --> 00:23:26,370
How am I going to do this?

398
00:23:27,810 --> 00:23:29,710
Because we know lots of numbers that are not solutions.

399
00:23:29,710 --> 00:23:32,330
Sorry, lots of ordered pairs.

400
00:23:37,990 --> 00:23:42,430
This definitely is

401
00:23:42,430 --> 00:23:43,450
a linear equation.

402
00:23:48,590 --> 00:23:52,230
Okay, in fact, you've written down the solution set to

403
00:23:52,230 --> 00:23:54,670
this many, many, many times.

404
00:23:56,390 --> 00:23:58,470
How do we do it in eighth grade?

405
00:24:01,490 --> 00:24:02,610
We

406
00:24:02,610 --> 00:24:07,370
graphed

407
00:24:07,370 --> 00:24:08,810
the solution set.

408
00:24:11,410 --> 00:24:12,530
Okay.

409
00:24:15,050 --> 00:24:17,610
We draw a coordinate plane.

410
00:24:18,410 --> 00:24:22,990
So when you graph linear equations in two variables in

411
00:24:22,990 --> 00:24:24,890
eighth grade, and you probably did a bunch last year

412
00:24:24,890 --> 00:24:28,850
too, you're really just graphing the solution set.

413
00:24:29,870 --> 00:24:32,930
All the ordered pairs that were replaced for the variable

414
00:24:32,930 --> 00:24:34,310
make the equation true.

415
00:24:35,230 --> 00:24:39,570
There's an This line is simply an infinite

416
00:24:39,570 --> 00:24:43,850
number of ordered pairs Every single ordered pair on that

417
00:24:43,850 --> 00:24:47,790
line is a solution all of them together is the

418
00:24:47,790 --> 00:24:48,690
solution set

419
00:24:49,690 --> 00:24:54,150
Okay We could have graphed the solutions

420
00:24:54,150 --> 00:24:58,230
for these these are actually also called linear equations but

421
00:24:58,230 --> 00:24:59,150
only in one variable.

422
00:24:59,910 --> 00:25:00,650
We could graph it.

423
00:25:00,910 --> 00:25:03,070
There's only one variable we need only one number line.

424
00:25:07,310 --> 00:25:10,930
We could put a dot right here on 12

425
00:25:10,930 --> 00:25:14,690
fifths and we graph the solution set.

426
00:25:14,970 --> 00:25:16,710
We put a dot on every solution.

427
00:25:17,490 --> 00:25:19,070
We don't do that because it's pointless.

428
00:25:19,390 --> 00:25:21,570
I mean it's a lot easier to do this.

429
00:25:22,630 --> 00:25:26,970
Okay we could have graphed this solution

430
00:25:26,970 --> 00:25:31,550
set here and we

431
00:25:31,550 --> 00:25:35,550
put a dot on every single solution which is all

432
00:25:35,550 --> 00:25:39,810
written numbers but we're gonna do that okay

433
00:25:39,810 --> 00:25:43,230
but we do graph the solution set to linear equations

434
00:25:43,230 --> 00:25:47,750
in two variables it's it's really the only way to

435
00:25:47,750 --> 00:25:50,950
write them all down there are some other notations that

436
00:25:50,950 --> 00:25:54,710
we don't to worry about that right now

437
00:25:54,710 --> 00:25:58,670
okay this tells you what is a solution what is

438
00:25:58,670 --> 00:25:59,570
not a solution

439
00:25:59,570 --> 00:26:05,770
okay

440
00:26:05,770 --> 00:26:09,890
so

441
00:26:09,890 --> 00:26:13,990
did we solve this equation

442
00:26:13,990 --> 00:26:18,550
yeah why

443
00:26:18,550 --> 00:26:22,010
do we solve it found the solution set.

444
00:26:23,470 --> 00:26:25,810
Okay, this is very important.

445
00:26:26,450 --> 00:26:28,910
We're always either going to start with an expression this

446
00:26:28,910 --> 00:26:29,930
class or an equation.

447
00:26:32,230 --> 00:26:32,880
Okay,

448
00:26:34,450 --> 00:26:35,790
this very first equation,

449
00:26:37,550 --> 00:26:39,430
sorry, very first Expression I wrote here.

450
00:26:39,910 --> 00:26:41,450
Can we solve this?

