Video: Determining the Solution Set of Linear Inequalities with Integer Numbers

Find the solution set of 4 < 𝑥 + 1 ≤ 9, where 𝑥 ∈ ℤ.

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Video Transcript

Find the solution set of four is less than 𝑥 plus one is less than or equal to nine, where 𝑥 is an element of the integers.

Let’s start this question by reminding ourselves that this symbol that looks like a 𝑧 signifies integers, which are whole number values. So let’s have a look at our inequality four is less than 𝑥 plus one is less than or equal to nine. If we look at the first part of our inequality four is less than 𝑥 plus one, this is the same as saying 𝑥 plus one is greater than four. Let’s see if we can represent our inequality on a number line.

We could illustrate this with a small hollow circle over four and a filled-in circle over nine. The four has a hollow circle since we know that our inequality cannot be equal to four and the nine is filled in because our inequality can be equal to nine. In our inequality — and in our diagram — we’re showing the values for 𝑥 plus one.

To find the solution set, we need to work towards finding the values for just 𝑥 by itself. And to do this, we would need to subtract one from our inequalities. This would give us a value that’s three is less than 𝑥 and that 𝑥 is less than or equal to eight. This means that our solution for 𝑥 must be greater than three and also less than or equal to eight.

So if we were to write a list of values that 𝑥 could take, then it wouldn’t include three since we know that our 𝑥 must be greater than three and it has to be a whole number integer value. Therefore, the first value in our list would be four. We know that five, six, and seven would also be in our range. And we can also include the value eight since 𝑥 can be less than or equal to eight.

So therefore, the answer for our solution set is four, five, six, seven, and eight.

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