## www.mathcentre.ac.uk/.../Solving%20Inequalities.mp4

• 0:01 - 0:04
The expression 5X
• 0:04 - 0:11
minus 4. Greater than
two X plus 3 looks like an
• 0:11 - 0:16
equation, but with the equal
• 0:17 - 0:19
This denotes that the.
• 0:20 - 0:25
Part on the left, 5X minus four
is greater than the part on the
• 0:25 - 0:26
right 2X plus 3.
• 0:27 - 0:30
We use four symbols to denote in
• 0:30 - 0:34
Equalities. This symbol
means is greater than.
• 0:37 - 0:41
This symbol means is greater
than or equal to.
• 0:42 - 0:45
This symbol means is less than.
• 0:46 - 0:50
On this symbol means is less
than or equal to.
• 0:51 - 0:56
always points to the
• 0:56 - 0:57
smaller expression.
• 0:59 - 1:02
In Equalities can be
manipulated like equations
• 1:02 - 1:04
rules.
• 1:05 - 1:07
But there is one
important exception.
• 1:09 - 1:14
If you add the same number to
both sides of an inequality, the
• 1:14 - 1:18
inequality remains true. If you
subtract the same number from
• 1:18 - 1:23
both sides of the inequality, it
remains true. If you multiply or
• 1:23 - 1:27
divide both sides of an
inequality by the same positive
• 1:27 - 1:28
number, it remains true.
• 1:30 - 1:34
But if you multiply or divide
both sides of an inequality by a
• 1:34 - 1:36
negative number. It's no longer
• 1:36 - 1:40
true. In fact, the inequality
becomes reversed. This is quite
• 1:40 - 1:45
easy to see because we can write
that four is greater than two.
• 1:46 - 1:51
But if we multiply both sides of
this inequality by minus one, we
• 1:51 - 1:52
get minus 4.
• 1:52 - 1:55
Is less than minus 2?
• 1:56 - 1:58
We have to reverse the
inequality.
• 1:59 - 2:05
when dealing with variables
• 2:05 - 2:12
because of variable can
be either positive or
• 2:12 - 2:15
negative. Look at these two
• 2:15 - 2:17
inequalities. X is greater than
• 2:17 - 2:20
one. And X squared.
• 2:20 - 2:22
Is greater than X.
• 2:23 - 2:28
Now clearly if X squared is
greater than ex, ex can't be 0.
• 2:28 - 2:32
So it looks as if we ought to be
able to divide both sides of
• 2:32 - 2:34
this inequality by X. Giving us.
• 2:35 - 2:38
X greater than one, which is
what we've got on the left.
• 2:40 - 2:43
But in fact we can't do this.
These two inequalities are not
• 2:43 - 2:47
the same. This is because X
can be negative.
• 2:48 - 2:53
Here we're saying that X is
greater than one, so X must be
• 2:53 - 2:56
positive. But here we have to
take into account the
• 2:56 - 2:58
possibility that X is negative.
• 2:58 - 3:05
In fact, the complete solution
for this is X is greater than
• 3:05 - 3:08
one or X less than 0.
• 3:08 - 3:11
Because obviously if X is
negative, then X squared is
• 3:11 - 3:15
always going to be greater than
X. I'll show you exactly how to
• 3:15 - 3:18
get the solution for this type
of inequality later on.
• 3:21 - 3:24
Great care really has to be
taken when solving inequalities
• 3:24 - 3:28
to make sure that you don't
multiply or divide by a negative
• 3:28 - 3:33
number by accident. For example,
saying that X is greater than Y.
• 3:34 - 3:41
Implies. That X
squared is greater than Y
• 3:41 - 3:44
squared only if X&Y are
• 3:44 - 3:51
positive. I'll start
with a very simple
• 3:51 - 3:58
inequality. X +3 is
greater than two.
• 3:59 - 4:03
To solve this, we simply need
to subtract 3 from both sides.
