0:00:00.990,0:00:04.056
The expression 5X

0:00:04.056,0:00:10.970
minus 4. Greater than[br]two X plus 3 looks like an

0:00:10.970,0:00:15.670
equation, but with the equal[br]sign replaced by an Arrowhead.

0:00:16.990,0:00:19.138
This denotes that the.

0:00:19.720,0:00:24.760
Part on the left, 5X minus four[br]is greater than the part on the

0:00:24.760,0:00:26.200
right 2X plus 3.

0:00:27.440,0:00:29.911
We use four symbols to denote in

0:00:29.911,0:00:34.358
Equalities. This symbol[br]means is greater than.

0:00:36.840,0:00:41.043
This symbol means is greater[br]than or equal to.

0:00:42.140,0:00:44.966
This symbol means is less than.

0:00:45.890,0:00:50.180
On this symbol means is less[br]than or equal to.

0:00:51.380,0:00:55.516
Notice that the Arrowhead[br]always points to the

0:00:55.516,0:00:56.550
smaller expression.

0:00:58.780,0:01:01.580
In Equalities can be[br]manipulated like equations

0:01:01.580,0:01:03.580
and follow very similar[br]rules.

0:01:04.940,0:01:06.998
But there is one[br]important exception.

0:01:08.790,0:01:13.925
If you add the same number to[br]both sides of an inequality, the

0:01:13.925,0:01:17.875
inequality remains true. If you[br]subtract the same number from

0:01:17.875,0:01:22.615
both sides of the inequality, it[br]remains true. If you multiply or

0:01:22.615,0:01:26.565
divide both sides of an[br]inequality by the same positive

0:01:26.565,0:01:28.145
number, it remains true.

0:01:29.860,0:01:33.955
But if you multiply or divide[br]both sides of an inequality by a

0:01:33.955,0:01:36.000
negative number. It's no longer

0:01:36.000,0:01:40.217
true. In fact, the inequality[br]becomes reversed. This is quite

0:01:40.217,0:01:45.066
easy to see because we can write[br]that four is greater than two.

0:01:45.860,0:01:50.800
But if we multiply both sides of[br]this inequality by minus one, we

0:01:50.800,0:01:51.940
get minus 4.

0:01:52.440,0:01:54.990
Is less than minus 2?

0:01:55.710,0:01:57.786
We have to reverse the[br]inequality.

0:01:58.920,0:02:05.336
This leads to difficulties[br]when dealing with variables

0:02:05.336,0:02:11.752
because of variable can[br]be either positive or

0:02:11.752,0:02:14.704
negative. Look at these two

0:02:14.704,0:02:17.244
inequalities. X is greater than

0:02:17.244,0:02:19.710
one. And X squared.

0:02:20.230,0:02:21.650
Is greater than X.

0:02:23.380,0:02:27.566
Now clearly if X squared is[br]greater than ex, ex can't be 0.

0:02:28.280,0:02:31.880
So it looks as if we ought to be[br]able to divide both sides of

0:02:31.880,0:02:34.230
this inequality by X. Giving us.

0:02:34.740,0:02:38.088
X greater than one, which is[br]what we've got on the left.

0:02:39.750,0:02:43.134
But in fact we can't do this.[br]These two inequalities are not

0:02:43.134,0:02:47.100
the same. This is because X[br]can be negative.

0:02:48.440,0:02:53.055
Here we're saying that X is[br]greater than one, so X must be

0:02:53.055,0:02:56.280
positive. But here we have to[br]take into account the

0:02:56.280,0:02:57.580
possibility that X is negative.

0:02:58.180,0:03:04.600
In fact, the complete solution[br]for this is X is greater than

0:03:04.600,0:03:07.810
one or X less than 0.

0:03:08.420,0:03:11.450
Because obviously if X is[br]negative, then X squared is

0:03:11.450,0:03:15.389
always going to be greater than[br]X. I'll show you exactly how to

0:03:15.389,0:03:18.419
get the solution for this type[br]of inequality later on.

