1
00:00:00,990 --> 00:00:04,056
The expression 5X

2
00:00:04,056 --> 00:00:10,970
minus 4. Greater than
two X plus 3 looks like an

3
00:00:10,970 --> 00:00:15,670
equation, but with the equal
sign replaced by an Arrowhead.

4
00:00:16,990 --> 00:00:19,138
This denotes that the.

5
00:00:19,720 --> 00:00:24,760
Part on the left, 5X minus four
is greater than the part on the

6
00:00:24,760 --> 00:00:26,200
right 2X plus 3.

7
00:00:27,440 --> 00:00:29,911
We use four symbols to denote in

8
00:00:29,911 --> 00:00:34,358
Equalities. This symbol
means is greater than.

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00:00:36,840 --> 00:00:41,043
This symbol means is greater
than or equal to.

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00:00:42,140 --> 00:00:44,966
This symbol means is less than.

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00:00:45,890 --> 00:00:50,180
On this symbol means is less
than or equal to.

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00:00:51,380 --> 00:00:55,516
Notice that the Arrowhead
always points to the

13
00:00:55,516 --> 00:00:56,550
smaller expression.

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In Equalities can be
manipulated like equations

15
00:01:01,580 --> 00:01:03,580
and follow very similar
rules.

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00:01:04,940 --> 00:01:06,998
But there is one
important exception.

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00:01:08,790 --> 00:01:13,925
If you add the same number to
both sides of an inequality, the

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00:01:13,925 --> 00:01:17,875
inequality remains true. If you
subtract the same number from

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00:01:17,875 --> 00:01:22,615
both sides of the inequality, it
remains true. If you multiply or

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divide both sides of an
inequality by the same positive

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00:01:26,565 --> 00:01:28,145
number, it remains true.

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But if you multiply or divide
both sides of an inequality by a

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00:01:33,955 --> 00:01:36,000
negative number. It's no longer

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00:01:36,000 --> 00:01:40,217
true. In fact, the inequality
becomes reversed. This is quite

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easy to see because we can write
that four is greater than two.

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But if we multiply both sides of
this inequality by minus one, we

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get minus 4.

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Is less than minus 2?

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We have to reverse the
inequality.

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This leads to difficulties
when dealing with variables

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00:02:05,336 --> 00:02:11,752
because of variable can
be either positive or

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00:02:11,752 --> 00:02:14,704
negative. Look at these two

33
00:02:14,704 --> 00:02:17,244
inequalities. X is greater than

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00:02:17,244 --> 00:02:19,710
one. And X squared.

35
00:02:20,230 --> 00:02:21,650
Is greater than X.

36
00:02:23,380 --> 00:02:27,566
Now clearly if X squared is
greater than ex, ex can't be 0.

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So it looks as if we ought to be
able to divide both sides of

38
00:02:31,880 --> 00:02:34,230
this inequality by X. Giving us.

39
00:02:34,740 --> 00:02:38,088
X greater than one, which is
what we've got on the left.

40
00:02:39,750 --> 00:02:43,134
But in fact we can't do this.
These two inequalities are not

41
00:02:43,134 --> 00:02:47,100
the same. This is because X
can be negative.

42
00:02:48,440 --> 00:02:53,055
Here we're saying that X is
greater than one, so X must be

43
00:02:53,055 --> 00:02:56,280
positive. But here we have to
take into account the

44
00:02:56,280 --> 00:02:57,580
possibility that X is negative.

45
00:02:58,180 --> 00:03:04,600
In fact, the complete solution
for this is X is greater than

46
00:03:04,600 --> 00:03:07,810
one or X less than 0.

47
00:03:08,420 --> 00:03:11,450
Because obviously if X is
negative, then X squared is

48
00:03:11,450 --> 00:03:15,389
always going to be greater than
X. I'll show you exactly how to

49
00:03:15,389 --> 00:03:18,419
get the solution for this type
of inequality later on.

50
00:03:20,930 --> 00:03:23,930
Great care really has to be
taken when solving inequalities

51
00:03:23,930 --> 00:03:27,530
to make sure that you don't
multiply or divide by a negative

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00:03:27,530 --> 00:03:33,157
number by accident. For example,
saying that X is greater than Y.