• 4:03 - 4:08
If we subtract 3 from the left
hand side were left with X. If
• 4:08 - 4:12
we subtract 3 from the right
hand side were left with minus
• 4:12 - 4:15
one and that is the solution
to the inequality.
• 4:16 - 4:19
In Equalities can be represented
on the number line.
• 4:21 - 4:25
Here are solution is X is
greater than minus one.
• 4:26 - 4:28
So we start at minus one.
• 4:30 - 4:33
And this line shows the range of
• 4:33 - 4:35
values. The decks can take.
• 4:36 - 4:40
I'm going to put an open circle
there. That open circle denotes
• 4:40 - 4:42
that although the line goes to
• 4:42 - 4:47
minus one. X cannot actually
equal minus. 1X has to be
• 4:47 - 4:48
greater than minus one.
• 4:49 - 4:55
Let's have a
look at another
• 4:55 - 4:56
one.
• 4:58 - 5:01
4X plus 6.
• 5:02 - 5:06
Is greater than 3X plus 7.
• 5:07 - 5:12
First of all, I'm going to
subtract 6 from both sides, so
• 5:12 - 5:17
we get 4X on the left, greater
than 3X plus one.
• 5:18 - 5:23
And now I'm going to subtract 3
X from both sides, which gives
• 5:23 - 5:24
me X greater than one.
• 5:25 - 5:29
And again, I can represent this
on the number line.
• 5:30 - 5:32
X has to be greater than one.
• 5:34 - 5:35
But X cannot equal 1.
• 5:37 - 5:44
Another example is 3X minus
five is less than or
• 5:44 - 5:47
equal to 3 minus X.
• 5:49 - 5:54
This time I need to add 5 to
both sides which gives me 3X is
• 5:54 - 5:56
less than or equal to.
• 5:57 - 5:59
8 minus X.
• 5:59 - 6:04
And then I need to add extra
both sides, which gives me 4X
• 6:04 - 6:06
less than or equal to 8.
• 6:07 - 6:12
Finally, I can divide both sides
by two, which gives me X is less
• 6:12 - 6:14
than or equal to two.
• 6:15 - 6:16
And on the number line.
• 6:19 - 6:23
X is less than or equal to two,
so we go this way.
• 6:23 - 6:26
And this time I'm going to do a
• 6:26 - 6:30
closed circle. This denotes that
X can be equal to two.
• 6:33 - 6:40
Now I'd like to look at
the inequality minus 2X is
• 6:40 - 6:42
greater than 4.
• 6:43 - 6:47
In order to solve this
inequality, we're going to have
• 6:47 - 6:49
to divide both sides by minus 2.
• 6:52 - 6:57
So we get minus two X divided by
minus two is X.
• 6:58 - 7:02
I've got to remember because I'm
dividing by a negative number to
• 7:02 - 7:03
reverse the inequality.
• 7:04 - 7:09
And four divided by minus
two is minus 2, so I get
• 7:09 - 7:11
X is less than minus 2.
• 7:14 - 7:17
There's often more than one way
to solve an inequality.
• 7:19 - 7:22
And I can just solve this
one again by using a
• 7:22 - 7:25
different method, so we have
-2 X is greater than 4.
• 7:26 - 7:29
If we add 2X to both sides we
• 7:29 - 7:35
get. Zero is greater than
4 + 2 X.
• 7:37 - 7:43
And then if we subtract 4 from
both sides, we get minus four is
• 7:43 - 7:44
greater than two X.
• 7:45 - 7:51
And we can divide through by two
again getting minus two is
• 7:51 - 7:52
greater than X.
• 7:52 - 7:57
And saying that X is less than
minus two is the same thing as
• 7:57 - 8:01
saying minus two is greater than
X, so we've solved this
• 8:01 - 8:04
inequality by do different
methods. The second one avoids
• 8:04 - 8:06
dividing by a negative number.
• 8:08 - 8:14
In Equalities often appear in
conjunction with the modulus
• 8:14 - 8:17
symbol. For instance.
• 8:19 - 8:23
We say MoD X is less than two.
