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Find the solution set of two 𝑥 plus 10 is less than negative six but greater than negative 16, where 𝑥 is in the set of natural numbers.
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Well, first of all, we want to have a look at what the natural numbers are.
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And the natural numbers are also known as the counting numbers.
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And these are positive integers.
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What’s worth mentioning, which is quite interesting, is the number zero because sometimes zero is included within the natural numbers, but some arguments say that zero is not.
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It won’t affect this question, but it is sometimes seen within or not in the natural numbers.
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So the first thing we want to do is subtract 10 from each section of our inequality cause we’re gonna deal with that double inequality as one whole unit together.
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And when we do that, what we’re gonna get is two 𝑥 is less than negative 16 but greater than negative 26.
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And then if we divide each part of our inequality by two, because we’ve got two 𝑥 and we want to find one 𝑥, then we’re gonna get 𝑥 is less than negative eight but greater than negative 13.
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Well, at this point, we might think, “Wait!
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We can pick out what our value’s gonna be.”
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Because if we want to satisfy the inequality, we need to choose the values of 𝑥 that will do that.
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And what we’d get is negative nine, negative 10, negative 11, and negative 12.
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So is this going to be our answer?
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Well, no, it’s not.
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And the reason is this last bit here, which we spoke about at the beginning.
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And that is 𝑥 is in the set of natural numbers.
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So therefore, 𝑥 needs to be a positive integer.
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Well, we can see that all of the solutions we’ve found for the inequality are, in fact, negative integer values.
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So therefore, what we can say is that the solution set of two 𝑥 plus 10 is less than negative six but greater than negative 16 when 𝑥 is in the set of natural numbers is going to be the null or empty set because there aren’t any values that satisfy this.