53
00:03:34,140 --> 00:03:40,954
Implies. That X
squared is greater than Y

54
00:03:40,954 --> 00:03:44,064
squared only if X&Y are

55
00:03:44,064 --> 00:03:51,126
positive. I'll start
with a very simple

56
00:03:51,126 --> 00:03:57,958
inequality. X +3 is
greater than two.

57
00:03:59,040 --> 00:04:03,048
To solve this, we simply need
to subtract 3 from both sides.

58
00:04:03,048 --> 00:04:07,724
If we subtract 3 from the left
hand side were left with X. If

59
00:04:07,724 --> 00:04:11,732
we subtract 3 from the right
hand side were left with minus

60
00:04:11,732 --> 00:04:14,738
one and that is the solution
to the inequality.

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00:04:15,930 --> 00:04:19,269
In Equalities can be represented
on the number line.

62
00:04:21,320 --> 00:04:25,340
Here are solution is X is
greater than minus one.

63
00:04:26,240 --> 00:04:28,478
So we start at minus one.

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00:04:30,280 --> 00:04:32,667
And this line shows the range of

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00:04:32,667 --> 00:04:35,108
values. The decks can take.

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I'm going to put an open circle
there. That open circle denotes

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that although the line goes to

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00:04:42,078 --> 00:04:46,637
minus one. X cannot actually
equal minus. 1X has to be

69
00:04:46,637 --> 00:04:47,969
greater than minus one.

70
00:04:49,200 --> 00:04:55,404
Let's have a
look at another

71
00:04:55,404 --> 00:04:56,438
one.

72
00:04:58,440 --> 00:05:01,428
4X plus 6.

73
00:05:02,060 --> 00:05:05,798
Is greater than 3X plus 7.

74
00:05:07,210 --> 00:05:12,310
First of all, I'm going to
subtract 6 from both sides, so

75
00:05:12,310 --> 00:05:16,985
we get 4X on the left, greater
than 3X plus one.

76
00:05:17,920 --> 00:05:22,535
And now I'm going to subtract 3
X from both sides, which gives

77
00:05:22,535 --> 00:05:24,310
me X greater than one.

78
00:05:25,080 --> 00:05:29,040
And again, I can represent this
on the number line.

79
00:05:29,780 --> 00:05:31,810
X has to be greater than one.

80
00:05:33,680 --> 00:05:35,400
But X cannot equal 1.

81
00:05:36,990 --> 00:05:43,770
Another example is 3X minus
five is less than or

82
00:05:43,770 --> 00:05:47,160
equal to 3 minus X.

83
00:05:48,860 --> 00:05:54,230
This time I need to add 5 to
both sides which gives me 3X is

84
00:05:54,230 --> 00:05:56,020
less than or equal to.

85
00:05:56,530 --> 00:05:58,798
8 minus X.

86
00:05:59,440 --> 00:06:04,003
And then I need to add extra
both sides, which gives me 4X

87
00:06:04,003 --> 00:06:06,109
less than or equal to 8.

88
00:06:06,890 --> 00:06:11,944
Finally, I can divide both sides
by two, which gives me X is less

89
00:06:11,944 --> 00:06:13,749
than or equal to two.

90
00:06:14,980 --> 00:06:16,160
And on the number line.

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00:06:18,830 --> 00:06:22,535
X is less than or equal to two,
so we go this way.

92
00:06:23,290 --> 00:06:26,194
And this time I'm going to do a

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00:06:26,194 --> 00:06:30,355
closed circle. This denotes that
X can be equal to two.

94
00:06:33,130 --> 00:06:40,291
Now I'd like to look at
the inequality minus 2X is

95
00:06:40,291 --> 00:06:42,244
greater than 4.

96
00:06:43,260 --> 00:06:46,640
In order to solve this
inequality, we're going to have

97
00:06:46,640 --> 00:06:49,006
to divide both sides by minus 2.

98
00:06:51,930 --> 00:06:56,742
So we get minus two X divided by
minus two is X.

99
00:06:58,060 --> 00:07:02,092
I've got to remember because I'm
dividing by a negative number to

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00:07:02,092 --> 00:07:03,100
reverse the inequality.

101
00:07:04,140 --> 00:07:09,012
And four divided by minus
two is minus 2, so I get

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00:07:09,012 --> 00:07:11,448
X is less than minus 2.

103
00:07:14,390 --> 00:07:17,330
There's often more than one way
to solve an inequality.

