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Question Video: Ordering Rational and Irrational Numbers Mathematics • 8th Grade

For a real number π‘₯, determine whether π‘₯ is positive or negative in each of the following cases. 1) π‘₯ = βˆ’7. 2) π‘₯ > 2. 3) βˆ’3 > π‘₯.

03:06

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‘Ž sub one, π‘Ž sub two, and so on, up to π‘Ž sub 𝑛 is said to be in ascending order if π‘Ž sub one is less than π‘Ž sub two, and so on, which is less than π‘Ž sub 𝑛. In other words, the numbers are getting larger. In the same way, a list of real numbers π‘Ž sub one, π‘Ž sub two, and so on, up to π‘Ž sub 𝑛 is said to be in descending order if π‘Ž sub one is greater than π‘Ž sub two, and so on, which is greater than π‘Ž sub 𝑛. In this case, the numbers are getting smaller.

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