• 8:24 - 8:27
The modular symbol denotes that
we have to take the absolute
• 8:27 - 8:32
value of X regardless of sign.
This is just the magnitude of X.
• 8:33 - 8:37
And it is always
positive. So for
• 8:37 - 8:40
instance, MoD 2 is
equal to 2.
• 8:41 - 8:45
And MoD minus two is
also equal to two.
• 8:47 - 8:53
If the absolute value of X is
less than two, then X must lie
• 8:53 - 8:59
between 2:00 and minus two. We
write minus two is less than X,
• 8:59 - 9:01
is less than two.
• 9:01 - 9:05
We can show this on the
number line.
• 9:07 - 9:15
X has to lie between minus two
and two, but it can't be too
• 9:15 - 9:22
itself. This shows the range
of values that ex can take.
• 9:25 - 9:31
If MoD X is greater than or
equal to five, we have the
• 9:31 - 9:36
absolute value of X must be
greater than or equal to five,
• 9:36 - 9:42
which means that X is going to
itself is going to be greater
• 9:42 - 9:49
than or equal to five or less
than or equal to minus five. We
• 9:49 - 9:55
write X less than or equal to
minus five or X greater than or
• 9:55 - 9:56
equal to 5.
• 9:56 - 9:58
And on the number line.
• 9:59 - 10:04
X can take the value 5, so we do
a closed circle.
• 10:05 - 10:08
And it can take the
value minus 5.
• 10:10 - 10:16
Now I want to look at
another slightly more
• 10:16 - 10:18
complicated modulus one.
• 10:19 - 10:22
We have MoD X minus 4.
• 10:23 - 10:24
Less than three.
• 10:25 - 10:30
The modulus sign shows that
the absolute value of X minus
• 10:30 - 10:36
four is less than three. This
means that X minus four must
• 10:36 - 10:40
lie between minus three and
three, so we write minus
• 10:40 - 10:44
three less than X minus four
less than three.
• 10:46 - 10:51
This is what we call a double
inequality of women's treated as
• 10:51 - 10:56
two separate inequalities. So on
the left we have minus three is
• 10:56 - 10:58
less than X minus 4.
• 11:00 - 11:07
By adding four to both sides, we
get one is less than X. On the
• 11:07 - 11:11
right we have X minus four is
less than three.
• 11:12 - 11:17
And again we had four to both
sides to get. X is less than 7.
• 11:18 - 11:22
So the solution to this
particular inequality is X is
• 11:22 - 11:26
greater than One X is less
than Seven. We write 1 less
• 11:26 - 11:31
than X less than Seven, and
again I'll show you that on
• 11:31 - 11:32
the number line.
• 11:35 - 11:38
X lies between one and Seven,
but it can't be either.
• 11:43 - 11:49
Now let's solve
MoD. 5X. Minus 8
• 11:49 - 11:56
is less than or
equal to 12.
• 11:58 - 12:02
We're saying here that the
absolute value of 5X minus 8 is
• 12:02 - 12:04
less than or equal to 12.
• 12:05 - 12:07
So 5X minus 8.
• 12:08 - 12:09
Must be less than 12.
• 12:11 - 12:13
Or greater than minus 12.
• 12:14 - 12:21
We write minus 12 is less than
or equal to 5X minus 8.
• 12:21 - 12:24
Is less than or equal to 12?
• 12:25 - 12:30
Again, we have a double
inequality on the left, we have
• 12:30 - 12:35
minus 12 is less than or equal
to 5X minus 8.
• 12:36 - 12:42
We add it to both sides, which
gives us minus four is less than
• 12:42 - 12:44
or equal to 5X.
• 12:45 - 12:49
And then we divide both
sides by 5, which gives
• 12:49 - 12:53
us minus four fifths is
less than or equal to X.
• 12:54 - 12:59
On the right we have the
inequality 5X minus 8 is less
• 12:59 - 13:00
than or equal to 12.
• 13:01 - 13:07
So we write 5X minus 8 less than
or equal to 12.
• 13:07 - 13:12
We had eight to both sides,
which gives us 5X is less than
• 13:12 - 13:14
or equal to 20.
