Video Transcript
So in this lesson, what we’re going
to do is learn how to interpret and write simple inequalities to represent verbal
expressions. Inequalities have a lot in common
with equations because they give us information about the relationship between two
numbers or expressions, which can include variables. And we often solve them in a very
similar way. However, the difference is they are
not in fact an inequality as the name suggests.
An inequality states that the value
of one expression is greater or less than that of another. So before we come on to look at
some problems about inequalities, what we’re gonna have a look at first of all is
the notation. So while in an equation the equal
signs are used between the two expressions, when we’re looking at inequalities,
there are four different symbols that we can use.
Now, in the first of these, what we
can see is we’ve got our inequality sign between the 𝑎 and the 𝑏. We’ll see that the small end or
closed end is pointed towards the 𝑎 and the open or wide end is pointing towards
the 𝑏. Now, the wider end is always
pointing towards the thing that is greater or bigger. And the pointy end is always
pointing towards the thing that is less than or smaller.
So therefore, what we’ve got here,
we’ve got 𝑎 and then we have our sign and then 𝑏 means that 𝑎 is less than
𝑏. And then what we have are our other
inequality signs. So we’ve got 𝑎 is greater than 𝑏,
and that’s where we’ve got the 𝑎. And then we have the open side of
the inequality sign and then 𝑏. And then what in fact we have,
let’s see more signs.
What we can see is, well, they look
similar to the ones that we had before. And they are, except we have this
line underneath. And what this line means is it
means or equal to. So, for instance, we’ve got 𝑎 is
less than or equal to 𝑏 and 𝑎 is greater than or equal to 𝑏. It is worth mentioning that there’s
also another sign you might see. And I’ve shown you here because we
have 𝑎 and then we’ve got the equal sign, but with a line through it, then 𝑏. And this means that 𝑎 is not equal
to 𝑏. This is an, in fact, an inequality
sign, but it’s worth knowing cause you might sometimes see it written down. So now let’s take a look at our
first question to see how we would identify an inequality.
Which of the following is an
inequality? We have two 𝑥 minus seven is less
than or equal to five, seven minus two multiplied by 𝑥 minus one is equal to four,
or 12𝑛 squared plus four.
So what we’re trying to find out in
this question is which one of (A), (B), and (C) is in fact an inequality. Well, for it to be an inequality,
it has to be a mathematical sentence that includes one of our inequality signs. Well, first of all, we’ll quickly
remind ourselves of what the inequality signs are. We have less than, greater than,
less than or equal to, or greater than or equal to.
Well, we can see that (A) is two 𝑥
minus seven and then is less than or equal to five. So this has one of our inequality
signs. So therefore, this is in fact an
inequality. Okay, but let’s have a quick look
at the other two and see what these are. Well, if we take a look at seven
minus two multiplied by 𝑥 minus one is equal to four, well, this has an equals
sign. So therefore, this is an
equation. Well, if we look at 12𝑛 squared
plus four, this neither has an inequality sign or an equals sign or in fact an
identity sign. So therefore, this is an
expression.
So now we’re going to see with the
next questions how to write inequalities. What we’re gonna do is start with a
simple example.
Rewrite the statement one is
greater than or equal to 𝑥 using less than, less than or equal to, greater than, or
greater than or equal to inequality signs.
Well, with the first part of our
statement, we’re told that one is greater than 𝑥. But we have two signs that involve
greater than. And we know that that is when the
wider end is facing towards the one. So we’ve got one is greater than
𝑥. However, it’s the second part that
is key since it’s greater than or equal to 𝑥. And if it’s gonna be or equal to
𝑥, then it means that we must have the line underneath our inequality sign that
means or equal to as well. So therefore, we can show that
statement one is greater than or equal to 𝑥 using our inequality notation.
So now what we’re gonna do is look
at a question which gets us to represent a real-life situation with
inequalities. So for the question, what we’re
gonna need to do is read carefully everything that’s said to understand all the
information. And then we have to translate it
into a mathematical sentence.
Matthew walked for 29 minutes
before reaching the university campus. He took at least another five
minutes to reach the lecture hall. Write an inequality that represents
the amount of time it took Matthew to reach the lecture hall.
So what we’re gonna do here to help
us visualize what’s actually happening is draw a simple diagram. So what we’ve done first of all is
drawn a line, and this line represents time. And then what we’ve done is, along
that time line, marked the two points that Matthew’s trying to get to, so first of
all the university campus then second of all the lecture hall.
Well, first of all, we know that it
takes Matthew 29 minutes after walking to arrive at the university campus. And we also know that it takes him
five minutes or more to walk between the campus and the hall. Well, if it took Matthew exactly
five minutes once he was on the university campus to walk across to the lecture
hall, then we know that it’d be 29 plus five, which would be 34 minutes for his
total journey.
