Video: Finding the Solution Set of Two-Step Linear Inequalities over the Set of Natural Numbers

Find the solution set of 3π‘₯ βˆ’ 7 < βˆ’4 given that π‘₯ ∈ β„•.

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Video Transcript

Find the solution set of three π‘₯ minus seven is less than negative four given that π‘₯ is a natural number.

Before trying to solve this inequality, it is worth recalling what a natural number is. The natural numbers are the nonnegative integers, for example, zero, one, two, three, four, and so on. We will now solve the inequality given and find which of these numbers satisfy the inequality. Our inequality states that three π‘₯ minus seven is less than negative four. We can solve this using inverse operations. Our first step is to add seven to both sides of the inequality, as the opposite of subtracting seven is adding seven. Negative four plus seven is equal to three, so three π‘₯ is less than three.

Our second and final step is to divide both sides of this new inequality by three. Three π‘₯ divided by three is equal to π‘₯, and three divided by three is equal to one. The solution to our inequality is π‘₯ is less than one. This answer can be written in interval notation, where π‘₯ can take any value less than one down to negative ∞. In this question, however, we’re asked for the solution set. π‘₯ also needed to be a natural number. The only natural number that is less than one is zero. This means that the solution set of the inequality three π‘₯ minus seven is less than negative four where π‘₯ is a natural number is the set containing the number zero.

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