### Video Transcript

For a real number π₯, determine
whether π₯ is positive or negative in each of the following cases. Firstly, π₯ is equal to negative
seven; secondly, π₯ is greater than two; and thirdly, negative three is greater than
π₯.

We begin by recalling that positive
numbers lie to the right of zero on a number line, whilst negative numbers lie to
the left of zero. We can therefore determine the
signs of π₯ in each case by considering the possible positions of π₯ on a number
line.

In the first part of the question,
we are told that π₯ is equal to negative seven. We know that negative seven will
lie to the left of zero as shown. And we can therefore conclude that
when π₯ is equal to negative seven, π₯ is negative. In the second part of the question,
we are told that π₯ is greater than two. And this means that π₯ lies to the
right of two on a number line. Since π₯ lies to the right of two
and two lies to the right of zero, we can conclude that π₯ lies to the right of zero
and is therefore positive.

In the final part of the question,
we have negative three is greater than π₯, which can also be read as π₯ is less than
negative three. Marking negative three on our
number line, we know that π₯ lies to the left of this. And since all values to the left of
negative three are negative, we can conclude that if negative three is greater than
π₯, π₯ is negative.

Now that we can compare any two
real numbers, we can use this to order any list of any real numbers. This can be done in one of two
ways: either from least to greatest, which is called ascending order, or from
greatest to least, which is called descending order. A list of real numbers π sub one,
π sub two, and so on, up to π sub π is said to be in ascending order if π sub
one is less than π sub two, and so on, which is less than π sub π. In other words, the numbers are
getting larger. In the same way, a list of real
numbers π sub one, π sub two, and so on, up to π sub π is said to be in
descending order if π sub one is greater than π sub two, and so on, which is
greater than π sub π. In this case, the numbers are
getting smaller.