• 13:15 - 13:18
And we divide both sides
by 5, which gives us X is
• 13:18 - 13:20
less than or equal to 4.
• 13:22 - 13:29
So our final answer is minus 4
over 5 is less than or equal to
• 13:29 - 13:32
X. Which in turn is less
than or equal to 4.
• 13:33 - 13:36
And we can show this
on the number line.
• 13:37 - 13:40
Minus four fifths is about here.
• 13:41 - 13:42
Let me go through to four.
• 13:43 - 13:45
And because it's less than or
• 13:45 - 13:49
equal to. We use
a closed circle.
• 13:51 - 13:55
In Equalities can be solved
very easily using graphs,
• 13:55 - 14:00
and if you're in any way
• 14:00 - 14:06
can could be a good idea to
do a graph to check. Let me
• 14:06 - 14:08
show you how this works.
• 14:10 - 14:15
We take the inequality 2X, plus
three is less than 0.
• 14:16 - 14:19
Now this inequality can be
solved very easily doing
• 14:19 - 14:21
algebra, but it makes a good
• 14:21 - 14:27
example. The first thing that we
need to do is to draw the graph
• 14:27 - 14:30
of Y equals 2X plus 3.
• 14:32 - 14:34
And I've got this graph here.
• 14:34 - 14:40
Note that it's the equation of
a straight line.
• 14:40 - 14:44
It has a slope of two
and then intercept on
• 14:44 - 14:46
the Y axis of three.
• 14:47 - 14:51
On the X axis.
• 14:52 - 14:56
Why is equal to 0 so that
where the line cuts the X
• 14:56 - 14:58
axis Y is equal to 0?
• 14:59 - 15:02
Above the X axis Y is greater
• 15:02 - 15:06
than 0. And below the X axis Y
is less than 0.
• 15:08 - 15:12
So when we say that we want 2X
plus three less than 0.
• 15:13 - 15:17
On this graph, that means why is
less than zero, so we're looking
• 15:17 - 15:20
for the points where the line is
below the X axis.
• 15:21 - 15:26
In other words, where X is less
than minus one and a half, and
• 15:26 - 15:28
this is the solution to the
• 15:28 - 15:35
inequality. And we can mark
this on the graph using the
• 15:35 - 15:39
X axis as the number line.
• 15:40 - 15:46
This technique can also be
used with modulus inequalities
• 15:46 - 15:53
and here using a graph
• 15:54 - 15:56
Take for example the inequality.
• 15:57 - 16:01
MoD X minus two is less than 0.
• 16:02 - 16:08
Again, we need to plot the graph
of Y equals MoD X minus 2.
• 16:09 - 16:15
This is the graph of Y equals
MoD X minus 2.
• 16:16 - 16:18
For those of you who are not
familiar with modulus functions,
• 16:18 - 16:20
it might look a little bit
• 16:20 - 16:24
strange. On the right we have
part of the graph of Y equals X
• 16:24 - 16:30
minus 2. And on the left,
where X is less than zero, we
• 16:30 - 16:34
have part of the graph of Y
equals minus X minus two.
• 16:34 - 16:37
This is because the modulus
function changes the sign of
• 16:37 - 16:39
X when X is negative.
• 16:41 - 16:46
Again, we're looking for MoD X.
Minus two is less than 0.
• 16:47 - 16:52
So we want the places where Y is
less than zero, which is between
• 16:52 - 16:57
X equals minus two and X equals
+2, and again this is the
• 16:57 - 16:59
solution to our problem.
• 16:59 - 17:05
So we say minus two less than
X less than two.
• 17:06 - 17:11
Again, we can mark this on the
graph using the X axis as the
• 17:11 - 17:15
inequalities need
• 17:15 - 17:22
handling with care.
Let's solve X
• 17:22 - 17:29
squared minus three
X +2 is
• 17:29 - 17:32
greater than 0.
• 17:36 - 17:39
Note that all the terms are on
the left hand side.
