1
00:00:00,000 --> 00:00:00,960


2
00:00:00,960 --> 00:00:03,529
Welcome to the presentation on quadratic inequalities.

3
00:00:03,529 --> 00:00:06,449
I know that sounds very complicated, but hopefully

4
00:00:06,450 --> 00:00:09,140
you'll see it's actually not that difficult.

5
00:00:09,140 --> 00:00:10,830
Or at least, maybe the problems we're going to work on

6
00:00:10,830 --> 00:00:12,169
aren't that difficult.

7
00:00:12,169 --> 00:00:15,890
Well, let's get started with some problems and hopefully

8
00:00:15,890 --> 00:00:17,730
you'll see where this is kind of slightly different

9
00:00:17,730 --> 00:00:20,050
than solving regular quadratic equations.

10
00:00:20,050 --> 00:00:27,730
So let's say I had the inequality x squared plus

11
00:00:27,730 --> 00:00:33,480
3x is greater than 10.

12
00:00:33,479 --> 00:00:36,969
And remember, whenever you solve a quadratic or I guess

13
00:00:36,969 --> 00:00:38,640
you would call it a second degree equation-- I guess

14
00:00:38,640 --> 00:00:39,500
this is an inequality.

15
00:00:39,500 --> 00:00:41,460
I shouldn't use the word equation.

16
00:00:41,460 --> 00:00:43,679
It's tempting to sometimes do it the same way you'd do a

17
00:00:43,679 --> 00:00:45,469
linear equation, kind of getting all the x terms

18
00:00:45,469 --> 00:00:47,070
on one side and all the constants on the other.

19
00:00:47,070 --> 00:00:49,950
But it never works because you actually have an x term and

20
00:00:49,950 --> 00:00:51,570
then you have an x squared term.

21
00:00:51,570 --> 00:00:53,950
So you actually want to get it in kind of what I would call

22
00:00:53,950 --> 00:00:55,750
the-- I don't know if it's actually called this-- the

23
00:00:55,750 --> 00:00:58,729
standard form where you actually have all of the terms

24
00:00:58,729 --> 00:01:00,709
on one side and then a 0 on the other side.

25
00:01:00,710 --> 00:01:03,030
And then you can either factor it or use the

26
00:01:03,030 --> 00:01:03,910
quadratic equation.

27
00:01:03,909 --> 00:01:05,200
So let's do that.

28
00:01:05,200 --> 00:01:06,409
Well this is pretty easy.

29
00:01:06,409 --> 00:01:10,890
We just have to subtract 10 from both sides and we get x

30
00:01:10,890 --> 00:01:22,040
squared plus 3x minus 10 is greater than 0.

31
00:01:22,040 --> 00:01:23,920
Now let's see if we can factor it.

32
00:01:23,920 --> 00:01:27,049
Are there two numbers that when you multiply it become negative

33
00:01:27,049 --> 00:01:31,189
10 and when you add it become positive 3?

34
00:01:31,189 --> 00:01:31,840
Well, yeah.

35
00:01:31,840 --> 00:01:33,670
Positive 5 and negative 2.

36
00:01:33,670 --> 00:01:36,510
And once again, at this point I think you already know how to

37
00:01:36,510 --> 00:01:41,450
do factoring, so this should be hopefully, obvious to you.

38
00:01:41,450 --> 00:01:53,060
So it's x plus 5 times x minus 2 is greater than 0.

39
00:01:53,060 --> 00:01:55,840
Now this is the part where it's going to become a little bit

40
00:01:55,840 --> 00:02:00,810
more difficult than just your traditional factoring problem.

41
00:02:00,810 --> 00:02:02,650
We have two numbers, I guess you could view it.

42
00:02:02,650 --> 00:02:03,670
We have x plus 5.

43
00:02:03,670 --> 00:02:04,850
I view that as one number.

44
00:02:04,849 --> 00:02:06,489
Or I guess we have two expressions.

45
00:02:06,489 --> 00:02:10,299
We have x plus 5 and we have x minus 2.

46
00:02:10,300 --> 00:02:12,320
And when we're multiplying them we're getting

47
00:02:12,319 --> 00:02:15,019
something greater than 0.

48
00:02:15,020 --> 00:02:16,909
Now let's think about what happens when you

49
00:02:16,909 --> 00:02:19,460
multiply numbers.

