Video: AQA GCSE Mathematics Foundation Tier Pack 4 • Paper 3 • Question 1

Circle the correct inequality. [A] −6 ≥ −5 [B] −4 ≤ 3 [C] 2 < −1 [D] −1 > 0

04:35

Video Transcript

Circle the correct inequality. Negative six is greater than or equal to negative five, negative four is less than or equal to three, two is less than negative one, or negative one is greater than zero.

So before we start the question, let’s just recall what all these different signs mean. 𝑎 is less than 𝑏 means 𝑎 is strictly less than 𝑏. And we can represent that on a number line. Here’s 𝑏, and the values that are strictly less than 𝑏 are to the left of 𝑏. And we’ve used a hollow circle to represent the fact that the value 𝑏 is not included in that region. And if that inequality is true, then the value of 𝑎 is anywhere to the left of 𝑏.

And finally, remember that the smaller end of the symbol goes with the smaller number and the larger end of the symbol goes against the larger number. We read this as 𝑎 is strictly greater than 𝑏 or just 𝑎 is greater than 𝑏. And we represent it like this on the number line. And the value of 𝑎 is greater than the value of 𝑏, can’t be equal to 𝑏, so 𝑎 is going to be to the right of 𝑏 on that number line.

And when we have this symbol, we say 𝑎 is less than or equal to 𝑏. And when we represent that on our number line, 𝑎 could be in line with 𝑏 or anywhere to the left of 𝑏. And we use a solid dot on that diagram to represent the fact that it could be equal to 𝑏. The value 𝑏 is in the region.

And finally, we have 𝑎 is greater than or equal to 𝑏. 𝑎 could be to the right of 𝑏 on the number line or it could be directly in line with 𝑏. 𝑎 could be equal to 𝑏.

Okay then, let’s run through our inequalities and see which one we think is correct. Let’s start with negative six is greater than or equal to negative five. So to work out whether this is a correct inequality or not, we need to work out is negative six in the region of numbers which is greater than or equal to negative five. Well, negative five is here. And the region including all the numbers which are greater than or equal to negative five is here. I’ve done a solid dot because it is allowed to be negative five if it’s in that region.

But look, negative six is over here. It’s outside the region. That inequality is not correct. Negative six is not in the region of numbers which are greater than or equal to negative five. So let’s put a little cross by that inequality and move on to the next one.

Now the second one: negative four is less than or equal to three. Is negative four in the region of numbers which are less than or equal to three? Well, three is here, and the numbers which are less than or equal to three are here. Now the dot was solid because three is included in the region. We are allowed to be equal to three. Now negative four is here. That is in the region. So this is a correct inequality. And the question told us to circle the correct inequality. But let’s go on and check the other two anyway.

Two is less than negative one. Is the number two in the region of numbers that are less than negative one? Well, negative one is here, and the numbers that are less than negative one are off in this direction. And we’ve got a hollow dot because the number negative one isn’t included in the region. We’re not allowed to be equal to negative one. Now the number two is over here. That’s not in the region, so this is an incorrect inequality. Two is not less than negative one.

And finally, negative one is greater than zero. Is negative one in the region of numbers which are greater than zero? These are all the numbers which are greater than zero. Hollow dot because zero wasn’t included. And negative one isn’t in that region, so it’s an incorrect inequality. And we’ve confirmed that the only correct inequality is negative four is less than or equal to three.

Now just before we go, let’s take a moment to address a common issue that lots of people have with this question. Negative four is less than or equal to three. But negative four is not equal to three. So that looks wrong. But looking at the question in this way and thinking about less than or equal to three as being a region, we can see that negative four is in the region of numbers which are less than or equal to three. It doesn’t have to be equal to three to be in that region. So we can confidently say that the only correct inequality out of those four is negative four is less than or equal to three.

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