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

The expression 5X

minus 4. Greater than
two X plus 3 looks like an

equation, but with the equal
sign replaced by an Arrowhead.

This denotes that the.

Part on the left, 5X minus four
is greater than the part on the

right 2X plus 3.

We use four symbols to denote in

Equalities. This symbol
means is greater than.

This symbol means is greater
than or equal to.

This symbol means is less than.

On this symbol means is less
than or equal to.

Notice that the Arrowhead
always points to the

smaller expression.

In Equalities can be
manipulated like equations

and follow very similar
rules.

But there is one
important exception.

If you add the same number to
both sides of an inequality, the

inequality remains true. If you
subtract the same number from

both sides of the inequality, it
remains true. If you multiply or

divide both sides of an
inequality by the same positive

number, it remains true.

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

negative number. It's no longer

true. In fact, the inequality
becomes reversed. This is quite

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

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

get minus 4.

Is less than minus 2?

We have to reverse the
inequality.

This leads to difficulties
when dealing with variables

because of variable can
be either positive or

negative. Look at these two

inequalities. X is greater than

one. And X squared.

Is greater than X.

Now clearly if X squared is
greater than ex, ex can't be 0.

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

this inequality by X. Giving us.

X greater than one, which is
what we've got on the left.

But in fact we can't do this.
These two inequalities are not

the same. This is because X
can be negative.

Here we're saying that X is
greater than one, so X must be

positive. But here we have to
take into account the

possibility that X is negative.

In fact, the complete solution
for this is X is greater than

one or X less than 0.

Because obviously if X is
negative, then X squared is

always going to be greater than
X. I'll show you exactly how to

get the solution for this type
of inequality later on.

Great care really has to be
taken when solving inequalities

to make sure that you don't
multiply or divide by a negative

number by accident. For example,
saying that X is greater than Y.

Implies. That X
squared is greater than Y

squared only if X&Y are

positive. I'll start
with a very simple

inequality. X +3 is
greater than two.

To solve this, we simply need
to subtract 3 from both sides.

If we subtract 3 from the left
hand side were left with X. If

we subtract 3 from the right
hand side were left with minus

one and that is the solution
to the inequality.

In Equalities can be represented
on the number line.

Here are solution is X is
greater than minus one.

So we start at minus one.

And this line shows the range of

values. The decks can take.

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

that although the line goes to

minus one. X cannot actually
equal minus. 1X has to be

greater than minus one.

Let's have a
look at another

one.

4X plus 6.

Is greater than 3X plus 7.

First of all, I'm going to
subtract 6 from both sides, so

we get 4X on the left, greater
than 3X plus one.

And now I'm going to subtract 3
X from both sides, which gives

me X greater than one.

And again, I can represent this
on the number line.

X has to be greater than one.

But X cannot equal 1.

Another example is 3X minus
five is less than or

equal to 3 minus X.

This time I need to add 5 to
both sides which gives me 3X is

less than or equal to.

8 minus X.

And then I need to add extra
both sides, which gives me 4X

less than or equal to 8.

Finally, I can divide both sides
by two, which gives me X is less

than or equal to two.

And on the number line.

X is less than or equal to two,
so we go this way.

And this time I'm going to do a

closed circle. This denotes that
X can be equal to two.

Now I'd like to look at
the inequality minus 2X is

greater than 4.

In order to solve this
inequality, we're going to have

to divide both sides by minus 2.

So we get minus two X divided by
minus two is X.

I've got to remember because I'm
dividing by a negative number to

reverse the inequality.

And four divided by minus
two is minus 2, so I get

X is less than minus 2.

There's often more than one way
to solve an inequality.

And I can just solve this
one again by using a

different method, so we have
-2 X is greater than 4.

If we add 2X to both sides we

get. Zero is greater than
4 + 2 X.

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

greater than two X.

And we can divide through by two
again getting minus two is

greater than X.

And saying that X is less than
minus two is the same thing as

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

inequality by do different
methods. The second one avoids

dividing by a negative number.

In Equalities often appear in
conjunction with the modulus

symbol. For instance.

We say MoD X is less than two.