104
00:07:18,550 --> 00:07:21,542
And I can just solve this
one again by using a

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00:07:21,542 --> 00:07:24,534
different method, so we have
-2 X is greater than 4.

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00:07:25,890 --> 00:07:28,786
If we add 2X to both sides we

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get. Zero is greater than
4 + 2 X.

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And then if we subtract 4 from
both sides, we get minus four is

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00:07:42,594 --> 00:07:44,278
greater than two X.

110
00:07:44,900 --> 00:07:50,504
And we can divide through by two
again getting minus two is

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00:07:50,504 --> 00:07:51,905
greater than X.

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00:07:52,450 --> 00:07:57,364
And saying that X is less than
minus two is the same thing as

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saying minus two is greater than
X, so we've solved this

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inequality by do different
methods. The second one avoids

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dividing by a negative number.

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In Equalities often appear in
conjunction with the modulus

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symbol. For instance.

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We say MoD X is less than two.

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The modular symbol denotes that
we have to take the absolute

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00:08:27,407 --> 00:08:31,788
value of X regardless of sign.
This is just the magnitude of X.

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And it is always
positive. So for

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00:08:36,564 --> 00:08:39,658
instance, MoD 2 is
equal to 2.

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And MoD minus two is
also equal to two.

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If the absolute value of X is
less than two, then X must lie

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between 2:00 and minus two. We
write minus two is less than X,

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is less than two.

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00:09:01,260 --> 00:09:05,100
We can show this on the
number line.

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X has to lie between minus two
and two, but it can't be too

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itself. This shows the range
of values that ex can take.

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00:09:25,320 --> 00:09:31,118
If MoD X is greater than or
equal to five, we have the

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absolute value of X must be
greater than or equal to five,

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which means that X is going to
itself is going to be greater

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than or equal to five or less
than or equal to minus five. We

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write X less than or equal to
minus five or X greater than or

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00:09:54,756 --> 00:09:56,094
equal to 5.

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00:09:56,270 --> 00:09:57,710
And on the number line.

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X can take the value 5, so we do
a closed circle.

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00:10:04,900 --> 00:10:08,004
And it can take the
value minus 5.

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00:10:10,210 --> 00:10:15,871
Now I want to look at
another slightly more

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complicated modulus one.

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00:10:18,890 --> 00:10:21,620
We have MoD X minus 4.

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00:10:22,830 --> 00:10:24,498
Less than three.

143
00:10:25,390 --> 00:10:30,329
The modulus sign shows that
the absolute value of X minus

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four is less than three. This
means that X minus four must

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00:10:35,717 --> 00:10:40,207
lie between minus three and
three, so we write minus

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00:10:40,207 --> 00:10:44,248
three less than X minus four
less than three.

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00:10:45,910 --> 00:10:50,914
This is what we call a double
inequality of women's treated as

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00:10:50,914 --> 00:10:55,918
two separate inequalities. So on
the left we have minus three is

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00:10:55,918 --> 00:10:58,003
less than X minus 4.

150
00:11:00,220 --> 00:11:06,955
By adding four to both sides, we
get one is less than X. On the

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00:11:06,955 --> 00:11:11,445
right we have X minus four is
less than three.

152
00:11:12,110 --> 00:11:17,090
And again we had four to both
sides to get. X is less than 7.

153
00:11:17,750 --> 00:11:21,610
So the solution to this
particular inequality is X is

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greater than One X is less
than Seven. We write 1 less

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00:11:26,242 --> 00:11:30,874
than X less than Seven, and
again I'll show you that on

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00:11:30,874 --> 00:11:32,032
the number line.

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00:11:34,510 --> 00:11:38,481
X lies between one and Seven,
but it can't be either.

158
00:11:42,950 --> 00:11:49,229
Now let's solve
MoD. 5X. Minus 8

159
00:11:49,229 --> 00:11:55,508
is less than or
equal to 12.

160
00:11:58,000 --> 00:12:02,140
We're saying here that the
absolute value of 5X minus 8 is

161
00:12:02,140 --> 00:12:04,210
less than or equal to 12.

162
00:12:05,080 --> 00:12:07,268
So 5X minus 8.

163
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Must be less than 12.

164
00:12:10,850 --> 00:12:13,020
Or greater than minus 12.