0:03:20.930,0:03:23.930
Great care really has to be[br]taken when solving inequalities

0:03:23.930,0:03:27.530
to make sure that you don't[br]multiply or divide by a negative

0:03:27.530,0:03:33.157
number by accident. For example,[br]saying that X is greater than Y.

0:03:34.140,0:03:40.954
Implies. That X[br]squared is greater than Y

0:03:40.954,0:03:44.064
squared only if X&Y are

0:03:44.064,0:03:51.126
positive. I'll start[br]with a very simple

0:03:51.126,0:03:57.958
inequality. X +3 is[br]greater than two.

0:03:59.040,0:04:03.048
To solve this, we simply need[br]to subtract 3 from both sides.

0:04:03.048,0:04:07.724
If we subtract 3 from the left[br]hand side were left with X. If

0:04:07.724,0:04:11.732
we subtract 3 from the right[br]hand side were left with minus

0:04:11.732,0:04:14.738
one and that is the solution[br]to the inequality.

0:04:15.930,0:04:19.269
In Equalities can be represented[br]on the number line.

0:04:21.320,0:04:25.340
Here are solution is X is[br]greater than minus one.

0:04:26.240,0:04:28.478
So we start at minus one.

0:04:30.280,0:04:32.667
And this line shows the range of

0:04:32.667,0:04:35.108
values. The decks can take.

0:04:36.300,0:04:40.152
I'm going to put an open circle[br]there. That open circle denotes

0:04:40.152,0:04:42.078
that although the line goes to

0:04:42.078,0:04:46.637
minus one. X cannot actually[br]equal minus. 1X has to be

0:04:46.637,0:04:47.969
greater than minus one.

0:04:49.200,0:04:55.404
Let's have a[br]look at another

0:04:55.404,0:04:56.438
one.

0:04:58.440,0:05:01.428
4X plus 6.

0:05:02.060,0:05:05.798
Is greater than 3X plus 7.

0:05:07.210,0:05:12.310
First of all, I'm going to[br]subtract 6 from both sides, so

0:05:12.310,0:05:16.985
we get 4X on the left, greater[br]than 3X plus one.

0:05:17.920,0:05:22.535
And now I'm going to subtract 3[br]X from both sides, which gives

0:05:22.535,0:05:24.310
me X greater than one.

0:05:25.080,0:05:29.040
And again, I can represent this[br]on the number line.

0:05:29.780,0:05:31.810
X has to be greater than one.

0:05:33.680,0:05:35.400
But X cannot equal 1.

0:05:36.990,0:05:43.770
Another example is 3X minus[br]five is less than or

0:05:43.770,0:05:47.160
equal to 3 minus X.

0:05:48.860,0:05:54.230
This time I need to add 5 to[br]both sides which gives me 3X is

0:05:54.230,0:05:56.020
less than or equal to.

0:05:56.530,0:05:58.798
8 minus X.

0:05:59.440,0:06:04.003
And then I need to add extra[br]both sides, which gives me 4X

0:06:04.003,0:06:06.109
less than or equal to 8.

0:06:06.890,0:06:11.944
Finally, I can divide both sides[br]by two, which gives me X is less

0:06:11.944,0:06:13.749
than or equal to two.

0:06:14.980,0:06:16.160
And on the number line.

0:06:18.830,0:06:22.535
X is less than or equal to two,[br]so we go this way.

0:06:23.290,0:06:26.194
And this time I'm going to do a

0:06:26.194,0:06:30.355
closed circle. This denotes that[br]X can be equal to two.

0:06:33.130,0:06:40.291
Now I'd like to look at[br]the inequality minus 2X is

0:06:40.291,0:06:42.244
greater than 4.

0:06:43.260,0:06:46.640
In order to solve this[br]inequality, we're going to have

0:06:46.640,0:06:49.006
to divide both sides by minus 2.

0:06:51.930,0:06:56.742
So we get minus two X divided by[br]minus two is X.

0:06:58.060,0:07:02.092
I've got to remember because I'm[br]dividing by a negative number to

0:07:02.092,0:07:03.100
reverse the inequality.

0:07:04.140,0:07:09.012
And four divided by minus[br]two is minus 2, so I get

0:07:09.012,0:07:11.448
X is less than minus 2.

