WEBVTT

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The expression 5X

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

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

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This denotes that the.

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

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right 2X plus 3.

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We use four symbols to denote in

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Equalities. This symbol
means is greater than.

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This symbol means is greater
than or equal to.

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This symbol means is less than.

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On this symbol means is less
than or equal to.

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Notice that the Arrowhead
always points to the

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smaller expression.

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In Equalities can be
manipulated like equations

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and follow very similar
rules.

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But there is one
important exception.

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

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inequality remains true. If you
subtract the same number from

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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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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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negative number. It's no longer

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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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because of variable can
be either positive or

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negative. Look at these two

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inequalities. X is greater than

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one. And X squared.

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Is greater than X.

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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

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this inequality by X. Giving us.

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

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

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the same. This is because X
can be negative.

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

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positive. But here we have to
take into account the

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possibility that X is negative.

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

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one or X less than 0.

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Because obviously if X is
negative, then X squared is

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

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get the solution for this type
of inequality later on.

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Great care really has to be
taken when solving inequalities

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

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

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Implies. That X
squared is greater than Y

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squared only if X&Y are

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positive. I'll start
with a very simple

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inequality. X +3 is
greater than two.

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

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

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

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one and that is the solution
to the inequality.

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In Equalities can be represented
on the number line.

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Here are solution is X is
greater than minus one.

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So we start at minus one.

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And this line shows the range of

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

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greater than minus one.

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Let's have a
look at another

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one.

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4X plus 6.

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Is greater than 3X plus 7.

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

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

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

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me X greater than one.

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

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X has to be greater than one.

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But X cannot equal 1.

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Another example is 3X minus
five is less than or

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equal to 3 minus X.

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

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less than or equal to.

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8 minus X.

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

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less than or equal to 8.

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

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than or equal to two.

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And on the number line.

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

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And this time I'm going to do a

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

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

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greater than 4.

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

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to divide both sides by minus 2.

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

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

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reverse the inequality.

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

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X is less than minus 2.

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

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And I can just solve this
one again by using a

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

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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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greater than two X.

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

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greater than X.

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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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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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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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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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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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equal to 5.

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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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And it can take the
value minus 5.

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Now I want to look at
another slightly more

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complicated modulus one.

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We have MoD X minus 4.

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Less than three.

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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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lie between minus three and
three, so we write minus

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three less than X minus four
less than three.

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

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

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less than X minus 4.

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

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right we have X minus four is
less than three.

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

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

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the number line.

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

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

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is less than or
equal to 12.

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

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less than or equal to 12.

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So 5X minus 8.

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Must be less than 12.

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Or greater than minus 12.

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

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Is less than or equal to 12?

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

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

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

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or equal to 5X.

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And then we divide both
sides by 5, which gives

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

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

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than or equal to 12.

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

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

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or equal to 20.

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

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less than or equal to 4.

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

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

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And we can show this
on the number line.

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Minus four fifths is about here.

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Let me go through to four.

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And because it's less than or

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equal to. We use
a closed circle.

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In Equalities can be solved
very easily using graphs,

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

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

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show you how this works.

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

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Now this inequality can be
solved very easily doing

00:14:18.992 --> 00:14:20.960
algebra, but it makes a good

00:14:20.960 --> 00:14:27.313
example. The first thing that we
need to do is to draw the graph

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of Y equals 2X plus 3.

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And I've got this graph here.

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

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It has a slope of two
and then intercept on

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the Y axis of three.

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On the X axis.

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

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axis Y is equal to 0?

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Above the X axis Y is greater

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

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

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

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

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

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this is the solution to the

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

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X axis as the number line.

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This technique can also be
used with modulus inequalities

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and here using a graph
can be very helpful.

00:15:53.750 --> 00:15:56.440
Take for example the inequality.

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MoD X minus two is less than 0.

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

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

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

00:16:18.236 --> 00:16:19.592
it might look a little bit

00:16:19.592 --> 00:16:24.438
strange. On the right we have
part of the graph of Y equals X

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

00:16:29.602 --> 00:16:33.706
have part of the graph of Y
equals minus X minus two.

00:16:33.706 --> 00:16:37.126
This is because the modulus
function changes the sign of

00:16:37.126 --> 00:16:38.836
X when X is negative.

00:16:40.660 --> 00:16:45.580
Again, we're looking for MoD X.
Minus two is less than 0.

00:16:46.760 --> 00:16:52.122
So we want the places where Y is
less than zero, which is between

00:16:52.122 --> 00:16:57.101
X equals minus two and X equals
+2, and again this is the

00:16:57.101 --> 00:16:58.633
solution to our problem.

00:16:59.460 --> 00:17:05.213
So we say minus two less than
X less than two.

00:17:05.920 --> 00:17:10.526
Again, we can mark this on the
graph using the X axis as the

00:17:10.526 --> 00:17:15.290
number line. Quadratic
inequalities need

00:17:15.290 --> 00:17:22.130
handling with care.
Let's solve X

00:17:22.130 --> 00:17:28.970
squared minus three
X +2 is

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greater than 0.