The modular symbol denotes that
we have to take the absolute

value of X regardless of sign.
This is just the magnitude of X.

And it is always
positive. So for

instance, MoD 2 is
equal to 2.

And MoD minus two is
also equal to two.

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

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

is less than two.

We can show this on the
number line.

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

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

If MoD X is greater than or
equal to five, we have the

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

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

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

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

equal to 5.

And on the number line.

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

And it can take the
value minus 5.

Now I want to look at
another slightly more

complicated modulus one.

We have MoD X minus 4.

Less than three.

The modulus sign shows that
the absolute value of X minus

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

lie between minus three and
three, so we write minus

three less than X minus four
less than three.

This is what we call a double
inequality of women's treated as

two separate inequalities. So on
the left we have minus three is

less than X minus 4.

By adding four to both sides, we
get one is less than X. On the

right we have X minus four is
less than three.

And again we had four to both
sides to get. X is less than 7.

So the solution to this
particular inequality is X is

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

than X less than Seven, and
again I'll show you that on

the number line.

X lies between one and Seven,
but it can't be either.

Now let's solve
MoD. 5X. Minus 8

is less than or
equal to 12.

We're saying here that the
absolute value of 5X minus 8 is

less than or equal to 12.

So 5X minus 8.

Must be less than 12.

Or greater than minus 12.

We write minus 12 is less than
or equal to 5X minus 8.

Is less than or equal to 12?

Again, we have a double
inequality on the left, we have

minus 12 is less than or equal
to 5X minus 8.

We add it to both sides, which
gives us minus four is less than

or equal to 5X.

And then we divide both
sides by 5, which gives

us minus four fifths is
less than or equal to X.

On the right we have the
inequality 5X minus 8 is less

than or equal to 12.

So we write 5X minus 8 less than
or equal to 12.

We had eight to both sides,
which gives us 5X is less than

or equal to 20.

And we divide both sides
by 5, which gives us X is

less than or equal to 4.

So our final answer is minus 4
over 5 is less than or equal to

X. Which in turn is less
than or equal to 4.

And we can show this
on the number line.

Minus four fifths is about here.

Let me go through to four.

And because it's less than or

equal to. We use
a closed circle.

In Equalities can be solved
very easily using graphs,

and if you're in any way
unsure about the algebra it

can could be a good idea to
do a graph to check. Let me

show you how this works.

We take the inequality 2X, plus
three is less than 0.

Now this inequality can be
solved very easily doing

algebra, but it makes a good

example. The first thing that we
need to do is to draw the graph

of Y equals 2X plus 3.

And I've got this graph here.

Note that it's the equation of
a straight line.

It has a slope of two
and then intercept on

the Y axis of three.

On the X axis.

Why is equal to 0 so that
where the line cuts the X

axis Y is equal to 0?

Above the X axis Y is greater

than 0. And below the X axis Y
is less than 0.

So when we say that we want 2X
plus three less than 0.

On this graph, that means why is
less than zero, so we're looking

for the points where the line is
below the X axis.

In other words, where X is less
than minus one and a half, and

this is the solution to the

inequality. And we can mark
this on the graph using the

X axis as the number line.

This technique can also be
used with modulus inequalities

and here using a graph
can be very helpful.

Take for example the inequality.

MoD X minus two is less than 0.

Again, we need to plot the graph
of Y equals MoD X minus 2.

This is the graph of Y equals
MoD X minus 2.

For those of you who are not
familiar with modulus functions,

it might look a little bit

strange. On the right we have
part of the graph of Y equals X

minus 2. And on the left,
where X is less than zero, we

have part of the graph of Y
equals minus X minus two.

This is because the modulus
function changes the sign of

X when X is negative.

Again, we're looking for MoD X.
Minus two is less than 0.

So we want the places where Y is
less than zero, which is between

X equals minus two and X equals
+2, and again this is the

solution to our problem.

So we say minus two less than
X less than two.

Again, we can mark this on the
graph using the X axis as the

number line. Quadratic
inequalities need

handling with care.
Let's solve X

squared minus three
X +2 is

greater than 0.

Note that all the terms are on
the left hand side.

And on the right hand side we
just had zero, exactly as with

the quadratic equation before

you solve it. This expression

factorizes too. X minus
two X minus one.