451
00:26:42,250 --> 00:26:44,730
Why can we not solve It's

452
00:26:45,850 --> 00:26:49,730
it's not an equation to solve needs to find the

453
00:26:49,730 --> 00:26:50,720
solution set a

454
00:26:51,970 --> 00:26:55,670
solution set that all solutions a solutions a number that

455
00:26:55,670 --> 00:26:58,950
replace for the variable makes the equation true There's no

456
00:26:58,950 --> 00:26:59,600
equation

457
00:27:01,050 --> 00:27:03,110
Expressions are neither true or false

458
00:27:04,090 --> 00:27:08,250
Okay, just is what it is Okay, you cannot solve

459
00:27:08,250 --> 00:27:12,410
this you can't solve this an equation okay,

460
00:27:12,410 --> 00:27:15,850
so when we When you look at something around the

461
00:27:15,850 --> 00:27:17,690
board it's either going to be an expression or an

462
00:27:17,690 --> 00:27:18,340
equation

463
00:27:19,290 --> 00:27:23,470
We do totally different stuff Almost always we solve these

464
00:27:23,470 --> 00:27:27,270
Expressions we can do we can manipulate them You

465
00:27:27,270 --> 00:27:31,230
could rewrite them.

466
00:27:31,250 --> 00:27:35,470
You could factor out a number.

467
00:27:37,170 --> 00:27:41,350
You can change it, manipulate it, simplify it, complexify it,

468
00:27:41,350 --> 00:27:42,770
but you cannot solve it.

469
00:27:42,950 --> 00:27:47,430
Very important. A new definition.

470
00:27:48,230 --> 00:27:52,010
The definition of a relation is simply a set of

471
00:27:52,010 --> 00:27:52,670
ordered pairs.

472
00:27:53,830 --> 00:27:55,270
It's called a relation.

473
00:27:56,070 --> 00:27:57,950
Is this a relation?

474
00:27:59,890 --> 00:28:02,890
Yes. Yes. Yes, why?

475
00:28:03,670 --> 00:28:04,670
Because it's a set of ordered pairs.

476
00:28:05,140 --> 00:28:07,850
Curly brackets mean it's a set.

477
00:28:07,850 --> 00:28:10,770
and the inside separated by commas are?

478
00:28:11,070 --> 00:28:12,630
Ordered pairs. Ordered pairs, this set of order

479
00:28:12,630 --> 00:28:13,280
of pairs.

480
00:28:14,790 --> 00:28:17,670
Okay. This has a very clear pattern.

481
00:28:17,990 --> 00:28:19,510
Well, it should be clear.

482
00:28:21,730 --> 00:28:25,050
1, 3, 2, 5, 3, 7, what's the next one?

483
00:28:25,750 --> 00:28:28,230
4, 9. 4, 9, so on and so forth.

484
00:28:28,670 --> 00:28:30,850
If there's a clear pattern, you just put dot dot

485
00:28:30,850 --> 00:28:31,930
dot, the ellipses.

486
00:28:32,890 --> 00:28:35,890
Okay, so here this is a relation.

487
00:28:35,910 --> 00:28:37,190
Why is this a relation?

488
00:28:37,990 --> 00:28:40,170
Why is this a relation?

489
00:28:40,970 --> 00:28:43,370
Because it's a set of ordered pairs.

490
00:28:43,570 --> 00:28:46,890
That's right Because there's the brackets and there's everything inside

491
00:28:46,890 --> 00:28:49,590
is ordered. This has an infinite of ordered pairs in

492
00:28:49,590 --> 00:28:54,190
it Okay Okay, there are three ways

493
00:28:54,190 --> 00:28:58,190
to

494
00:28:58,190 --> 00:29:02,770
denote relations.

495
00:29:03,570 --> 00:29:04,450
First of

496
00:29:04,450 --> 00:29:08,990
all, you list

497
00:29:08,990 --> 00:29:13,590
the ordered pairs. Just like we did

498
00:29:13,590 --> 00:29:15,790
here and here. Okay.

499
00:29:15,790 --> 00:29:17,950
It could be in a table too.

500
00:29:18,190 --> 00:29:21,670
Doesn't necessarily have to look exactly...

501
00:29:22,470 --> 00:29:23,120
Okay.

502
00:29:24,830 --> 00:29:26,850
You can list them like that in the correct...

503
00:29:26,850 --> 00:29:31,430
This is called roster notation you put them

504
00:29:31,430 --> 00:29:35,810
in a table you can go 1 5 negative 4

505
00:29:35,810 --> 00:29:37,390
and put arrows

506
00:29:37,390 --> 00:29:42,530
2

507
00:29:42,530 --> 00:29:47,030
ok doesn't really matter as long as you listing your

508
00:29:47,030 --> 00:29:51,590
ordered pairs. The second way what What else can we

509
00:29:51,590 --> 00:29:52,370
do with ordered pairs?

510
00:29:53,170 --> 00:29:56,670
Just like we did up here, we graphed them.

511
00:29:57,170 --> 00:29:59,790
We graphed the relation.

512
00:30:00,590 --> 00:30:04,190
Thank ya. Okay, it could be like this or like

513
00:30:04,190 --> 00:30:07,190
we did earlier. We can have an infinite number of

514
00:30:07,190 --> 00:30:08,290
ordered pairs.

