Find the solution set of 𝑥 minus 12 is greater than seven but less than 11, where 𝑥 is in the set of natural numbers.
Well, the first thing we wanna take a look at is this bit of notation here, and this is set notation. And it means the natural numbers. And what the natural numbers are is sometimes known as the counting numbers. They are the positive integers, so one, two, three, four, etc. So, if we take a look at our inequality, it’s called a double inequality. And it’s called that because we have two inequality signs. And what we remember about our inequality signs is that the open end always points to the greater value. So, that’s why we read it as 𝑥 minus 12 is greater than seven. And that’s because the 𝑥 minus 12 is on the right-hand side and next to the open side of our inequality sign. But it’s less than 11, and again it’s because the open side is pointing towards the 11.
So, now, to solve our double inequality, what we want to do is the same thing to each part of our inequality, and that is add 12. And we’re gonna do that because we want to get the 𝑥 on its own. And when we do that, what we get is that 𝑥 is greater than 19 but less than 23.
So, have we solved the problem? Have we finished there? Well, the answer is no. And the reason it’s no is because what we’re asked to do is find the solution set. Well, therefore, our solution set is gonna be the values of 𝑥 that will actually satisfy our inequality. And they’re going to be 20, 21, and 22. But you might ask yourself, well, why is it not 19 and 23? Well, the reason we know that it doesn’t include 19 or 23 is cause if we look at our inequality signs, they don’t have a line underneath. So, therefore, it just means greater than or less than. If they had a line underneath, then it would be greater or less than or equal to. So, in that case, it would actually include those values themselves.
Another way we could’ve shown this is by looking on a number line. Because if we look at a number line, what we’ve got here is a number line showing the inequality 𝑥 is greater than 19 but less than 23. And we know it means that because if we see our circles on our number line, they’re not colored in. If they were colored in, then this would mean greater than or less than or equal to. And we can see from within our number line that the only three possible solutions are in fact 20, 21, and 22.
So, therefore, our solution set of 20, 21, and 22 is the correct set.