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