00:17:35.610 --> 00:17:38.734
Note that all the terms are on
the left hand side.

00:17:39.240 --> 00:17:42.867
And on the right hand side we
just had zero, exactly as with

00:17:42.867 --> 00:17:43.983
the quadratic equation before

00:17:43.983 --> 00:17:47.654
you solve it. This expression

00:17:47.654 --> 00:17:53.746
factorizes too. X minus
two X minus one.

00:17:54.530 --> 00:17:58.310
Now this is a quadratic
equation. We would simply say

00:17:58.310 --> 00:18:02.468
right X equals 2 or X equals 1
and that's it.

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But we've got a bit more work to

00:18:04.682 --> 00:18:10.120
do here. Weather this expression
is greater than zero is going to

00:18:10.120 --> 00:18:15.450
depend on the sign of each of
these two factors. We sort this

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out by using a grid.

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The points
that were

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checks equals.
X minus 2 equals 0 and X minus

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

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We put the two factors on the

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left. And their product.

00:18:43.280 --> 00:18:47.000
Now.

00:18:48.210 --> 00:18:53.700
When X is less than one, both X
minus one and X minus two are

00:18:53.700 --> 00:18:55.164
going to be negative.

00:18:56.580 --> 00:18:59.950
So when you multiply them
together, their product is going

00:18:59.950 --> 00:19:00.961
to be positive.

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When X is greater than one but

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

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But X minus two is going to be

00:19:13.096 --> 00:19:15.350
negative. So when you multiply

00:19:15.350 --> 00:19:17.386
them together. The product will

00:19:17.386 --> 00:19:23.420
be negative. Finally, when X is
greater than two, both X minus

00:19:23.420 --> 00:19:26.556
one and X minus two will be

00:19:26.556 --> 00:19:30.282
positive. And if you multiply
them together, their product

00:19:30.282 --> 00:19:31.578
will also be positive.

00:19:34.070 --> 00:19:35.798
We are looking for.

00:19:36.300 --> 00:19:39.900
X minus two times X minus one to
be greater than 0.

00:19:40.890 --> 00:19:42.620
This occurs when it's positive.

00:19:43.500 --> 00:19:47.140
And our grid shows that this
happens when X is less than one.

00:19:47.640 --> 00:19:49.866
Or when X is greater than two?

00:19:50.450 --> 00:19:52.418
So we write in our answer.

00:19:53.660 --> 00:20:00.849
Which is X is less than one
or X is greater than two.

00:20:03.950 --> 00:20:06.590
And on the number line.

00:20:07.210 --> 00:20:09.388
X must be less than one.

00:20:09.980 --> 00:20:12.536
So I put a circle to show
that it can't be 1.

00:20:14.280 --> 00:20:16.520
And X can also be greater
than two.

00:20:20.050 --> 00:20:23.976
Here's another

00:20:23.976 --> 00:20:30.116
quadratic. Minus two
X squared plus 5X

00:20:30.116 --> 00:20:35.480
plus 12 is greater
than or equal to 0.

00:20:36.570 --> 00:20:40.674
I don't like having a negative
coefficient of X squared, so I'm

00:20:40.674 --> 00:20:44.094
going to multiply this whole
thing through by minus one,

00:20:44.094 --> 00:20:47.514
remembering to change the
direction of the inequality as I

00:20:47.514 --> 00:20:48.882
do. This gives us.

00:20:49.410 --> 00:20:57.278
Two X squared minus 5X minus 12
is less than or equal to 0.

00:20:58.680 --> 00:21:04.906
This expression factorizes to 2X
plus three times X minus four,

00:21:04.906 --> 00:21:08.868
so that is less than or equal

00:21:08.868 --> 00:21:12.955
to 0. Again, I'm going to
do a grid.

00:21:18.150 --> 00:21:25.590
This factor is zero
when X is minus

00:21:25.590 --> 00:21:28.380
three over 2.

00:21:29.450 --> 00:21:31.858
This fact is zero when X is 4.

00:21:32.770 --> 00:21:35.698
We write in the two factors.

00:21:36.380 --> 00:21:39.938
And we right in the product.

00:21:43.460 --> 00:21:50.530
When X is less than minus three
over 2, both 2X plus three and

00:21:50.530 --> 00:21:53.055
X minus four and negative.

00:21:53.860 --> 00:21:56.110
So their product is positive.

00:21:57.580 --> 00:22:01.350
When X lies between minus three
over two and four.

00:22:02.540 --> 00:22:04.670
2X plus three is positive.

00:22:05.410 --> 00:22:09.590
But X minus four is still
negative, so their product

00:22:09.590 --> 00:22:10.426
is negative.

00:22:11.480 --> 00:22:16.797
When X is greater than four,
both 2X plus three and X minus

00:22:16.797 --> 00:22:18.024
four are positive.