• 17:39 - 17:43
And on the right hand side we
just had zero, exactly as with
• 17:43 - 17:44
• 17:44 - 17:48
you solve it. This expression
• 17:48 - 17:54
factorizes too. X minus
two X minus one.
• 17:55 - 17:58
equation. We would simply say
• 17:58 - 18:02
right X equals 2 or X equals 1
and that's it.
• 18:03 - 18:05
But we've got a bit more work to
• 18:05 - 18:10
do here. Weather this expression
is greater than zero is going to
• 18:10 - 18:15
depend on the sign of each of
these two factors. We sort this
• 18:15 - 18:18
out by using a grid.
• 18:18 - 18:25
The points
that were
• 18:25 - 18:31
checks equals.
X minus 2 equals 0 and X minus
• 18:31 - 18:35
one equals 0 and marked in, so
this is one and two.
• 18:36 - 18:40
We put the two factors on the
• 18:40 - 18:43
left. And their product.
• 18:43 - 18:47
Now.
• 18:48 - 18:54
When X is less than one, both X
minus one and X minus two are
• 18:54 - 18:55
going to be negative.
• 18:57 - 19:00
So when you multiply them
together, their product is going
• 19:00 - 19:01
to be positive.
• 19:03 - 19:06
When X is greater than one but
• 19:06 - 19:10
less than two. X minus one is
going to be positive.
• 19:11 - 19:13
But X minus two is going to be
• 19:13 - 19:15
negative. So when you multiply
• 19:15 - 19:17
them together. The product will
• 19:17 - 19:23
be negative. Finally, when X is
greater than two, both X minus
• 19:23 - 19:27
one and X minus two will be
• 19:27 - 19:30
positive. And if you multiply
them together, their product
• 19:30 - 19:32
will also be positive.
• 19:34 - 19:36
We are looking for.
• 19:36 - 19:40
X minus two times X minus one to
be greater than 0.
• 19:41 - 19:43
This occurs when it's positive.
• 19:44 - 19:47
And our grid shows that this
happens when X is less than one.
• 19:48 - 19:50
Or when X is greater than two?
• 19:50 - 19:52
So we write in our answer.
• 19:54 - 20:01
Which is X is less than one
or X is greater than two.
• 20:04 - 20:07
And on the number line.
• 20:07 - 20:09
X must be less than one.
• 20:10 - 20:13
So I put a circle to show
that it can't be 1.
• 20:14 - 20:17
And X can also be greater
than two.
• 20:20 - 20:24
Here's another
• 20:24 - 20:30
X squared plus 5X
• 20:30 - 20:35
plus 12 is greater
than or equal to 0.
• 20:37 - 20:41
I don't like having a negative
coefficient of X squared, so I'm
• 20:41 - 20:44
going to multiply this whole
thing through by minus one,
• 20:44 - 20:48
remembering to change the
direction of the inequality as I
• 20:48 - 20:49
do. This gives us.
• 20:49 - 20:57
Two X squared minus 5X minus 12
is less than or equal to 0.
• 20:59 - 21:05
This expression factorizes to 2X
plus three times X minus four,
• 21:05 - 21:09
so that is less than or equal
• 21:09 - 21:13
to 0. Again, I'm going to
do a grid.
• 21:18 - 21:26
This factor is zero
when X is minus
• 21:26 - 21:28
three over 2.
• 21:29 - 21:32
This fact is zero when X is 4.
• 21:33 - 21:36
We write in the two factors.
• 21:36 - 21:40
And we right in the product.
• 21:43 - 21:51
When X is less than minus three
over 2, both 2X plus three and
• 21:51 - 21:53
X minus four and negative.
• 21:54 - 21:56
So their product is positive.
• 21:58 - 22:01
When X lies between minus three
over two and four.
• 22:03 - 22:05
2X plus three is positive.
• 22:05 - 22:10
But X minus four is still
negative, so their product
• 22:10 - 22:10
is negative.
• 22:11 - 22:17
When X is greater than four,
both 2X plus three and X minus
• 22:17 - 22:18
four are positive.
