Question Video: Solving Two-Step Double Linear Inequalities over the Set of Natural Numbers Mathematics

Find the solution set of −6 > 2𝑥 + 10 > −16, where 𝑥 ∈ ℕ.

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Video Transcript

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.

Well, first of all, we want to have a look at what the natural numbers are. And the natural numbers are also known as the counting numbers. And these are positive integers. 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. It won’t affect this question, but it is sometimes seen within or not in the natural numbers.

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. And when we do that, what we’re gonna get is two 𝑥 is less than negative 16 but greater than negative 26. 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.

Well, at this point, we might think, “Wait! We can pick out what our value’s gonna be.” Because if we want to satisfy the inequality, we need to choose the values of 𝑥 that will do that. And what we’d get is negative nine, negative 10, negative 11, and negative 12.

So is this going to be our answer? Well, no, it’s not. And the reason is this last bit here, which we spoke about at the beginning. And that is 𝑥 is in the set of natural numbers. So therefore, 𝑥 needs to be a positive integer. Well, we can see that all of the solutions we’ve found for the inequality are, in fact, negative integer values. 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.

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