Video: AF5P2-Q24-508141650673

𝑦 is greater than or equal to nine and less than 13. Circle the inequality that shows this. [A] 9 < 𝑦 ≤ 13 [B] 9 ≤ 𝑦 < 13 [C] 9 < 𝑦 ≥ 13 [D] 9 ≥ 𝑦 < 13

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Video Transcript

𝑦 is greater than or equal to nine and less than 13. Circle the inequality that shows this. We’ve got 𝑦 is greater than nine less than or equal to 13. 𝑦 is greater than or equal to nine and less than 13. 𝑦 is greater than nine or greater than or equal to 13. Or 𝑦 is less than or equal to nine and less than 13.

So, the first thing we need to do in this question is recap our inequality signs. So, here, we got 𝑎 and then we’ve got our inequality sign and 𝑏. We could see that the larger end of the inequality sign is facing the 𝑎 and the smaller, or pointy end, is facing the 𝑏. So, this means 𝑎 is greater than 𝑏 because the wider end is always pointing towards the value that is greater.

So therefore, this inequality is gonna be 𝑎 is less than 𝑏 because the pointy end is towards 𝑎 and the open end towards 𝑏. However, it means 𝑎 is less than or equal to 𝑏. And this is because we have the line underneath inequality sign. And it can either be a diagonal line or horizontal line, depending on how it’s written. But it means the same thing. It means it can be or equal to. So, it means that 𝑎 is not only less than 𝑏, 𝑎 could be 𝑏. So, now that we know what our inequality signs mean, let’s go on and solve the problem.

So, to help us do this, what I wanna do is draw a number line to help us visualize what’s going on. Well, first of all, we’re told that 𝑦 is greater than or equal to nine. So, if I’m going to represent this on our number line, what I’m gonna do is put a colored-in dot above the nine. And it’s colored-in because colored-in means or equal to as well as greater than or less than. Well, we know that it’s going to be greater than or equal to nine. So therefore, we’re looking to the right-hand side of this dot. But we’ll draw the line in in a second because we’ve also got another parameter.

We’re also told that 𝑦 is less than 13. So therefore, this time I draw an open circle above 13. That’s because it’s less than. It’s not less than or equal to. It’s just less than. So, we’ve been told that it’s greater than or equal to nine. So, it’s gonna be right of our dot over nine that’s colored-in. But less than 13, so it’s going to be to the left of our open dot. So therefore, 𝑦 is gonna lie in the region that’s shown here by the two dots and the blue line between them.

Well, therefore, if we wanna think about this using symbols, we’ve got 𝑦 and then it’s greater than one value and less than another value. But it’s greater than or equal to one value and less than another value. So, I’ve added the line underneath the inequality sign. So, now we know that 𝑦 is greater than or equal to one value and less than another value.

And then, we can see that these values are nine and 13. And we can get them from the question or by looking at our number line. So therefore, the correct inequality is nine is less than or equal to 𝑦 which is less than 13. Which can be said also as 𝑦 is greater than or equal to nine but less than 13. So therefore, the second answer is the correct answer. And I’ve circled this. And what we can do is have a little look at the other answers, just to see what they’d look like on our number line.

Well, if we have a look at the first inequality, we’ve got 𝑦 is greater than nine. So, we’d have an open circle because it’s just greater than nine not greater than or equal to. And we got that open circle above the nine. So, we’re looking to the right-hand side because 𝑦 is greater than it. It’s then less than or equal to 13. So, we put a colored-in dot above 13. So therefore, 𝑦 would lie within this region I have marked here in orange. The only difference between this region and the region marked in blue, which is the correct region, is the fact that we’ve got or equal to with the 13 and just greater than with the nine.

Again, with the third inequality, we’d have 𝑦 is greater than nine. So, we put an open circle above the nine. But we’re also told that 𝑦 is greater than or equal to 13. So, we’d have our colored-in dot above 13. Well, in fact, for this inequality, the first part would be irrelevant because 𝑦 is greater than nine, but also 𝑦 is greater than or equal to 13. Well, therefore, we can ignore the first open dot because it’s the second part which takes precedence because we need 𝑦 to be greater than or equal to 13. So, we draw an arrow to the right of 13.

And then, finally, for the final inequality, we’ve got 𝑦 is less than or equal to nine. So, we draw a closed dot over the nine. But this time we’re looking to the left of it. But we’re also told that 𝑦 is less than 13. So, this time we’d have an open dot over the 13. But we can ignore this because 𝑦 is less than or equal to 9 take precedence because nine is less than 13. So, what we do is we draw an arrow to the left of our colored-in dot at nine to represent our inequality. So therefore, the final answer is that the inequality that shows that 𝑦 is greater than or equal to nine and less than 13 is the one that I’ve circled.

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