Video Transcript
Which of the following inequalities have
been represented on the number line? We’ve got (A) 𝑥 is less than or equal to
negative one or 𝑥 is greater than two. (B) 𝑥 is less than negative one or 𝑥 is
greater than two. (C) 𝑥 is less than or equal to negative
one and 𝑥 is greater than two. Or (D) 𝑥 is greater than or equal to
negative one but less than two. Or (E) 𝑥 is greater than negative one
but less than two.
So, first of all, what we can see is that
our inequality on a number line has two regions. So, we have the region on the left and
the region on the right. So, first of all, we’re gonna have a look
at the region on the left. Well, the region on the left, we can see
that we’ve got an arrow pointing to the left. And what this means is less that because
it means anything less than a value cause it’s moving down the number line. And then, we can quickly remind ourselves
of the number-line notation because we’ve got a colored-in dot. So, what this means is it’s going to be
greater or less than or equal to because an open dot would mean just greater or less
than.
So, therefore, we can say that the
left-hand part of our inequality on our number line is represented by 𝑥 is less than or
equal to negative one. That’s cause our colored-in circle or dot
is on negative one, and then we’ve got an arrow to the left. So, then, if we take a look at the
right-hand side, then what we’ve got is an open circle. So, this means that it’s gonna be greater
or less than, that it’s not gonna be or equal to. And then, we’ve got an arrow to the
right, which means greater than. So, therefore, the right-hand side of our
inequality on the number line is represented by 𝑥 is greater than two.
Okay, so great. So now, let’s give our final answer,
where we can say that 𝑥 is less than or equal to negative one or 𝑥 is greater than
two. And it’s this word “or” which is key
because it’s telling us that 𝑥 can be less than or equal to negative one or 𝑥 can be
greater than two. It cannot be and because you cannot have
a value that is less than or equal to negative one and greater than two. That’s why answer (C) would be ruled
out. Because here you can see clearly that the
word “and” is used; however, what we want is the word “or” to be used. So, therefore, the correct answer would
be answer (A), as this is the inequality that’s been represented on our number line.
Okay, great. So, now, let’s have a quick look at the
other answers that we’ve been given and why they may be misconceptions and why they’re
incorrect. Well, if we take a look at answer (B),
the reason answer (B) is incorrect — and this is a common mistake — is that if we look at
the first inequality notation, we haven’t got a line underneath our inequality. Which means, therefore, that it’ll just
be 𝑥 is less than negative one. Well, if it was 𝑥 is less than negative
one, we’d have an open circle on the negative one. But, in fact, we’ve got a colored-in or
closed circle there. So, therefore, it’s less than or equal
to.
And if we look at (D) and (E), these both
are incorrect because (D) and (E) are both — we’ve got here double inequalities. And what these would mean is we’d
actually have a region between two values. And the way we’d represent these on a
number line is by using two other bits of notation. So, for instance, if we had (D), this
would be a colored-in dot above our negative one or on our negative one. And then, we’d have an open circle or dot
on our two. And then, we’d have a line joining
between them. And what this would mean is that 𝑥 could
take any value that was between negative one and two, but also including negative one, but
not including two. And for (E), we’d just have an open
circle on negative one and an open circle on two and then a line joining them.