00:22:18.590 --> 00:22:20.180
So their product is positive.

00:22:20.780 --> 00:22:26.576
We are looking for 2X plus three
times X minus four to be less

00:22:26.576 --> 00:22:28.646
than or equal to 0.

00:22:29.330 --> 00:22:33.110
In other words, this expression
has to be either 0 or negative.

00:22:34.300 --> 00:22:35.270
This occurs.

00:22:36.520 --> 00:22:41.824
When X lies between minus three
over two and four, and it can

00:22:41.824 --> 00:22:47.128
equal either number. So we have
minus three over 2 is less than

00:22:47.128 --> 00:22:51.616
or equal to X is less than or
equal to 4.

00:22:53.890 --> 00:22:56.370
And on the number line.

00:22:58.220 --> 00:23:00.326
Minus three over 2 is here.

00:23:01.760 --> 00:23:05.738
Four is here.

00:23:08.840 --> 00:23:12.040
And I've done filled
circles because we have

00:23:12.040 --> 00:23:14.040
less than or equal to.

00:23:17.260 --> 00:23:22.783
Quadratic inequalities can
also be solved graphically.

00:23:22.783 --> 00:23:30.673
Let's solve X squared minus
three X +2 is greater

00:23:30.673 --> 00:23:32.251
than 0.

00:23:34.130 --> 00:23:38.710
As with the linear equalities
inequalities, we have to plot

00:23:38.710 --> 00:23:43.748
the graph of Y equals X squared
minus three X +2.

00:23:44.650 --> 00:23:51.527
This factorizes to give Y equals
X minus one times X minus 2.

00:23:52.800 --> 00:23:54.600
The graph looks like this.

00:23:55.960 --> 00:24:01.174
Because it's a quadratic, it's a
parabola. Are U shaped curve?

00:24:02.210 --> 00:24:04.275
And it crosses the X axis where

00:24:04.275 --> 00:24:08.729
X equals 1. Because of the
factor X minus one and where

00:24:08.729 --> 00:24:12.139
X equals 2 because of the
factor X minus 2.

00:24:13.490 --> 00:24:18.963
Now we're looking for X squared
minus three X +2 to be greater

00:24:18.963 --> 00:24:23.662
than 0. This is where Y
is greater than zero. In

00:24:23.662 --> 00:24:27.042
other words, the part of
the graph that is above

00:24:27.042 --> 00:24:31.098
the X axis, which are the
two arms of the you here.

00:24:32.710 --> 00:24:36.021
This occurs where X is less than

00:24:36.021 --> 00:24:41.218
one. And where X is greater
than two, so we can write

00:24:41.218 --> 00:24:43.058
that in as our solution.

00:24:46.140 --> 00:24:52.040
And we can mark this
in using the X axis

00:24:52.040 --> 00:24:54.400
as the number line.

00:24:55.600 --> 00:25:00.222
I'll
do

00:25:00.222 --> 00:25:04.844
one
more

00:25:04.844 --> 00:25:07.155
quadratic

00:25:07.155 --> 00:25:09.466
inequality.

00:25:10.470 --> 00:25:14.040
X squared Minus X

00:25:14.040 --> 00:25:18.419
minus 6. So less than or
equal to 0.

00:25:22.680 --> 00:25:27.146
Again, we need to plot
the graph of Y equals X

00:25:27.146 --> 00:25:29.176
squared minus X minus 6.

00:25:30.360 --> 00:25:32.058
The expression factorizes.

00:25:32.830 --> 00:25:35.998
To X minus three.

00:25:36.070 --> 00:25:40.029
X +2 And the graph

00:25:40.029 --> 00:25:46.756
looks like this. Similar
to the previous

00:25:46.756 --> 00:25:48.040
graph.

00:25:49.210 --> 00:25:54.716
We have The factor X +2 the line
crosses the point at X equals

00:25:54.716 --> 00:25:58.832
minus two and for the factor X
minus three, the curve crosses

00:25:58.832 --> 00:26:00.890
the point at X equals 3.

00:26:01.750 --> 00:26:06.046
And we're looking for where X
squared minus X minus six is

00:26:06.046 --> 00:26:08.194
less than or equal to 0.

00:26:09.470 --> 00:26:14.189
In other words, why must lie on
the X axis or below it?

00:26:14.920 --> 00:26:19.509
This part of the curve and that
occurs between the points of X

00:26:19.509 --> 00:26:24.804
equals minus two and X equals 3.
So we can say that minus two is

00:26:24.804 --> 00:26:29.746
less than or equal to X, which
is less than or equal to 3.

00:26:31.260 --> 00:26:36.746
And we can put this in again
using the X axis is the

00:26:36.746 --> 00:26:40.966
number line from minus 2
using a closed circle because

00:26:40.966 --> 00:26:43.920
2 - 2 is included to +3.