50
00:02:19,460 --> 00:02:21,980
If they're both positive and you multiply them, then

51
00:02:21,979 --> 00:02:23,319
you get a positive number.

52
00:02:23,319 --> 00:02:26,709
And if they're both negative and you multiply them, then you

53
00:02:26,710 --> 00:02:28,610
also get a positive number.

54
00:02:28,610 --> 00:02:32,590
So we know that either both of these expressions are the same

55
00:02:32,590 --> 00:02:35,870
sign, that they're both greater than 0, they're both positive.

56
00:02:35,870 --> 00:02:38,650
Or we know that they're both negative.

57
00:02:38,650 --> 00:02:40,240
And I know this might be a little confusing, but just

58
00:02:40,240 --> 00:02:43,840
think of it as-- if I told you that-- I'll do something

59
00:02:43,840 --> 00:02:44,530
slightly separate out here.

60
00:02:44,530 --> 00:02:50,330
If I told you that a times b is greater than 0 we know that

61
00:02:50,330 --> 00:02:57,469
either a is greater than 0 and b is greater than 0.

62
00:02:57,469 --> 00:02:59,419
Which just means that they're both positive.

63
00:02:59,419 --> 00:03:07,229
Or a is less than 0 and b is less than 0.

64
00:03:07,229 --> 00:03:08,699
Which means that they're both negative.

65
00:03:08,699 --> 00:03:11,199
All we know is that they both have to be the same sign in

66
00:03:11,199 --> 00:03:14,399
order for their product to be greater than 0.

67
00:03:14,400 --> 00:03:16,270
Now we just do the same thing here.

68
00:03:16,270 --> 00:03:21,140
So we know that either both of these are positive, so x plus

69
00:03:21,139 --> 00:03:30,889
5 is greater than 0 and x minus 2 is greater than 0.

70
00:03:30,889 --> 00:03:37,239
Or-- now this is a little confusing, but if you work

71
00:03:37,240 --> 00:03:39,189
through these problems it actually makes a lot of sense.

72
00:03:39,189 --> 00:03:41,750
Or they're both negative.

73
00:03:41,750 --> 00:03:50,169
Or x plus 5 is less than 0 and x minus 2 is less than 0.

74
00:03:50,169 --> 00:03:53,349
I know that's confusing, but just think of it in terms of we

75
00:03:53,349 --> 00:03:55,349
have two expressions: they're either both positive or

76
00:03:55,349 --> 00:03:56,699
they're either both negative.

77
00:03:56,699 --> 00:03:58,109
Because when you multiple them you get something

78
00:03:58,110 --> 00:03:59,450
larger than 0.

79
00:03:59,449 --> 00:04:00,439
Well, let's solve this side.

80
00:04:00,439 --> 00:04:04,879
So this says that x is greater than negative 5

81
00:04:04,879 --> 00:04:11,639
and x is greater than 2.

82
00:04:11,639 --> 00:04:13,559
We just 2 both sides of this equation.

83
00:04:13,560 --> 00:04:16,100


84
00:04:16,100 --> 00:04:22,540
Or, and if we solve this side-- x is less than negative

85
00:04:22,540 --> 00:04:27,819
5 and x is less than 2.

86
00:04:27,819 --> 00:04:30,519
I just solved both of these inequalities right here.

87
00:04:30,519 --> 00:04:33,529
Now we can actually simplify this because here we say that

88
00:04:33,529 --> 00:04:37,859
x is greater than negative 5 and x is greater than 2.

89
00:04:37,860 --> 00:04:42,600
So in order for x ti be greater than negative 5 and for x to be

90
00:04:42,600 --> 00:04:45,450
greater than 2, this just simplifies as saying, well,

91
00:04:45,449 --> 00:04:47,000
x is just greater than 2.

92
00:04:47,000 --> 00:04:48,870
Because if x is greater than 2, it's definitely

93
00:04:48,870 --> 00:04:50,230
greater than negative 5.

94
00:04:50,230 --> 00:04:52,710
So it just simplifies to this.

95
00:04:52,709 --> 00:04:57,370
And we'd say or-- and here we said x is less than negative

96
00:04:57,370 --> 00:05:00,530
5 or x is less than 2.

97
00:05:00,529 --> 00:05:03,329
Well, we know if x is less than negative 5, then x is

98
00:05:03,329 --> 00:05:04,919
definitely less than 2.

