Video: Writing the Solution Set of Linear Inequalities with the Unknown on Both Sides over the Integer Numbers Set

Given that 𝑥 ∈ ℤ, write the solution set of 2𝑥 − 6 ≤ 𝑥 − 1.

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Video Transcript

Given that 𝑥 is an integer, write the solution set of two 𝑥 minus six is less than or equal to 𝑥 minus one.

First of all, before we actually go ahead and solve this problem, we just want have a look at this notation here. And we wanna see what they actually mean. Well, a giant ∈ actually means belongs to. So it’s telling us that 𝑥 belongs to something. And, well, 𝑥 belongs to and it’s like a giant ℤ. And this means actually the integers and they can be positive or negative up to infinity. So therefore, we can say that this means that 𝑥 belongs to the set of integers positive or negative up to infinity.

Okay, great! So now, we know the parameters. Let’s go on and solve the inequality. So our inequality states that two 𝑥 minus six is less than or equal to 𝑥 minus one. So the first thing we’re gonna do here is actually we’re going to add six to each side cause we’re gonna solve that as we would solve an equation. So this gives us that two 𝑥 is less than or equal to 𝑥 plus five.

Okay, so now our next step is going to be subtracting 𝑥 from each side. So this gives us an answer that says that 𝑥 is less than or equal to five. Okay, great! So now we can solve our inequality. But if we look back to the question, it’s actually asked us to write the solution set. So let’s have a go at doing that.

Well, I’ve drawn a little sketch here of a number line and actually shown on that so you can see that it’s actually we want all the values that are less than and including five. And I put that because I’ve got the colored-in dot that says and including five. So we want all of these values.

So now, we need to actually write them as a solution set. So therefore, we can say the solution set will be as we’ve shown here. So we’ve got five, four, three, et cetera. And that’s if you think again what the parameters were, we said that 𝑥 belongs to the set of integers. And they can be positive or negative up to infinity. So that’s why we get the final solution set written here.

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