165
00:12:13,810 --> 00:12:20,609
We write minus 12 is less than
or equal to 5X minus 8.

166
00:12:21,260 --> 00:12:23,710
Is less than or equal to 12?

167
00:12:25,030 --> 00:12:30,200
Again, we have a double
inequality on the left, we have

168
00:12:30,200 --> 00:12:35,370
minus 12 is less than or equal
to 5X minus 8.

169
00:12:36,480 --> 00:12:42,178
We add it to both sides, which
gives us minus four is less than

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00:12:42,178 --> 00:12:43,806
or equal to 5X.

171
00:12:44,960 --> 00:12:48,970
And then we divide both
sides by 5, which gives

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00:12:48,970 --> 00:12:53,381
us minus four fifths is
less than or equal to X.

173
00:12:54,460 --> 00:12:58,708
On the right we have the
inequality 5X minus 8 is less

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00:12:58,708 --> 00:13:00,478
than or equal to 12.

175
00:13:01,480 --> 00:13:06,628
So we write 5X minus 8 less than
or equal to 12.

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00:13:07,360 --> 00:13:12,261
We had eight to both sides,
which gives us 5X is less than

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00:13:12,261 --> 00:13:13,769
or equal to 20.

178
00:13:14,510 --> 00:13:18,374
And we divide both sides
by 5, which gives us X is

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00:13:18,374 --> 00:13:20,306
less than or equal to 4.

180
00:13:22,070 --> 00:13:28,685
So our final answer is minus 4
over 5 is less than or equal to

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00:13:28,685 --> 00:13:32,240
X. Which in turn is less
than or equal to 4.

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00:13:33,440 --> 00:13:35,834
And we can show this
on the number line.

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00:13:37,190 --> 00:13:40,010
Minus four fifths is about here.

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00:13:40,930 --> 00:13:42,460
Let me go through to four.

185
00:13:43,160 --> 00:13:45,176
And because it's less than or

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00:13:45,176 --> 00:13:48,860
equal to. We use
a closed circle.

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00:13:50,700 --> 00:13:54,678
In Equalities can be solved
very easily using graphs,

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00:13:54,678 --> 00:13:59,540
and if you're in any way
unsure about the algebra it

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00:13:59,540 --> 00:14:05,728
can could be a good idea to
do a graph to check. Let me

190
00:14:05,728 --> 00:14:07,938
show you how this works.

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00:14:09,700 --> 00:14:15,365
We take the inequality 2X, plus
three is less than 0.

192
00:14:16,040 --> 00:14:18,992
Now this inequality can be
solved very easily doing

193
00:14:18,992 --> 00:14:20,960
algebra, but it makes a good

194
00:14:20,960 --> 00:14:27,313
example. The first thing that we
need to do is to draw the graph

195
00:14:27,313 --> 00:14:29,719
of Y equals 2X plus 3.

196
00:14:32,180 --> 00:14:33,638
And I've got this graph here.

197
00:14:34,200 --> 00:14:39,735
Note that it's the equation of
a straight line.

198
00:14:40,440 --> 00:14:43,820
It has a slope of two
and then intercept on

199
00:14:43,820 --> 00:14:45,510
the Y axis of three.

200
00:14:47,450 --> 00:14:51,278
On the X axis.

201
00:14:52,460 --> 00:14:56,308
Why is equal to 0 so that
where the line cuts the X

202
00:14:56,308 --> 00:14:58,084
axis Y is equal to 0?

203
00:14:59,280 --> 00:15:01,632
Above the X axis Y is greater

204
00:15:01,632 --> 00:15:06,390
than 0. And below the X axis Y
is less than 0.

205
00:15:08,260 --> 00:15:11,978
So when we say that we want 2X
plus three less than 0.

206
00:15:13,420 --> 00:15:17,203
On this graph, that means why is
less than zero, so we're looking

207
00:15:17,203 --> 00:15:20,404
for the points where the line is
below the X axis.

208
00:15:21,090 --> 00:15:25,682
In other words, where X is less
than minus one and a half, and

209
00:15:25,682 --> 00:15:27,650
this is the solution to the

210
00:15:27,650 --> 00:15:35,240
inequality. And we can mark
this on the graph using the

211
00:15:35,240 --> 00:15:39,128
X axis as the number line.