0:07:14.390,0:07:17.330
There's often more than one way[br]to solve an inequality.

0:07:18.550,0:07:21.542
And I can just solve this[br]one again by using a

0:07:21.542,0:07:24.534
different method, so we have[br]-2 X is greater than 4.

0:07:25.890,0:07:28.786
If we add 2X to both sides we

0:07:28.786,0:07:34.850
get. Zero is greater than[br]4 + 2 X.

0:07:36.700,0:07:42.594
And then if we subtract 4 from[br]both sides, we get minus four is

0:07:42.594,0:07:44.278
greater than two X.

0:07:44.900,0:07:50.504
And we can divide through by two[br]again getting minus two is

0:07:50.504,0:07:51.905
greater than X.

0:07:52.450,0:07:57.364
And saying that X is less than[br]minus two is the same thing as

0:07:57.364,0:08:01.225
saying minus two is greater than[br]X, so we've solved this

0:08:01.225,0:08:04.384
inequality by do different[br]methods. The second one avoids

0:08:04.384,0:08:06.139
dividing by a negative number.

0:08:07.760,0:08:13.907
In Equalities often appear in[br]conjunction with the modulus

0:08:13.907,0:08:17.150
symbol. For instance.

0:08:18.840,0:08:22.608
We say MoD X is less than two.

0:08:23.700,0:08:27.407
The modular symbol denotes that[br]we have to take the absolute

0:08:27.407,0:08:31.788
value of X regardless of sign.[br]This is just the magnitude of X.

0:08:33.470,0:08:36.564
And it is always[br]positive. So for

0:08:36.564,0:08:39.658
instance, MoD 2 is[br]equal to 2.

0:08:41.010,0:08:45.393
And MoD minus two is[br]also equal to two.

0:08:46.850,0:08:53.038
If the absolute value of X is[br]less than two, then X must lie

0:08:53.038,0:08:58.784
between 2:00 and minus two. We[br]write minus two is less than X,

0:08:58.784,0:09:00.552
is less than two.

0:09:01.260,0:09:05.100
We can show this on the[br]number line.

0:09:06.980,0:09:14.792
X has to lie between minus two[br]and two, but it can't be too

0:09:14.792,0:09:22.370
itself. This shows the range[br]of values that ex can take.

0:09:25.320,0:09:31.118
If MoD X is greater than or[br]equal to five, we have the

0:09:31.118,0:09:36.470
absolute value of X must be[br]greater than or equal to five,

0:09:36.470,0:09:42.268
which means that X is going to[br]itself is going to be greater

0:09:42.268,0:09:48.512
than or equal to five or less[br]than or equal to minus five. We

0:09:48.512,0:09:54.756
write X less than or equal to[br]minus five or X greater than or

0:09:54.756,0:09:56.094
equal to 5.

0:09:56.270,0:09:57.710
And on the number line.

0:09:59.300,0:10:03.584
X can take the value 5, so we do[br]a closed circle.

0:10:04.900,0:10:08.004
And it can take the[br]value minus 5.

0:10:10.210,0:10:15.871
Now I want to look at[br]another slightly more

0:10:15.871,0:10:17.758
complicated modulus one.

0:10:18.890,0:10:21.620
We have MoD X minus 4.

0:10:22.830,0:10:24.498
Less than three.

0:10:25.390,0:10:30.329
The modulus sign shows that[br]the absolute value of X minus

0:10:30.329,0:10:35.717
four is less than three. This[br]means that X minus four must

0:10:35.717,0:10:40.207
lie between minus three and[br]three, so we write minus

0:10:40.207,0:10:44.248
three less than X minus four[br]less than three.

0:10:45.910,0:10:50.914
This is what we call a double[br]inequality of women's treated as

0:10:50.914,0:10:55.918
two separate inequalities. So on[br]the left we have minus three is

0:10:55.918,0:10:58.003
less than X minus 4.

0:11:00.220,0:11:06.955
By adding four to both sides, we[br]get one is less than X. On the

0:11:06.955,0:11:11.445
right we have X minus four is[br]less than three.

