1
00:00:00,000 --> 00:00:00,410


2
00:00:00,410 --> 00:00:03,669
What I want to do in this video is a handful of fairly

3
00:00:03,669 --> 00:00:05,469
simple inequality videos.

4
00:00:05,469 --> 00:00:07,710
But the real value of it, I think, will be just to get you

5
00:00:07,710 --> 00:00:10,750
warmed up in the notation of inequality.

6
00:00:10,750 --> 00:00:12,390
So let's just start with one.

7
00:00:12,390 --> 00:00:19,589
So we have x minus 5 is less than 35.

8
00:00:19,589 --> 00:00:22,480
So let's see if we can find all of the x's that will

9
00:00:22,480 --> 00:00:23,940
satisfy this equation.

10
00:00:23,940 --> 00:00:26,370
And that's one of the distinctions of an inequality.

11
00:00:26,370 --> 00:00:30,000
In an equation, you typically have one solution, or at least

12
00:00:30,000 --> 00:00:31,219
the ones we've solved so far.

13
00:00:31,219 --> 00:00:33,149
In the future, we'll see equations where they have more

14
00:00:33,149 --> 00:00:34,250
than one solution.

15
00:00:34,250 --> 00:00:36,149
But in the ones we've solved so far, you solved for a

16
00:00:36,149 --> 00:00:37,309
particular x.

17
00:00:37,310 --> 00:00:40,350
In the inequalities, there's a whole set of x's that will

18
00:00:40,350 --> 00:00:41,719
satisfy this inequality.

19
00:00:41,719 --> 00:00:43,929
So they're saying, what are all the x's, that when you

20
00:00:43,929 --> 00:00:48,119
subtract 5 from them, it's going to be less than 35?

21
00:00:48,119 --> 00:00:49,169
And we can already think about it.

22
00:00:49,170 --> 00:00:50,690
I mean 0 minus 5.

23
00:00:50,689 --> 00:00:52,289
That's less than 35.

24
00:00:52,289 --> 00:00:54,310
Minus 100 minus 5.

25
00:00:54,310 --> 00:00:55,750
That's less than 35.

26
00:00:55,750 --> 00:00:57,159
5 minus 5.

27
00:00:57,159 --> 00:00:58,299
That's less than 35.

28
00:00:58,299 --> 00:01:01,750
So there's clearly a lot of x's that will satisfy that.

29
00:01:01,750 --> 00:01:04,609
But what we want to do is come up with a solution that

30
00:01:04,609 --> 00:01:07,060
essentially encompasses all of the x's.

31
00:01:07,060 --> 00:01:09,640
So the way we do that is essentially the same way that

32
00:01:09,640 --> 00:01:10,959
we solved any equations.

33
00:01:10,959 --> 00:01:13,609
We want to get just the x terms, in this case, on the

34
00:01:13,609 --> 00:01:14,719
left-hand side.

35
00:01:14,719 --> 00:01:17,280
So I want to get rid of this negative 5, and I can do that

36
00:01:17,280 --> 00:01:20,400
by adding 5 to both sides of this equation.

37
00:01:20,400 --> 00:01:24,350
So I can add 5 to both sides of this equation.

38
00:01:24,349 --> 00:01:25,929
That won't change the inequality.

39
00:01:25,930 --> 00:01:28,710
It won't change the less than sign.

40
00:01:28,709 --> 00:01:31,469
If something is less than something else, something plus

41
00:01:31,469 --> 00:01:35,719
5 is still going to be less than the other thing plus 5.

42
00:01:35,719 --> 00:01:38,579
So on the left-hand side, we just have an x.

43
00:01:38,579 --> 00:01:41,500
This negative 5 and this positive 5 cancel out.

44
00:01:41,500 --> 00:01:45,730
x is less than 35 plus 5, which is 40.

45
00:01:45,730 --> 00:01:47,189
And that's our solution.

46
00:01:47,189 --> 00:01:50,009
And to just visualize the set of all numbers that

47
00:01:50,010 --> 00:01:52,340
represents, let me draw a number line here.

48
00:01:52,340 --> 00:01:56,760


49
00:01:56,760 --> 00:02:01,280
And I'll do it around-- let's say that's 40,

50
00:02:01,280 --> 00:02:05,250
this is 40, 41, 42.

51
00:02:05,250 --> 00:02:06,730
And then we could go below 40.

52
00:02:06,730 --> 00:02:09,389
39, 38.

53
00:02:09,389 --> 00:02:12,240
You can just keep going below 40.

54
00:02:12,240 --> 00:02:14,700
It just keeps going on in both directions.

55
00:02:14,699 --> 00:02:18,609
And any x that is less than 40 will satisfy this.

56
00:02:18,610 --> 00:02:22,950
So it can't be equal to 40, because if x is equal to 40,

57
00:02:22,949 --> 00:02:25,009
40 minus 5 is 35.

58
00:02:25,009 --> 00:02:27,129
It's not less than 35.

59
00:02:27,129 --> 00:02:28,819
So x has to be less than 40.