212
00:15:39,850 --> 00:15:46,330
This technique can also be
used with modulus inequalities

213
00:15:46,330 --> 00:15:52,810
and here using a graph
can be very helpful.

214
00:15:53,750 --> 00:15:56,440
Take for example the inequality.

215
00:15:57,010 --> 00:16:00,690
MoD X minus two is less than 0.

216
00:16:01,820 --> 00:16:08,148
Again, we need to plot the graph
of Y equals MoD X minus 2.

217
00:16:08,720 --> 00:16:14,924
This is the graph of Y equals
MoD X minus 2.

218
00:16:15,750 --> 00:16:18,236
For those of you who are not
familiar with modulus functions,

219
00:16:18,236 --> 00:16:19,592
it might look a little bit

220
00:16:19,592 --> 00:16:24,438
strange. On the right we have
part of the graph of Y equals X

221
00:16:24,438 --> 00:16:29,602
minus 2. And on the left,
where X is less than zero, we

222
00:16:29,602 --> 00:16:33,706
have part of the graph of Y
equals minus X minus two.

223
00:16:33,706 --> 00:16:37,126
This is because the modulus
function changes the sign of

224
00:16:37,126 --> 00:16:38,836
X when X is negative.

225
00:16:40,660 --> 00:16:45,580
Again, we're looking for MoD X.
Minus two is less than 0.

226
00:16:46,760 --> 00:16:52,122
So we want the places where Y is
less than zero, which is between

227
00:16:52,122 --> 00:16:57,101
X equals minus two and X equals
+2, and again this is the

228
00:16:57,101 --> 00:16:58,633
solution to our problem.

229
00:16:59,460 --> 00:17:05,213
So we say minus two less than
X less than two.

230
00:17:05,920 --> 00:17:10,526
Again, we can mark this on the
graph using the X axis as the

231
00:17:10,526 --> 00:17:15,290
number line. Quadratic
inequalities need

232
00:17:15,290 --> 00:17:22,130
handling with care.
Let's solve X

233
00:17:22,130 --> 00:17:28,970
squared minus three
X +2 is

234
00:17:28,970 --> 00:17:32,390
greater than 0.

235
00:17:35,610 --> 00:17:38,734
Note that all the terms are on
the left hand side.

236
00:17:39,240 --> 00:17:42,867
And on the right hand side we
just had zero, exactly as with

237
00:17:42,867 --> 00:17:43,983
the quadratic equation before

238
00:17:43,983 --> 00:17:47,654
you solve it. This expression

239
00:17:47,654 --> 00:17:53,746
factorizes too. X minus
two X minus one.

240
00:17:54,530 --> 00:17:58,310
Now this is a quadratic
equation. We would simply say

241
00:17:58,310 --> 00:18:02,468
right X equals 2 or X equals 1
and that's it.

242
00:18:03,250 --> 00:18:04,682
But we've got a bit more work to

243
00:18:04,682 --> 00:18:10,120
do here. Weather this expression
is greater than zero is going to

244
00:18:10,120 --> 00:18:15,450
depend on the sign of each of
these two factors. We sort this

245
00:18:15,450 --> 00:18:17,500
out by using a grid.

246
00:18:18,240 --> 00:18:24,744
The points
that were

247
00:18:24,744 --> 00:18:31,370
checks equals.
X minus 2 equals 0 and X minus

248
00:18:31,370 --> 00:18:35,390
one equals 0 and marked in, so
this is one and two.

249
00:18:36,170 --> 00:18:39,579
We put the two factors on the

250
00:18:39,579 --> 00:18:42,698
left. And their product.

251
00:18:43,280 --> 00:18:47,000
Now.

252
00:18:48,210 --> 00:18:53,700
When X is less than one, both X
minus one and X minus two are

253
00:18:53,700 --> 00:18:55,164
going to be negative.

254
00:18:56,580 --> 00:18:59,950
So when you multiply them
together, their product is going

255
00:18:59,950 --> 00:19:00,961
to be positive.

256
00:19:03,390 --> 00:19:05,525
When X is greater than one but

257
00:19:05,525 --> 00:19:09,688
less than two. X minus one is
going to be positive.