0:11:12.110,0:11:17.090
And again we had four to both[br]sides to get. X is less than 7.

0:11:17.750,0:11:21.610
So the solution to this[br]particular inequality is X is

0:11:21.610,0:11:26.242
greater than One X is less[br]than Seven. We write 1 less

0:11:26.242,0:11:30.874
than X less than Seven, and[br]again I'll show you that on

0:11:30.874,0:11:32.032
the number line.

0:11:34.510,0:11:38.481
X lies between one and Seven,[br]but it can't be either.

0:11:42.950,0:11:49.229
Now let's solve[br]MoD. 5X. Minus 8

0:11:49.229,0:11:55.508
is less than or[br]equal to 12.

0:11:58.000,0:12:02.140
We're saying here that the[br]absolute value of 5X minus 8 is

0:12:02.140,0:12:04.210
less than or equal to 12.

0:12:05.080,0:12:07.268
So 5X minus 8.

0:12:07.820,0:12:09.460
Must be less than 12.

0:12:10.850,0:12:13.020
Or greater than minus 12.

0:12:13.810,0:12:20.609
We write minus 12 is less than[br]or equal to 5X minus 8.

0:12:21.260,0:12:23.710
Is less than or equal to 12?

0:12:25.030,0:12:30.200
Again, we have a double[br]inequality on the left, we have

0:12:30.200,0:12:35.370
minus 12 is less than or equal[br]to 5X minus 8.

0:12:36.480,0:12:42.178
We add it to both sides, which[br]gives us minus four is less than

0:12:42.178,0:12:43.806
or equal to 5X.

0:12:44.960,0:12:48.970
And then we divide both[br]sides by 5, which gives

0:12:48.970,0:12:53.381
us minus four fifths is[br]less than or equal to X.

0:12:54.460,0:12:58.708
On the right we have the[br]inequality 5X minus 8 is less

0:12:58.708,0:13:00.478
than or equal to 12.

0:13:01.480,0:13:06.628
So we write 5X minus 8 less than[br]or equal to 12.

0:13:07.360,0:13:12.261
We had eight to both sides,[br]which gives us 5X is less than

0:13:12.261,0:13:13.769
or equal to 20.

0:13:14.510,0:13:18.374
And we divide both sides[br]by 5, which gives us X is

0:13:18.374,0:13:20.306
less than or equal to 4.

0:13:22.070,0:13:28.685
So our final answer is minus 4[br]over 5 is less than or equal to

0:13:28.685,0:13:32.240
X. Which in turn is less[br]than or equal to 4.

0:13:33.440,0:13:35.834
And we can show this[br]on the number line.

0:13:37.190,0:13:40.010
Minus four fifths is about here.

0:13:40.930,0:13:42.460
Let me go through to four.

0:13:43.160,0:13:45.176
And because it's less than or

0:13:45.176,0:13:48.860
equal to. We use[br]a closed circle.

0:13:50.700,0:13:54.678
In Equalities can be solved[br]very easily using graphs,

0:13:54.678,0:13:59.540
and if you're in any way[br]unsure about the algebra it

0:13:59.540,0:14:05.728
can could be a good idea to[br]do a graph to check. Let me

0:14:05.728,0:14:07.938
show you how this works.

0:14:09.700,0:14:15.365
We take the inequality 2X, plus[br]three is less than 0.

0:14:16.040,0:14:18.992
Now this inequality can be[br]solved very easily doing

0:14:18.992,0:14:20.960
algebra, but it makes a good

0:14:20.960,0:14:27.313
example. The first thing that we[br]need to do is to draw the graph

0:14:27.313,0:14:29.719
of Y equals 2X plus 3.

0:14:32.180,0:14:33.638
And I've got this graph here.

0:14:34.200,0:14:39.735
Note that it's the equation of[br]a straight line.

0:14:40.440,0:14:43.820
It has a slope of two[br]and then intercept on

0:14:43.820,0:14:45.510
the Y axis of three.

0:14:47.450,0:14:51.278
On the X axis.

0:14:52.460,0:14:56.308
Why is equal to 0 so that[br]where the line cuts the X

0:14:56.308,0:14:58.084
axis Y is equal to 0?