515
00:30:09,230 --> 00:30:11,390
That's typically when we graph a relation.

516
00:30:11,730 --> 00:30:13,230
When there's an infinite number of them.

517
00:30:14,630 --> 00:30:16,450
And then what's the third way?

518
00:30:17,850 --> 00:30:19,310
Do you want to take a guess what the third

519
00:30:19,310 --> 00:30:21,140
way is? This

520
00:30:23,240 --> 00:30:26,540
Was the solution set this guy, right

521
00:30:28,140 --> 00:30:31,480
Okay, so the third way is we write down an

522
00:30:31,480 --> 00:30:32,130
equation

523
00:30:33,860 --> 00:30:37,940
Okay However, an equation is not

524
00:30:37,940 --> 00:30:40,780
a set of ordered pairs.

525
00:30:42,560 --> 00:30:46,800
Okay, so if we write down y

526
00:30:46,800 --> 00:30:50,920
equals 2x plus 3, this is not a set of

527
00:30:50,920 --> 00:30:52,900
ordered pairs. But,

528
00:30:53,940 --> 00:30:56,580
what about this is a set of ordered pairs?

529
00:30:57,380 --> 00:31:00,640
The solution set. So in equation,

530
00:31:01,360 --> 00:31:04,160
it's the solution set.

531
00:31:04,960 --> 00:31:09,180
OK. It's the solution set that is the set of

532
00:31:09,180 --> 00:31:11,040
ordered pairs. It is the relation.

533
00:31:11,880 --> 00:31:14,000
So we'll call this a relation

534
00:31:14,900 --> 00:31:17,600
Technically, it's not the solution set that this is the

535
00:31:17,600 --> 00:31:18,250
equation

536
00:31:20,480 --> 00:31:23,940
If the solution set is the set of ordered pairs,

537
00:31:23,940 --> 00:31:27,180
okay, so those are three ways we denote relations

538
00:31:28,950 --> 00:31:33,130
Is this an equation?

539
00:31:34,950 --> 00:31:35,600
Wait, will?

540
00:31:36,990 --> 00:31:38,370
Sometimes it's a yes.

541
00:31:38,870 --> 00:31:40,950
Definitely not. Is that an equal sign?

542
00:31:41,530 --> 00:31:43,750
No. No, definitely not an equation.

543
00:31:43,990 --> 00:31:44,850
What do we call this thing?

544
00:31:47,250 --> 00:31:47,900
An inequality.

545
00:31:48,910 --> 00:31:52,270
Okay. Can we solve inequalities?

546
00:31:53,970 --> 00:31:55,170
What's the need to solve?

547
00:31:55,590 --> 00:31:56,310
Find the solution set.

548
00:31:57,250 --> 00:32:00,170
Okay, it needs to find the solution set.

549
00:32:00,720 --> 00:32:03,530
What's the definition of a solution to an inequality?

550
00:32:04,330 --> 00:32:05,910
Actually let's go back.

551
00:32:05,910 --> 00:32:07,190
what is an inequality

552
00:32:07,190 --> 00:32:16,830
what

553
00:32:16,830 --> 00:32:20,790
is

554
00:32:20,790 --> 00:32:22,070
it

555
00:32:24,130 --> 00:32:26,570
So, what is the definition of an inequality?

556
00:32:27,370 --> 00:32:30,430
Umm, close. Not, well, not really.

557
00:32:31,190 --> 00:32:32,130
A question?

558
00:32:33,970 --> 00:32:35,230
Sort of.

559
00:32:36,170 --> 00:32:38,990
It's almost the same definition as equation.

560
00:32:39,790 --> 00:32:42,790
Or is equation? A mathematical statement that you use questions

561
00:32:42,790 --> 00:32:44,470
or a less than variable?

562
00:32:46,110 --> 00:32:49,550
Okay, so instead of equations, it's going to be an

563
00:32:49,550 --> 00:32:52,190
inequality. We'll write this down in a second.

564
00:32:52,570 --> 00:32:56,490
It's a mathematical statement that one expression

565
00:32:56,490 --> 00:33:00,010
is either less than, greater than, less than or equal

566
00:33:00,010 --> 00:33:01,490
to, greater than or equal to another expression.

567
00:33:01,690 --> 00:33:02,340
Let me write that down.

