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.