0:14:59.280,0:15:01.632
Above the X axis Y is greater

0:15:01.632,0:15:06.390
than 0. And below the X axis Y[br]is less than 0.

0:15:08.260,0:15:11.978
So when we say that we want 2X[br]plus three less than 0.

0:15:13.420,0:15:17.203
On this graph, that means why is[br]less than zero, so we're looking

0:15:17.203,0:15:20.404
for the points where the line is[br]below the X axis.

0:15:21.090,0:15:25.682
In other words, where X is less[br]than minus one and a half, and

0:15:25.682,0:15:27.650
this is the solution to the

0:15:27.650,0:15:35.240
inequality. And we can mark[br]this on the graph using the

0:15:35.240,0:15:39.128
X axis as the number line.

0:15:39.850,0:15:46.330
This technique can also be[br]used with modulus inequalities

0:15:46.330,0:15:52.810
and here using a graph[br]can be very helpful.

0:15:53.750,0:15:56.440
Take for example the inequality.

0:15:57.010,0:16:00.690
MoD X minus two is less than 0.

0:16:01.820,0:16:08.148
Again, we need to plot the graph[br]of Y equals MoD X minus 2.

0:16:08.720,0:16:14.924
This is the graph of Y equals[br]MoD X minus 2.

0:16:15.750,0:16:18.236
For those of you who are not[br]familiar with modulus functions,

0:16:18.236,0:16:19.592
it might look a little bit

0:16:19.592,0:16:24.438
strange. On the right we have[br]part of the graph of Y equals X

0:16:24.438,0:16:29.602
minus 2. And on the left,[br]where X is less than zero, we

0:16:29.602,0:16:33.706
have part of the graph of Y[br]equals minus X minus two.

0:16:33.706,0:16:37.126
This is because the modulus[br]function changes the sign of

0:16:37.126,0:16:38.836
X when X is negative.

0:16:40.660,0:16:45.580
Again, we're looking for MoD X.[br]Minus two is less than 0.

0:16:46.760,0:16:52.122
So we want the places where Y is[br]less than zero, which is between

0:16:52.122,0:16:57.101
X equals minus two and X equals[br]+2, and again this is the

0:16:57.101,0:16:58.633
solution to our problem.

0:16:59.460,0:17:05.213
So we say minus two less than[br]X less than two.

0:17:05.920,0:17:10.526
Again, we can mark this on the[br]graph using the X axis as the

0:17:10.526,0:17:15.290
number line. Quadratic[br]inequalities need

0:17:15.290,0:17:22.130
handling with care.[br]Let's solve X

0:17:22.130,0:17:28.970
squared minus three[br]X +2 is

0:17:28.970,0:17:32.390
greater than 0.

0:17:35.610,0:17:38.734
Note that all the terms are on[br]the left hand side.

0:17:39.240,0:17:42.867
And on the right hand side we[br]just had zero, exactly as with

0:17:42.867,0:17:43.983
the quadratic equation before

0:17:43.983,0:17:47.654
you solve it. This expression

0:17:47.654,0:17:53.746
factorizes too. X minus[br]two X minus one.

0:17:54.530,0:17:58.310
Now this is a quadratic[br]equation. We would simply say

0:17:58.310,0:18:02.468
right X equals 2 or X equals 1[br]and that's it.

0:18:03.250,0:18:04.682
But we've got a bit more work to

0:18:04.682,0:18:10.120
do here. Weather this expression[br]is greater than zero is going to

0:18:10.120,0:18:15.450
depend on the sign of each of[br]these two factors. We sort this

0:18:15.450,0:18:17.500
out by using a grid.

0:18:18.240,0:18:24.744
The points[br]that were

0:18:24.744,0:18:31.370
checks equals.[br]X minus 2 equals 0 and X minus

0:18:31.370,0:18:35.390
one equals 0 and marked in, so[br]this is one and two.

0:18:36.170,0:18:39.579
We put the two factors on the

0:18:39.579,0:18:42.698
left. And their product.

0:18:43.280,0:18:47.000
Now.

