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

For a real number 𝑥, determine whether 𝑥 is positive or negative in each of the following cases. 1) 𝑥 = −7. 2) 𝑥 > 2. 3) −3 > 𝑥.

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Video Transcript

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

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 𝑥 in each case by considering the possible positions of 𝑥 on a number line.

In the first part of the question, we are told that 𝑥 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 𝑥 is equal to negative seven, 𝑥 is negative. In the second part of the question, we are told that 𝑥 is greater than two. And this means that 𝑥 lies to the right of two on a number line. Since 𝑥 lies to the right of two and two lies to the right of zero, we can conclude that 𝑥 lies to the right of zero and is therefore positive.

In the final part of the question, we have negative three is greater than 𝑥, which can also be read as 𝑥 is less than negative three. Marking negative three on our number line, we know that 𝑥 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 𝑥, 𝑥 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 𝑎 sub one, 𝑎 sub two, and so on, up to 𝑎 sub 𝑛 is said to be in ascending order if 𝑎 sub one is less than 𝑎 sub two, and so on, which is less than 𝑎 sub 𝑛. In other words, the numbers are getting larger. In the same way, a list of real numbers 𝑎 sub one, 𝑎 sub two, and so on, up to 𝑎 sub 𝑛 is said to be in descending order if 𝑎 sub one is greater than 𝑎 sub two, and so on, which is greater than 𝑎 sub 𝑛. In this case, the numbers are getting smaller.