258
00:19:10,600 --> 00:19:13,096
But X minus two is going to be

259
00:19:13,096 --> 00:19:15,350
negative. So when you multiply

260
00:19:15,350 --> 00:19:17,386
them together. The product will

261
00:19:17,386 --> 00:19:23,420
be negative. Finally, when X is
greater than two, both X minus

262
00:19:23,420 --> 00:19:26,556
one and X minus two will be

263
00:19:26,556 --> 00:19:30,282
positive. And if you multiply
them together, their product

264
00:19:30,282 --> 00:19:31,578
will also be positive.

265
00:19:34,070 --> 00:19:35,798
We are looking for.

266
00:19:36,300 --> 00:19:39,900
X minus two times X minus one to
be greater than 0.

267
00:19:40,890 --> 00:19:42,620
This occurs when it's positive.

268
00:19:43,500 --> 00:19:47,140
And our grid shows that this
happens when X is less than one.

269
00:19:47,640 --> 00:19:49,866
Or when X is greater than two?

270
00:19:50,450 --> 00:19:52,418
So we write in our answer.

271
00:19:53,660 --> 00:20:00,849
Which is X is less than one
or X is greater than two.

272
00:20:03,950 --> 00:20:06,590
And on the number line.

273
00:20:07,210 --> 00:20:09,388
X must be less than one.

274
00:20:09,980 --> 00:20:12,536
So I put a circle to show
that it can't be 1.

275
00:20:14,280 --> 00:20:16,520
And X can also be greater
than two.

276
00:20:20,050 --> 00:20:23,976
Here's another

277
00:20:23,976 --> 00:20:30,116
quadratic. Minus two
X squared plus 5X

278
00:20:30,116 --> 00:20:35,480
plus 12 is greater
than or equal to 0.

279
00:20:36,570 --> 00:20:40,674
I don't like having a negative
coefficient of X squared, so I'm

280
00:20:40,674 --> 00:20:44,094
going to multiply this whole
thing through by minus one,

281
00:20:44,094 --> 00:20:47,514
remembering to change the
direction of the inequality as I

282
00:20:47,514 --> 00:20:48,882
do. This gives us.

283
00:20:49,410 --> 00:20:57,278
Two X squared minus 5X minus 12
is less than or equal to 0.

284
00:20:58,680 --> 00:21:04,906
This expression factorizes to 2X
plus three times X minus four,

285
00:21:04,906 --> 00:21:08,868
so that is less than or equal

286
00:21:08,868 --> 00:21:12,955
to 0. Again, I'm going to
do a grid.

287
00:21:18,150 --> 00:21:25,590
This factor is zero
when X is minus

288
00:21:25,590 --> 00:21:28,380
three over 2.

289
00:21:29,450 --> 00:21:31,858
This fact is zero when X is 4.

290
00:21:32,770 --> 00:21:35,698
We write in the two factors.

291
00:21:36,380 --> 00:21:39,938
And we right in the product.

292
00:21:43,460 --> 00:21:50,530
When X is less than minus three
over 2, both 2X plus three and

293
00:21:50,530 --> 00:21:53,055
X minus four and negative.

294
00:21:53,860 --> 00:21:56,110
So their product is positive.

295
00:21:57,580 --> 00:22:01,350
When X lies between minus three
over two and four.

296
00:22:02,540 --> 00:22:04,670
2X plus three is positive.

297
00:22:05,410 --> 00:22:09,590
But X minus four is still
negative, so their product

298
00:22:09,590 --> 00:22:10,426
is negative.

299
00:22:11,480 --> 00:22:16,797
When X is greater than four,
both 2X plus three and X minus

300
00:22:16,797 --> 00:22:18,024
four are positive.

301
00:22:18,590 --> 00:22:20,180
So their product is positive.

302
00:22:20,780 --> 00:22:26,576
We are looking for 2X plus three
times X minus four to be less

303
00:22:26,576 --> 00:22:28,646
than or equal to 0.

304
00:22:29,330 --> 00:22:33,110
In other words, this expression
has to be either 0 or negative.

305
00:22:34,300 --> 00:22:35,270
This occurs.

306
00:22:36,520 --> 00:22:41,824
When X lies between minus three
over two and four, and it can

307
00:22:41,824 --> 00:22:47,128
equal either number. So we have
minus three over 2 is less than

308
00:22:47,128 --> 00:22:51,616
or equal to X is less than or
equal to 4.

309
00:22:53,890 --> 00:22:56,370
And on the number line.