However, we know that it took him
at least another five minutes to walk to the lecture once he’d arrived on
campus. So it’s five minutes or more. So if we take 𝑥 to be the time for
Matthew to reach the lecture hall, well then therefore what we can do is write how
long it takes Matthew to reach the lecture hall using inequality notation as 𝑥 is
greater than or equal to 34. And that’s because as we’re told,
it took him at least another five minutes. So it could take him five minutes,
which would mean that we could say that 𝑥 was equal to 34. However, it’s at least, so it could
take him more than five minutes. So therefore, that’s why we get 𝑥
is greater than or equal to 34.
So now we’ve looked at real-life
example. What we’re gonna have a look at now
is some inequalities that have variable conditions. So what we’re actually gonna have a
look at is some double inequalities. Well, before we take a look at any
questions, let’s look at the notation itself. So here we have 𝑥 is greater than
𝑦 but less than 𝑧. And as a double inequality, it
means we have two inequality signs. So we’ve got one here that
represents the fact that 𝑥 is greater than 𝑦. But for the other one here, it’s
representing the fact that 𝑥 is less than 𝑧.
So what we’ve in fact done is put
two inequalities together. And this can actually be formed
using any combination of the inequality signs that we’ve seen in this lesson. So, for example, on this one here,
we have 𝑥 is greater than or equal to 𝑦. And that’s because we have the
extra line underneath our inequality sign. But 𝑥 is also less than 𝑧. And as we already said, this can be
seen as a combination of two inequalities. And they’re the inequalities that
𝑥 is greater than or equal to 𝑦 and 𝑥 is less than 𝑧. Okay, great, so we’ve now seen
double inequalities. Let’s take a look at a question
that asks us to actually use this skill.
Rewrite the statement 𝑦 is between
eight and 10 using our inequality notation.
So what we’ve done here is drawn a
little sketch just to help us understand what’s going on. So we know that 𝑦 is between eight
and 10. However, as we can see in the
statement, we’re told that 𝑦 is between eight and 10. So therefore, it cannot be eight or
10. So therefore, we’ve got to say that
𝑦 is gonna be greater than eight, but it’s gotta be less than 10. And that’s because if we just said
that 𝑦 was greater than eight, then it can be anything that was greater than eight
at all. But we’re told that it’s between
eight and 10. But as we said, it doesn’t include
eight or 10. So that’s why we use our compound
inequality notation or double inequality notation we’ve used here.
And we’ve shown the 𝑦 between our
eight and 10. And then we’ve used our inequality
signs showing that 𝑦 is in fact greater than eight but less than 10. However, just to remind us, if we
thought that 𝑦 could be in fact eight or 10, then what we’d have to use is an
inequality sign with a line underneath it, which means or equal to.
So now we’re gonna move on to our
final example. And what we’re gonna do in this
example is complete an inequality, given variable conditions.
Complete using inequality
signs. If 𝑏 is less than or equal to
negative five, then 𝑏 plus one blank negative four.
So first of all, in this question,
what we’re told is that 𝑏 is less than or equal to negative five. But then what we want to do is use
this to help us fill in the blank. And we’re told that 𝑏 plus one
blank negative four. Well, if we took a look at our
original inequality, 𝑏 is less than or equal to negative five, then we can see that
what in fact has happened is we’ve added one to the left-hand side. And therefore, we have to add one
to the right-hand side because 𝑏 plus one would give us our 𝑏 plus one and
negative five plus one would give us our negative four.
So therefore, our inequality sign
must stay the same because in fact what we’re gonna have is the left-hand side and
the right-hand side are gonna have the same relation with each other. So therefore, we can say that 𝑏
plus one is gonna be less than or equal to negative four.
Okay, great, so we’ve now solved
this problem. And in fact, we’ve seen a wide
range of problems, looking at what an inequality is, rewriting one, changing a
real-life situation into an inequality, completing inequalities, and even looking at
double inequalities or compound inequalities. But now what we want to do is have
a look at a summary of what we’ve had in this lesson.
Well, the first key point is what
an inequality actually is. And an inequality is a mathematical
sentence stating that the value of one expression or number is less or greater than
the value of another expression or number. Then what we have are four symbols
that we use to write an inequality. We’ve got 𝑎 is less than than 𝑏,
𝑎 is greater than 𝑏, 𝑎 is less than or equal to 𝑏, or 𝑎 is greater than or
equal to 𝑏. And the difference between it being
less than or greater than or less than or equal to or greater than or equal to is
the line underneath the inequality sign.
And then we also know that we need
to read the question carefully when translating a real-life situation into an
inequality. And we also know that there’s a
double or compound inequality. And what this is is when we combine
inequalities. So, in fact, what we actually do is
we can use any one of our four inequality signs. So, for example, we’ve got one
here, which is 𝑥 is greater than or equal to three but less than eight.