60
00:02:28,819 --> 00:02:32,590
And to show this on the number line, we do a circle around 40

61
00:02:32,590 --> 00:02:34,620
to show that we're not including 40.

62
00:02:34,620 --> 00:02:37,020
But then we can shade in everything below 40.

63
00:02:37,020 --> 00:02:41,010
So everything that's just exactly below 40 is included

64
00:02:41,009 --> 00:02:41,769
in our solution set.

65
00:02:41,770 --> 00:02:45,240
So everything I've shaded in yellow is included in our

66
00:02:45,240 --> 00:02:46,040
solution set.

67
00:02:46,039 --> 00:02:52,530
So 39, 39.999999, repeating, which is about as close as you

68
00:02:52,530 --> 00:02:55,240
can get to 40 as possible, that's in our solution set.

69
00:02:55,240 --> 00:02:56,370
But 40 is not.

70
00:02:56,370 --> 00:02:58,629
That's why we put that open circle around it.

71
00:02:58,629 --> 00:03:00,849
Let's do another one.

72
00:03:00,849 --> 00:03:03,094
Let me do it in another color as well.

73
00:03:03,094 --> 00:03:06,859
So let's say we have x-- let me do it over on this top

74
00:03:06,860 --> 00:03:08,360
right corner.

75
00:03:08,360 --> 00:03:16,320
Say we have x plus 15 is greater than or equal to

76
00:03:16,319 --> 00:03:17,569
negative 60.

77
00:03:17,569 --> 00:03:20,219
Notice, now we have greater than or equal.

78
00:03:20,219 --> 00:03:22,109
So let's solve this the same way we solved

79
00:03:22,110 --> 00:03:22,830
that one over there.

80
00:03:22,830 --> 00:03:25,360
We can subtract 15 from both sides.

81
00:03:25,360 --> 00:03:26,800
And I like to switch up my notation.

82
00:03:26,800 --> 00:03:29,680
Here I added the 5, kind of, on the same line.

83
00:03:29,680 --> 00:03:32,460
You could also do your adding or subtracting below

84
00:03:32,460 --> 00:03:33,480
the line, like this.

85
00:03:33,479 --> 00:03:37,129
So if I subtract 15 from both sides, so I do a minus 15

86
00:03:37,129 --> 00:03:40,169
there, and I do a minus 15 there.

87
00:03:40,169 --> 00:03:44,449
Then the left-hand side just becomes an x.

88
00:03:44,449 --> 00:03:46,109
Because obviously you have 15 minus 15.

89
00:03:46,110 --> 00:03:47,420
That just cancels out.

90
00:03:47,419 --> 00:03:51,849
And you get x is greater than or equal to negative 60 minus

91
00:03:51,849 --> 00:03:56,000
15 is negative 75.

92
00:03:56,000 --> 00:03:59,050
If something is greater than or equal to something else, if

93
00:03:59,050 --> 00:04:01,830
I take 15 away from this and from that, the greater than or

94
00:04:01,830 --> 00:04:03,760
equal sign will still apply.

95
00:04:03,759 --> 00:04:06,319
So our solution is x is greater than or equal to

96
00:04:06,319 --> 00:04:07,949
negative 75.

97
00:04:07,949 --> 00:04:09,994
Let's graph it on the number line.

98
00:04:09,995 --> 00:04:13,060
So let me draw a number line here.

99
00:04:13,060 --> 00:04:18,310
I'll have-- let's say that's negative 75, that's negative

100
00:04:18,310 --> 00:04:24,310
74, that's negative 73, that's negative 76.

101
00:04:24,310 --> 00:04:25,449
And so on and so forth.

102
00:04:25,449 --> 00:04:26,870
I could keep plotting things.

103
00:04:26,870 --> 00:04:31,280
Now, x has to be greater than or equal to negative 75.

104
00:04:31,279 --> 00:04:34,589
So x can be equal to negative 75.

105
00:04:34,589 --> 00:04:38,469
So we can include the point, because we have this greater

106
00:04:38,470 --> 00:04:39,840
than or equal sign.

107
00:04:39,839 --> 00:04:41,539
Notice we're not making it hollow like we did there,

108
00:04:41,540 --> 00:04:44,090
we're making it filled in because it can equal negative

109
00:04:44,089 --> 00:04:46,869
75, or it needs to be greater than.

110
00:04:46,870 --> 00:04:48,014
So greater than or equal.

111
00:04:48,014 --> 00:04:52,339
We'll shade in everything above negative 75 as well.

112
00:04:52,339 --> 00:04:55,159
So in orange is the solution set.

113
00:04:55,160 --> 00:04:58,200
And this obviously, we could keep going to the right. x

114
00:04:58,199 --> 00:04:59,969
could be a million, it could be a billion,

115
00:04:59,970 --> 00:05:00,830
it could be a googol.

116
00:05:00,829 --> 00:05:03,219
It can be an arbitrarily large number as long as it's greater

117
00:05:03,220 --> 00:05:06,090
than or equal to negative 75.