• 22:19 - 22:20
So their product is positive.
• 22:21 - 22:27
We are looking for 2X plus three
times X minus four to be less
• 22:27 - 22:29
than or equal to 0.
• 22:29 - 22:33
In other words, this expression
has to be either 0 or negative.
• 22:34 - 22:35
This occurs.
• 22:37 - 22:42
When X lies between minus three
over two and four, and it can
• 22:42 - 22:47
equal either number. So we have
minus three over 2 is less than
• 22:47 - 22:52
or equal to X is less than or
equal to 4.
• 22:54 - 22:56
And on the number line.
• 22:58 - 23:00
Minus three over 2 is here.
• 23:02 - 23:06
Four is here.
• 23:09 - 23:12
And I've done filled
circles because we have
• 23:12 - 23:14
less than or equal to.
• 23:17 - 23:23
also be solved graphically.
• 23:23 - 23:31
Let's solve X squared minus
three X +2 is greater
• 23:31 - 23:32
than 0.
• 23:34 - 23:39
As with the linear equalities
inequalities, we have to plot
• 23:39 - 23:44
the graph of Y equals X squared
minus three X +2.
• 23:45 - 23:52
This factorizes to give Y equals
X minus one times X minus 2.
• 23:53 - 23:55
The graph looks like this.
• 23:56 - 24:01
Because it's a quadratic, it's a
parabola. Are U shaped curve?
• 24:02 - 24:04
And it crosses the X axis where
• 24:04 - 24:09
X equals 1. Because of the
factor X minus one and where
• 24:09 - 24:12
X equals 2 because of the
factor X minus 2.
• 24:13 - 24:19
Now we're looking for X squared
minus three X +2 to be greater
• 24:19 - 24:24
than 0. This is where Y
is greater than zero. In
• 24:24 - 24:27
other words, the part of
the graph that is above
• 24:27 - 24:31
the X axis, which are the
two arms of the you here.
• 24:33 - 24:36
This occurs where X is less than
• 24:36 - 24:41
one. And where X is greater
than two, so we can write
• 24:41 - 24:43
that in as our solution.
• 24:46 - 24:52
And we can mark this
in using the X axis
• 24:52 - 24:54
as the number line.
• 24:56 - 25:00
I'll
do
• 25:00 - 25:05
one
more
• 25:05 - 25:07
• 25:07 - 25:09
inequality.
• 25:10 - 25:14
X squared Minus X
• 25:14 - 25:18
minus 6. So less than or
equal to 0.
• 25:23 - 25:27
Again, we need to plot
the graph of Y equals X
• 25:27 - 25:29
squared minus X minus 6.
• 25:30 - 25:32
The expression factorizes.
• 25:33 - 25:36
To X minus three.
• 25:36 - 25:40
X +2 And the graph
• 25:40 - 25:47
looks like this. Similar
to the previous
• 25:47 - 25:48
graph.
• 25:49 - 25:55
We have The factor X +2 the line
crosses the point at X equals
• 25:55 - 25:59
minus two and for the factor X
minus three, the curve crosses
• 25:59 - 26:01
the point at X equals 3.
• 26:02 - 26:06
And we're looking for where X
squared minus X minus six is
• 26:06 - 26:08
less than or equal to 0.
• 26:09 - 26:14
In other words, why must lie on
the X axis or below it?
• 26:15 - 26:20
This part of the curve and that
occurs between the points of X
• 26:20 - 26:25
equals minus two and X equals 3.
So we can say that minus two is
• 26:25 - 26:30
less than or equal to X, which
is less than or equal to 3.
• 26:31 - 26:37
And we can put this in again
using the X axis is the
• 26:37 - 26:41
number line from minus 2
using a closed circle because
• 26:41 - 26:44
2 - 2 is included to +3.
Title:
www.mathcentre.ac.uk/.../Solving%20Inequalities.mp4
Video Language:
English
 mathcentre edited English subtitles for www.mathcentre.ac.uk/.../Solving%20Inequalities.mp4