568
00:33:02,330 --> 00:33:06,330
community and inequality

569
00:33:07,450 --> 00:33:11,970
a mathematical statement that one expression

570
00:33:12,770 --> 00:33:15,270
is less than

571
00:33:17,350 --> 00:33:21,950
or greater than or less than or

572
00:33:21,950 --> 00:33:26,450
equal to or greater than or equal to another

573
00:33:26,450 --> 00:33:37,650
I

574
00:33:37,650 --> 00:33:41,650
could have wrote the words less than or greater than

575
00:33:41,650 --> 00:33:53,150
or

576
00:33:53,150 --> 00:33:57,730
less than or this is it gonna be a number

577
00:33:57,730 --> 00:33:58,910
or an ordered pair

578
00:33:58,910 --> 00:34:03,430
a number

579
00:34:03,430 --> 00:34:07,330
what because only one variable

580
00:34:07,330 --> 00:34:11,250
okay

581
00:34:11,250 --> 00:34:15,530
how many

582
00:34:15,530 --> 00:34:17,510
solutions does this have

583
00:34:17,510 --> 00:34:23,710
okay

584
00:34:23,710 --> 00:34:28,210
is zero a solution

585
00:34:28,210 --> 00:34:32,230
yes is

586
00:34:32,230 --> 00:34:33,750
Is 3 a solution?

587
00:34:34,490 --> 00:34:35,140
Yeah.

588
00:34:36,610 --> 00:34:37,330
Is 4 a solution?

589
00:34:37,730 --> 00:34:40,230
Yes. Is 4.5 a solution?

590
00:34:40,730 --> 00:34:42,570
Yes. Is 5 a solution?

591
00:34:42,850 --> 00:34:45,110
Yes. Yes. Why is 5 a solution?

592
00:34:45,290 --> 00:34:46,090
Because it has a variable 2.

593
00:34:46,870 --> 00:34:51,290
When replaced for the variable, it makes the inequality true.

594
00:34:52,730 --> 00:34:55,230
Because of the or equal to, this is true.

595
00:34:56,370 --> 00:34:58,570
Okay. So it definitely has an infinite number of solutions.

596
00:35:00,470 --> 00:35:03,370
We can write them in something called interval notation.

597
00:35:03,630 --> 00:35:04,870
We'll learn that probably Monday.

598
00:35:06,170 --> 00:35:09,170
Right now we can just graph the solution set.

599
00:35:09,970 --> 00:35:12,490
Since there's only one variable, we need only one number

600
00:35:12,490 --> 00:35:16,490
line. Okay, and you simply put a dot

601
00:35:16,490 --> 00:35:20,310
on every single solution.

602
00:35:21,110 --> 00:35:22,910
You did this last year, correct?

603
00:35:24,650 --> 00:35:28,970
Solved inequalities. You typically graph a

604
00:35:28,970 --> 00:35:33,570
solution set. Let's put negative nine.

605
00:35:34,370 --> 00:35:37,190
What was our first step?

606
00:35:37,990 --> 00:35:42,450
and then divide by 2 do I have to flip

607
00:35:42,450 --> 00:35:43,430
the inequality sign?

608
00:35:44,730 --> 00:35:46,450
No. No. Because

609
00:35:46,450 --> 00:35:55,290
we're

610
00:35:55,290 --> 00:35:55,940
not

611
00:35:56,350 --> 00:35:57,410
No. No.

612
00:35:58,470 --> 00:36:01,650
All three of these are called equivalent inequalities.

613
00:36:03,530 --> 00:36:04,180
Okay.

614
00:36:06,230 --> 00:36:09,310
So we're going to graph the solution set.

615
00:36:10,110 --> 00:36:12,750
We've got to put a dot on every single solution.

616
00:36:14,330 --> 00:36:16,070
Is the number 3 a solution?

617
00:36:16,330 --> 00:36:18,250
Sorry, is the number negative 3 a solution?

618
00:36:18,590 --> 00:36:19,490
No. No.

619
00:36:20,890 --> 00:36:21,540
Okay.

620
00:36:22,970 --> 00:36:25,550
We have to just remove one tiny point.

621
00:36:26,120 --> 00:36:27,420
Okay, it's very hard.

622
00:36:27,560 --> 00:36:29,560
If you pull that one tiny point, you'll never see

623
00:36:29,560 --> 00:36:33,240
that it's gone. so we exaggerated it make a big

624
00:36:33,240 --> 00:36:37,100
open circle even though

625
00:36:37,100 --> 00:36:42,200
okay

626
00:36:42,200 --> 00:36:46,680
again there's gonna be a better way to

627
00:36:46,680 --> 00:36:48,680
do the solution set but for right now we're just

628
00:36:48,680 --> 00:36:52,760
gonna graph the solution set okay

629
00:36:52,760 --> 00:36:56,960
all these definitions are very important what's it

630
00:36:56,960 --> 00:37:00,500
Equation means we could make the solve with a solution

631
00:37:00,500 --> 00:37:04,720
set with a solution Put our equivalent equation

632
00:37:04,720 --> 00:37:06,980
or equation true false and open

633
00:37:08,200 --> 00:37:12,280
We got to memorize these Know them by heart real

634
00:37:12,280 --> 00:37:12,930
quick

635
00:37:13,920 --> 00:37:16,960
We will have a quiz probably in the next week

636
00:37:16,960 --> 00:37:18,200
on the definitions