Question Video: Verifying Inequality Statements Involving Rational Numbers

Which of the following inequalities is true? [A] −1.4 < −1.32 < 0 [B] 0 < −1.32 < −1.4 [C] 0 < −1.4 < −1.32 [D] −1.4 < 0 < −1.32 [E] −1.32 < −1.4 < 0

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Video Transcript

Which of the following inequalities is true? (A) Negative 1.4 is less than negative 1.32 which is less than zero. (B) Zero is less than negative 1.32 which is less than negative 1.4. (C) Zero is less than negative 1.4 which is less than negative 1.32. (D) Negative 1.4 is less than zero which is less than negative 1.32. Or (E) negative 1.32 is less than negative 1.4 which is less than zero.

Now, in each of our possible answers, what we have is a double inequality because we have two inequality signs. And what we also have is the same three numbers — negative 1.4, negative 1.32, and zero. So now if we consider these three numbers, we could find the result or the answer to this question by working out which order they’re going. And we’re gonna start by putting them in ascending order, which means we want to put the smallest value first.

And the reason we want to put in ascending order, so the smallest value first, is because if we look at our inequalities, we can see that all of the inequality signs we have are less than. So it means that the smallest answer or the least answer is gonna be on the left-hand side or the least value. And the reason we know it’s less than is because the pointy end is pointing towards the number on the left and the wide end is pointing towards the number on the right. So we’re saying that the number on the left is less than the number on the right.

So, first of all, what we can say is that we’re gonna have the greatest value is going to be zero. So we’re gonna start with the greatest value being zero. And we know that because the other two values are negative. So therefore they must be less than zero. Now in order to order the other values correctly, which is negative 1.4 and negative 1.32, there’s a little trick we can use to help us make sure that we don’t make any common mistakes. And that little trick is to add a zero after the four in the negative 1.4.

So now what we have is negative 1.40 and negative 1.32. And the reason we do that is to avoid a common mistake. And that is, if we had 1.32, often people think, “Oh! It’s greater than 1.4.” And the reason they think that is because they think it’s 1.32 and the other one is just 1.4. But this isn’t the case because, in fact, if we think about what we’ve got, we’ve got a unit which is the same in each. But then in the tenths column, we have three-tenths on the left-hand side but we have four-tenths on the right-hand side. So therefore this would be bigger.

So see here I’ve actually done this without negatives because I didn’t want to confuse things. But that is one of the common mistakes. So that’s why I’ve added the zero after the four. So we can sort of see that it looks like 40. So therefore we can say that the least value is gonna be negative 1.4. And then we’re gonna have negative 1.32. And, again, this is another mistake that we need to make sure that we avoid here, that the least value is negative 1.4. Because a common mistake would be to have these two the other way around. And that’s because they think, “Well, 1.4 is a bigger number than 1.32.”

However, because they’re negative, it means that negative 1.4 is more negative or is less than negative 1.32. So now what we need to do is add in our inequality signs. So we’ve got negative 1.4 is less than negative 1.32 which is less than zero. Well, if we check this, this is the same as our answer (A). So therefore we can say that the inequality which is true is answer A because it’s negative 1.4 is less than negative 1.32 which is less than zero.

Now, quickly, what we’re gonna do is check why the other inequalities are not true. What if we look at answer (B)? We can see it’s in descending order. So this is not true because what we’ve got here is that zero is less than negative 1.32 which is less than negative 1.4. Well, zero cannot be less than a negative number. And, again, in answer (C), we have zero being less than a negative number, which it cannot be. So this is incorrect.

In answer (D), we have zero being greater than one negative number but less than another negative number, which again cannot be true. And, finally, in answer (E), we could see that the negative 1.32 and the negative 1.4 are the wrong way round because the common mistake which we mentioned has been made. We’ve said that negative 1.4 is greater than negative 1.32 which is not. So therefore we can confirm the correct answer is answer (A).

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