Video: Creating Linear Inequalities with One Variable

Students in a grade four class were asked the distances in miles, 𝑑, which they travel to get to school. All of the students traveled farther than a quarter of a mile, and no one traveled farther than 3 miles. Which of the following inequalities represents the range of distances traveled to school? [A] 0.75 ≀ 𝑑 < 6 [B] 0.5 ≀ 𝑑 < 3 [C] 0.25 < 𝑑 ≀ 3 [D] 0.25 ≀ 𝑑 < 3 [E] 0.75 < 𝑑 ≀ 3

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Video Transcript

Students in a grade four class were asked the distances in miles, 𝑑, which they travel to get to school. All of the students traveled farther than a quarter of a mile, and no one traveled farther than three miles. Which of the following inequalities represents the range of distances traveled to school. (A) 𝑑 is greater than or equal to 0.75 but less than six. (B) 𝑑 is greater than or equal to 0.5 but less than three. (C) 𝑑 is greater than 0.25 but less than or equal to three. (D) 𝑑 is greater than or equal to 0.25 but less than three. And finally, (E) 𝑑 is greater than 0.75 but less than or equal to three.

So the first thing we want to do in a question like this is remind ourselves about our notation that we use for inequalities. So first of all, we have π‘₯ is less than 𝑦. And we know that it’s π‘₯ is less than 𝑦 because we’ve got the pointy end towards the π‘₯ and the open side of our inequality sign towards the 𝑦. And this open side always points towards the greater value. Then we have π‘₯ is greater than 𝑦. Then we have π‘₯ is greater than or equal to 𝑦. And the difference here is the fact we’ve got this line underneath our inequality sign. So this tells us that it can be that value as well. So π‘₯ is gonna be greater than or equal to 𝑦. So π‘₯ could be 𝑦 or any value greater. And then finally, we’ve got π‘₯ is less than or equal to 𝑦.

So for our inequality, we’ve got our variable 𝑑. And then our first bit of information is that all of the students traveled father than a quarter of a mile. So therefore, we know that 𝑑 is greater than 0.25. And that’s because 0.25 is the same as a quarter. And we know that it’s just 𝑑 is greater than not greater than or equal to because it says all of the students traveled father than a quarter of a mile. So they actually traveled more than a quarter of a mile. And then the next bit of useful information we get from question is that no one traveled farther than three miles. So therefore, we can say that 𝑑 is gonna be less than or equal to three.

You might think, well, why is it less than or equal to? Well, it’s because we’re told that no one traveled farther than three miles, which means that they could have traveled three miles. So therefore, we have to include the β€œor equals to” parts of our inequality. So therefore, if we take a look at our possible answers, this is the same as answer (C). So we can say that the correct inequality for our situation is 𝑑 is greater than 0.25 but less than or equal to three.

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