310
00:22:58,220 --> 00:23:00,326
Minus three over 2 is here.

311
00:23:01,760 --> 00:23:05,738
Four is here.

312
00:23:08,840 --> 00:23:12,040
And I've done filled
circles because we have

313
00:23:12,040 --> 00:23:14,040
less than or equal to.

314
00:23:17,260 --> 00:23:22,783
Quadratic inequalities can
also be solved graphically.

315
00:23:22,783 --> 00:23:30,673
Let's solve X squared minus
three X +2 is greater

316
00:23:30,673 --> 00:23:32,251
than 0.

317
00:23:34,130 --> 00:23:38,710
As with the linear equalities
inequalities, we have to plot

318
00:23:38,710 --> 00:23:43,748
the graph of Y equals X squared
minus three X +2.

319
00:23:44,650 --> 00:23:51,527
This factorizes to give Y equals
X minus one times X minus 2.

320
00:23:52,800 --> 00:23:54,600
The graph looks like this.

321
00:23:55,960 --> 00:24:01,174
Because it's a quadratic, it's a
parabola. Are U shaped curve?

322
00:24:02,210 --> 00:24:04,275
And it crosses the X axis where

323
00:24:04,275 --> 00:24:08,729
X equals 1. Because of the
factor X minus one and where

324
00:24:08,729 --> 00:24:12,139
X equals 2 because of the
factor X minus 2.

325
00:24:13,490 --> 00:24:18,963
Now we're looking for X squared
minus three X +2 to be greater

326
00:24:18,963 --> 00:24:23,662
than 0. This is where Y
is greater than zero. In

327
00:24:23,662 --> 00:24:27,042
other words, the part of
the graph that is above

328
00:24:27,042 --> 00:24:31,098
the X axis, which are the
two arms of the you here.

329
00:24:32,710 --> 00:24:36,021
This occurs where X is less than

330
00:24:36,021 --> 00:24:41,218
one. And where X is greater
than two, so we can write

331
00:24:41,218 --> 00:24:43,058
that in as our solution.

332
00:24:46,140 --> 00:24:52,040
And we can mark this
in using the X axis

333
00:24:52,040 --> 00:24:54,400
as the number line.

334
00:24:55,600 --> 00:25:00,222
I'll
do

335
00:25:00,222 --> 00:25:04,844
one
more

336
00:25:04,844 --> 00:25:07,155
quadratic

337
00:25:07,155 --> 00:25:09,466
inequality.

338
00:25:10,470 --> 00:25:14,040
X squared Minus X

339
00:25:14,040 --> 00:25:18,419
minus 6. So less than or
equal to 0.

340
00:25:22,680 --> 00:25:27,146
Again, we need to plot
the graph of Y equals X

341
00:25:27,146 --> 00:25:29,176
squared minus X minus 6.

342
00:25:30,360 --> 00:25:32,058
The expression factorizes.

343
00:25:32,830 --> 00:25:35,998
To X minus three.

344
00:25:36,070 --> 00:25:40,029
X +2 And the graph

345
00:25:40,029 --> 00:25:46,756
looks like this. Similar
to the previous

346
00:25:46,756 --> 00:25:48,040
graph.

347
00:25:49,210 --> 00:25:54,716
We have The factor X +2 the line
crosses the point at X equals

348
00:25:54,716 --> 00:25:58,832
minus two and for the factor X
minus three, the curve crosses

349
00:25:58,832 --> 00:26:00,890
the point at X equals 3.

350
00:26:01,750 --> 00:26:06,046
And we're looking for where X
squared minus X minus six is

351
00:26:06,046 --> 00:26:08,194
less than or equal to 0.

352
00:26:09,470 --> 00:26:14,189
In other words, why must lie on
the X axis or below it?

353
00:26:14,920 --> 00:26:19,509
This part of the curve and that
occurs between the points of X

354
00:26:19,509 --> 00:26:24,804
equals minus two and X equals 3.
So we can say that minus two is

355
00:26:24,804 --> 00:26:29,746
less than or equal to X, which
is less than or equal to 3.

356
00:26:31,260 --> 00:26:36,746
And we can put this in again
using the X axis is the

357
00:26:36,746 --> 00:26:40,966
number line from minus 2
using a closed circle because

358
00:26:40,966 --> 00:26:43,920
2 - 2 is included to +3.