0:18:48.210,0:18:53.700
When X is less than one, both X[br]minus one and X minus two are

0:18:53.700,0:18:55.164
going to be negative.

0:18:56.580,0:18:59.950
So when you multiply them[br]together, their product is going

0:18:59.950,0:19:00.961
to be positive.

0:19:03.390,0:19:05.525
When X is greater than one but

0:19:05.525,0:19:09.688
less than two. X minus one is[br]going to be positive.

0:19:10.600,0:19:13.096
But X minus two is going to be

0:19:13.096,0:19:15.350
negative. So when you multiply

0:19:15.350,0:19:17.386
them together. The product will

0:19:17.386,0:19:23.420
be negative. Finally, when X is[br]greater than two, both X minus

0:19:23.420,0:19:26.556
one and X minus two will be

0:19:26.556,0:19:30.282
positive. And if you multiply[br]them together, their product

0:19:30.282,0:19:31.578
will also be positive.

0:19:34.070,0:19:35.798
We are looking for.

0:19:36.300,0:19:39.900
X minus two times X minus one to[br]be greater than 0.

0:19:40.890,0:19:42.620
This occurs when it's positive.

0:19:43.500,0:19:47.140
And our grid shows that this[br]happens when X is less than one.

0:19:47.640,0:19:49.866
Or when X is greater than two?

0:19:50.450,0:19:52.418
So we write in our answer.

0:19:53.660,0:20:00.849
Which is X is less than one[br]or X is greater than two.

0:20:03.950,0:20:06.590
And on the number line.

0:20:07.210,0:20:09.388
X must be less than one.

0:20:09.980,0:20:12.536
So I put a circle to show[br]that it can't be 1.

0:20:14.280,0:20:16.520
And X can also be greater[br]than two.

0:20:20.050,0:20:23.976
Here's another

0:20:23.976,0:20:30.116
quadratic. Minus two[br]X squared plus 5X

0:20:30.116,0:20:35.480
plus 12 is greater[br]than or equal to 0.

0:20:36.570,0:20:40.674
I don't like having a negative[br]coefficient of X squared, so I'm

0:20:40.674,0:20:44.094
going to multiply this whole[br]thing through by minus one,

0:20:44.094,0:20:47.514
remembering to change the[br]direction of the inequality as I

0:20:47.514,0:20:48.882
do. This gives us.

0:20:49.410,0:20:57.278
Two X squared minus 5X minus 12[br]is less than or equal to 0.

0:20:58.680,0:21:04.906
This expression factorizes to 2X[br]plus three times X minus four,

0:21:04.906,0:21:08.868
so that is less than or equal

0:21:08.868,0:21:12.955
to 0. Again, I'm going to[br]do a grid.

0:21:18.150,0:21:25.590
This factor is zero[br]when X is minus

0:21:25.590,0:21:28.380
three over 2.

0:21:29.450,0:21:31.858
This fact is zero when X is 4.

0:21:32.770,0:21:35.698
We write in the two factors.

0:21:36.380,0:21:39.938
And we right in the product.

0:21:43.460,0:21:50.530
When X is less than minus three[br]over 2, both 2X plus three and

0:21:50.530,0:21:53.055
X minus four and negative.

0:21:53.860,0:21:56.110
So their product is positive.

0:21:57.580,0:22:01.350
When X lies between minus three[br]over two and four.

0:22:02.540,0:22:04.670
2X plus three is positive.

0:22:05.410,0:22:09.590
But X minus four is still[br]negative, so their product

0:22:09.590,0:22:10.426
is negative.

0:22:11.480,0:22:16.797
When X is greater than four,[br]both 2X plus three and X minus

0:22:16.797,0:22:18.024
four are positive.

0:22:18.590,0:22:20.180
So their product is positive.

0:22:20.780,0:22:26.576
We are looking for 2X plus three[br]times X minus four to be less

0:22:26.576,0:22:28.646
than or equal to 0.

0:22:29.330,0:22:33.110
In other words, this expression[br]has to be either 0 or negative.

0:22:34.300,0:22:35.270
This occurs.