99
00:05:04,920 --> 00:05:10,850
So we could just simplify it to or x is less than negative 5.

100
00:05:10,850 --> 00:05:13,710
So the solutions to this problem is x could be greater

101
00:05:13,709 --> 00:05:17,759
than 2 or x could be less than negative 5.

102
00:05:17,759 --> 00:05:19,500
And so let's just think about how that looks

103
00:05:19,500 --> 00:05:20,509
on the number line.

104
00:05:20,509 --> 00:05:24,454


105
00:05:24,454 --> 00:05:28,079
So if 2 is here, x could be greater than 2.

106
00:05:28,079 --> 00:05:29,189
So it's all of these numbers.

107
00:05:29,189 --> 00:05:32,469


108
00:05:32,470 --> 00:05:34,585
And if this is negative 5-- I shouldn't have done it

109
00:05:34,584 --> 00:05:37,250
so close to the bottom.

110
00:05:37,250 --> 00:05:38,610
x is less the negative 5.

111
00:05:38,610 --> 00:05:42,250
So these are the numbers that satisfy this equation.

112
00:05:42,250 --> 00:05:44,459
And I'll leave it up to you to try out to see that

113
00:05:44,459 --> 00:05:45,919
they actually work.

114
00:05:45,920 --> 00:05:48,370
Let's try another one and hopefully, I can

115
00:05:48,370 --> 00:05:49,519
confuse you even more.

116
00:05:49,519 --> 00:05:52,829


117
00:05:52,829 --> 00:06:07,069
Let's say I have minus x times 2x minus 14 is greater

118
00:06:07,069 --> 00:06:09,409
than or equal to 24.

119
00:06:09,410 --> 00:06:11,580
Well, the first thing we want to do is just manipulate this

120
00:06:11,579 --> 00:06:13,430
so it looks in the standard form.

121
00:06:13,430 --> 00:06:21,980
So we get negative 2x squared plus 14x-- I'm just

122
00:06:21,980 --> 00:06:27,050
distributing the minus x-- is greater than or equal to 24.

123
00:06:27,050 --> 00:06:29,895
I don't like any coefficient it front of my x squared term,

124
00:06:29,894 --> 00:06:32,909
so let's divide both sides of this equation by negative 2.

125
00:06:32,910 --> 00:06:36,220
So we get x squared-- we divided by negative

126
00:06:36,220 --> 00:06:39,440
2-- minus 7x.

127
00:06:39,439 --> 00:06:42,629
And remember, when you divide by a negative number you switch

128
00:06:42,629 --> 00:06:45,129
the sign on the inequality, or you switch the direction

129
00:06:45,129 --> 00:06:46,170
of the inequality.

130
00:06:46,170 --> 00:06:48,800
So we're dividing by negative 2, so we switched it.

131
00:06:48,800 --> 00:06:50,245
We went from greater than or equal to, to less

132
00:06:50,245 --> 00:06:51,120
than or equal to.

133
00:06:51,120 --> 00:06:55,780
And then 24 divided by negative 2 is minus 12.

134
00:06:55,779 --> 00:06:58,250
And now we can just bring this minus 12 onto the left-hand

135
00:06:58,250 --> 00:06:58,850
side of the equation.

136
00:06:58,850 --> 00:06:59,920
Add 12 to both sides.

137
00:06:59,920 --> 00:07:09,560
We get x squared minus 7x plus 12 is less than or equal to 0.

138
00:07:09,560 --> 00:07:12,670
And then we can just factor that and we get, what is that?

139
00:07:12,670 --> 00:07:21,680
It's x minus 3 times x minus 4 is less than or equal to 0.

140
00:07:21,680 --> 00:07:24,230
So now we know that when we multiply these two terms

141
00:07:24,230 --> 00:07:26,150
we get a negative number.

142
00:07:26,149 --> 00:07:29,349
So that means that these expressions have to be

143
00:07:29,350 --> 00:07:31,780
of different signs.

144
00:07:31,779 --> 00:07:32,369
Does that make sense?

145
00:07:32,370 --> 00:07:34,030
If I tell you I have two number and I multiply them,

146
00:07:34,029 --> 00:07:34,829
I get a negative number.

147
00:07:34,829 --> 00:07:37,029
You know that they have to be of different signs.