118
00:05:06,089 --> 00:05:08,079
Let's do a couple more.

119
00:05:08,079 --> 00:05:17,939
Let's do x minus 2 is less than or equal to 1.

120
00:05:17,939 --> 00:05:19,649
Once again, we want to get just our x on

121
00:05:19,649 --> 00:05:20,439
the left-hand side.

122
00:05:20,439 --> 00:05:21,829
Get rid of this negative 2.

123
00:05:21,829 --> 00:05:24,399
Let's add 2 to both sides of this equation.

124
00:05:24,399 --> 00:05:27,409
Plus 2.

125
00:05:27,410 --> 00:05:29,680
The left-hand side just becomes an x.

126
00:05:29,680 --> 00:05:31,400
You have a less than or equal sign.

127
00:05:31,399 --> 00:05:34,129
That won't change by adding or subtracting the same thing to

128
00:05:34,129 --> 00:05:36,139
both sides of the inequality.

129
00:05:36,139 --> 00:05:37,740
And then 1 plus 2 is 3.

130
00:05:37,740 --> 00:05:40,009
So x needs to be less than or equal to 3.

131
00:05:40,009 --> 00:05:42,959
Any x that is less than or equal to 3 will

132
00:05:42,959 --> 00:05:45,120
satisfy this equation.

133
00:05:45,120 --> 00:05:46,639
So let's plot it.

134
00:05:46,639 --> 00:05:49,180
And I'd try out any x that's less than or equal to 3 and

135
00:05:49,180 --> 00:05:52,160
verify for yourself that it does indeed satisfy this

136
00:05:52,160 --> 00:05:52,740
inequality.

137
00:05:52,740 --> 00:05:53,759
I shouldn't call it an equation.

138
00:05:53,759 --> 00:05:55,789
This inequality.

139
00:05:55,790 --> 00:05:59,520
So let me graph the solution set.

140
00:05:59,519 --> 00:06:05,139
So let's say this is 0, 1, 2, 3, 4.

141
00:06:05,139 --> 00:06:07,829
That's negative 1, negative 2.

142
00:06:07,829 --> 00:06:10,019
So x has to be less than or equal to 3.

143
00:06:10,019 --> 00:06:13,479
It can be equal to 3, so we fill in the dot,

144
00:06:13,480 --> 00:06:14,920
or less than 3.

145
00:06:14,920 --> 00:06:18,730
So the solution set for over here is in pink.

146
00:06:18,730 --> 00:06:20,286
Anything less than or equal to 3.

147
00:06:20,286 --> 00:06:21,639
And verify it for yourself.

148
00:06:21,639 --> 00:06:25,669
If x is equal to 3, you get 3 minus 2, which is equal to 1.

149
00:06:25,670 --> 00:06:29,170
And that is valid because it could be less than or equal.

150
00:06:29,170 --> 00:06:34,949
If you do 2.999999 minus 2, you get 0.999999, which is

151
00:06:34,949 --> 00:06:35,920
less than 1.

152
00:06:35,920 --> 00:06:38,500
And you can keep trying for any of these numbers in this

153
00:06:38,500 --> 00:06:40,279
pink solution set here.

154
00:06:40,279 --> 00:06:43,369
Let's do one more.

155
00:06:43,370 --> 00:06:49,829
Let's say we have x minus 32 is less than or equal to 0.

156
00:06:49,829 --> 00:06:51,159
Same drill as before.

157
00:06:51,160 --> 00:06:53,275
Let's add 32 to both sides of this equation.

158
00:06:53,274 --> 00:06:57,359


159
00:06:57,360 --> 00:07:00,819
The left-hand side just becomes x, and then the

160
00:07:00,819 --> 00:07:04,649
right-hand side is less than or equal to 32.

161
00:07:04,649 --> 00:07:05,799
Pretty straightforward.

162
00:07:05,800 --> 00:07:09,110
Same drill when we graph this equation.

163
00:07:09,110 --> 00:07:11,250
We draw the number line.

164
00:07:11,250 --> 00:07:16,199
If this is 32, this is 33, this is 31.

165
00:07:16,199 --> 00:07:20,139
I could keep adding things above and below 32.

166
00:07:20,139 --> 00:07:23,000
For the solution set is everything less than or equal,

167
00:07:23,000 --> 00:07:26,410
so we can-- it could be equal to 32, or less than.

168
00:07:26,410 --> 00:07:30,580
So we fill in everything below that.

169
00:07:30,579 --> 00:07:32,789
Remember, the reason why we're filling in this solid, the

170
00:07:32,790 --> 00:07:36,960
reason why 32 is an acceptable solution to this original

171
00:07:36,959 --> 00:07:41,089
inequality, is because of this less than or equal sign.

172
00:07:41,089 --> 00:07:43,039
Over here, you didn't have less than or equal, and that's

173
00:07:43,040 --> 00:07:45,890
why 40 wasn't part of the solution set.

174
00:07:45,889 --> 00:07:47,332