0:22:36.520,0:22:41.824
When X lies between minus three[br]over two and four, and it can

0:22:41.824,0:22:47.128
equal either number. So we have[br]minus three over 2 is less than

0:22:47.128,0:22:51.616
or equal to X is less than or[br]equal to 4.

0:22:53.890,0:22:56.370
And on the number line.

0:22:58.220,0:23:00.326
Minus three over 2 is here.

0:23:01.760,0:23:05.738
Four is here.

0:23:08.840,0:23:12.040
And I've done filled[br]circles because we have

0:23:12.040,0:23:14.040
less than or equal to.

0:23:17.260,0:23:22.783
Quadratic inequalities can[br]also be solved graphically.

0:23:22.783,0:23:30.673
Let's solve X squared minus[br]three X +2 is greater

0:23:30.673,0:23:32.251
than 0.

0:23:34.130,0:23:38.710
As with the linear equalities[br]inequalities, we have to plot

0:23:38.710,0:23:43.748
the graph of Y equals X squared[br]minus three X +2.

0:23:44.650,0:23:51.527
This factorizes to give Y equals[br]X minus one times X minus 2.

0:23:52.800,0:23:54.600
The graph looks like this.

0:23:55.960,0:24:01.174
Because it's a quadratic, it's a[br]parabola. Are U shaped curve?

0:24:02.210,0:24:04.275
And it crosses the X axis where

0:24:04.275,0:24:08.729
X equals 1. Because of the[br]factor X minus one and where

0:24:08.729,0:24:12.139
X equals 2 because of the[br]factor X minus 2.

0:24:13.490,0:24:18.963
Now we're looking for X squared[br]minus three X +2 to be greater

0:24:18.963,0:24:23.662
than 0. This is where Y[br]is greater than zero. In

0:24:23.662,0:24:27.042
other words, the part of[br]the graph that is above

0:24:27.042,0:24:31.098
the X axis, which are the[br]two arms of the you here.

0:24:32.710,0:24:36.021
This occurs where X is less than

0:24:36.021,0:24:41.218
one. And where X is greater[br]than two, so we can write

0:24:41.218,0:24:43.058
that in as our solution.

0:24:46.140,0:24:52.040
And we can mark this[br]in using the X axis

0:24:52.040,0:24:54.400
as the number line.

0:24:55.600,0:25:00.222
I'll[br]do

0:25:00.222,0:25:04.844
one[br]more

0:25:04.844,0:25:07.155
quadratic

0:25:07.155,0:25:09.466
inequality.

0:25:10.470,0:25:14.040
X squared Minus X

0:25:14.040,0:25:18.419
minus 6. So less than or[br]equal to 0.

0:25:22.680,0:25:27.146
Again, we need to plot[br]the graph of Y equals X

0:25:27.146,0:25:29.176
squared minus X minus 6.

0:25:30.360,0:25:32.058
The expression factorizes.

0:25:32.830,0:25:35.998
To X minus three.

0:25:36.070,0:25:40.029
X +2 And the graph

0:25:40.029,0:25:46.756
looks like this. Similar[br]to the previous

0:25:46.756,0:25:48.040
graph.

0:25:49.210,0:25:54.716
We have The factor X +2 the line[br]crosses the point at X equals

0:25:54.716,0:25:58.832
minus two and for the factor X[br]minus three, the curve crosses

0:25:58.832,0:26:00.890
the point at X equals 3.

0:26:01.750,0:26:06.046
And we're looking for where X[br]squared minus X minus six is

0:26:06.046,0:26:08.194
less than or equal to 0.

0:26:09.470,0:26:14.189
In other words, why must lie on[br]the X axis or below it?

0:26:14.920,0:26:19.509
This part of the curve and that[br]occurs between the points of X

0:26:19.509,0:26:24.804
equals minus two and X equals 3.[br]So we can say that minus two is

0:26:24.804,0:26:29.746
less than or equal to X, which[br]is less than or equal to 3.

0:26:31.260,0:26:36.746
And we can put this in again[br]using the X axis is the

0:26:36.746,0:26:40.966
number line from minus 2[br]using a closed circle because

0:26:40.966,0:26:43.920
2 - 2 is included to +3.