148
00:07:37,029 --> 00:07:45,039
So we know that either x minus 3 is less than or equal to 0

149
00:07:45,040 --> 00:07:50,379
and x minus 4 is greater than or equal to 0.

150
00:07:50,379 --> 00:07:51,750
So that's one case.

151
00:07:51,750 --> 00:07:57,779
And the other case is x minus 3 is greater than or equal to 0,

152
00:07:57,779 --> 00:08:00,039
which means x minus 3 is positive.

153
00:08:00,040 --> 00:08:04,689
And x minus 4 is less than or equal to 0--

154
00:08:04,689 --> 00:08:05,740
oh, I went to the edge.

155
00:08:05,740 --> 00:08:07,900
So let's solve this and hopefully it'll simplify more.

156
00:08:07,899 --> 00:08:12,039
So this just says that x is less than or equal to 3.

157
00:08:12,040 --> 00:08:18,340
And this says x is greater than or equal to 4.

158
00:08:18,339 --> 00:08:20,914
So both of these things have to be true. x has to be less than

159
00:08:20,915 --> 00:08:26,030
or equal to 3 and x has to be greater than or equal to 4.

160
00:08:26,029 --> 00:08:27,349
Well, let me ask you a question.

161
00:08:27,350 --> 00:08:32,389
Can something be both less than or equal to 3 and greater

162
00:08:32,389 --> 00:08:33,490
than or equal to 4?

163
00:08:33,490 --> 00:08:34,110
Well, no.

164
00:08:34,110 --> 00:08:37,519
So we know that this situation can't happen.

165
00:08:37,519 --> 00:08:40,149
There's no numbers that's less than or equal to 3 and

166
00:08:40,149 --> 00:08:41,559
greater than or equal to 4.

167
00:08:41,559 --> 00:08:42,699
So let's look at this situation.

168
00:08:42,700 --> 00:08:48,580
This says x is greater than or equal to 3 and x is

169
00:08:48,580 --> 00:08:50,770
less than or equal to 4.

170
00:08:50,769 --> 00:08:51,460
Can this happen?

171
00:08:51,460 --> 00:08:51,870
Sure.

172
00:08:51,870 --> 00:08:54,929
That just means that x is some number between 3 and 4.

173
00:08:54,929 --> 00:08:59,379
If we were to draw this on the number line, we would get--

174
00:08:59,379 --> 00:09:03,689
if this is 3, this is 4.

175
00:09:03,690 --> 00:09:06,750
And it's greater than or equal to so we fill it in.

176
00:09:06,750 --> 00:09:09,250
And less than or equal to so we'd fill it in.

177
00:09:09,250 --> 00:09:11,580
And it would be any number between 3 and 4 would

178
00:09:11,580 --> 00:09:12,485
satisfy this equation.

179
00:09:12,485 --> 00:09:14,730
And I'll leave it up to you to try it out.

180
00:09:14,730 --> 00:09:17,240
I know this is confusing at first, and this is actually

181
00:09:17,240 --> 00:09:19,159
something that they normally don't teach really well, I

182
00:09:19,159 --> 00:09:21,419
think, in most high schools until 10th or 11th grade.

183
00:09:21,419 --> 00:09:25,509
But just think about you're multiplying two expressions.

184
00:09:25,509 --> 00:09:27,629
If the answer is negative then they must be

185
00:09:27,629 --> 00:09:28,389
of different signs.

186
00:09:28,389 --> 00:09:31,360
If the answer is positive they must be the same sign.

187
00:09:31,360 --> 00:09:33,019
And then you just work through the logic.

188
00:09:33,019 --> 00:09:35,699
And you say, well, no number can be less than 3 and greater

189
00:09:35,700 --> 00:09:37,900
than 4, so this doesn't apply.

190
00:09:37,899 --> 00:09:39,490
And then you do this side and you're like, oh, this

191
00:09:39,490 --> 00:09:40,409
situation does work.

192
00:09:40,409 --> 00:09:42,579
It's any number between 3 and 4.

193
00:09:42,580 --> 00:09:44,090
Hopefully that gives you a sense of how to do

194
00:09:44,090 --> 00:09:45,110
these type of problems.

195
00:09:45,110 --> 00:09:47,000
I'll let you do the exercises now.

196
00:09:47,000 --> 00:09:48,539
Have fun.

197
00:09:48,539 --> 00:09:48,642