Now this is a quadratic
equation. We would simply say

right X equals 2 or X equals 1
and that's it.

But we've got a bit more work to

do here. Weather this expression
is greater than zero is going to

depend on the sign of each of
these two factors. We sort this

out by using a grid.

The points
that were

checks equals.
X minus 2 equals 0 and X minus

one equals 0 and marked in, so
this is one and two.

We put the two factors on the

left. And their product.

Now.

When X is less than one, both X
minus one and X minus two are

going to be negative.

So when you multiply them
together, their product is going

to be positive.

When X is greater than one but

less than two. X minus one is
going to be positive.

But X minus two is going to be

negative. So when you multiply

them together. The product will

be negative. Finally, when X is
greater than two, both X minus

one and X minus two will be

positive. And if you multiply
them together, their product

will also be positive.

We are looking for.

X minus two times X minus one to
be greater than 0.

This occurs when it's positive.

And our grid shows that this
happens when X is less than one.

Or when X is greater than two?

So we write in our answer.

Which is X is less than one
or X is greater than two.

And on the number line.

X must be less than one.

So I put a circle to show
that it can't be 1.

And X can also be greater
than two.

Here's another

quadratic. Minus two
X squared plus 5X

plus 12 is greater
than or equal to 0.

I don't like having a negative
coefficient of X squared, so I'm

going to multiply this whole
thing through by minus one,

remembering to change the
direction of the inequality as I

do. This gives us.

Two X squared minus 5X minus 12
is less than or equal to 0.

This expression factorizes to 2X
plus three times X minus four,

so that is less than or equal

to 0. Again, I'm going to
do a grid.

This factor is zero
when X is minus

three over 2.

This fact is zero when X is 4.

We write in the two factors.

And we right in the product.

When X is less than minus three
over 2, both 2X plus three and

X minus four and negative.

So their product is positive.

When X lies between minus three
over two and four.

2X plus three is positive.

But X minus four is still
negative, so their product

is negative.

When X is greater than four,
both 2X plus three and X minus

four are positive.

So their product is positive.

We are looking for 2X plus three
times X minus four to be less

than or equal to 0.

In other words, this expression
has to be either 0 or negative.

This occurs.

When X lies between minus three
over two and four, and it can

equal either number. So we have
minus three over 2 is less than

or equal to X is less than or
equal to 4.

And on the number line.

Minus three over 2 is here.

Four is here.

And I've done filled
circles because we have

less than or equal to.

Quadratic inequalities can
also be solved graphically.

Let's solve X squared minus
three X +2 is greater

than 0.

As with the linear equalities
inequalities, we have to plot

the graph of Y equals X squared
minus three X +2.

This factorizes to give Y equals
X minus one times X minus 2.

The graph looks like this.

Because it's a quadratic, it's a
parabola. Are U shaped curve?

And it crosses the X axis where

X equals 1. Because of the
factor X minus one and where

X equals 2 because of the
factor X minus 2.

Now we're looking for X squared
minus three X +2 to be greater

than 0. This is where Y
is greater than zero. In

other words, the part of
the graph that is above

the X axis, which are the
two arms of the you here.

This occurs where X is less than

one. And where X is greater
than two, so we can write

that in as our solution.

And we can mark this
in using the X axis

as the number line.

I'll
do

one
more

quadratic

inequality.

X squared Minus X

minus 6. So less than or
equal to 0.

Again, we need to plot
the graph of Y equals X

squared minus X minus 6.

The expression factorizes.

To X minus three.

X +2 And the graph

looks like this. Similar
to the previous

graph.

We have The factor X +2 the line
crosses the point at X equals

minus two and for the factor X
minus three, the curve crosses

the point at X equals 3.

And we're looking for where X
squared minus X minus six is

less than or equal to 0.

In other words, why must lie on
the X axis or below it?

This part of the curve and that
occurs between the points of X

equals minus two and X equals 3.
So we can say that minus two is

less than or equal to X, which
is less than or equal to 3.

And we can put this in again
using the X axis is the

number line from minus 2
using a closed circle because

2 - 2 is included to +3.