Video: Solving Double Inequalities with Variables on Three Sides

Find all values of 𝑥 that satisfy −20 − 𝑥 ≤ 3𝑥 + 2< 18 − 2𝑥. Write your answer as an interval.

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Video Transcript

Find all values of 𝑥 that satisfy the inequality negative 20 minus 𝑥 is less than or equal to three 𝑥 plus two, which is less than 18 minus two 𝑥. Write your answer as an interval.

To start off with, we’re gonna solve this problem in two parts: this first inequality on the left and then this second inequality on the right. Okay, so we can get straight on and solve our first inequality. So we’ve got negative 20 minus 𝑥 is less than or equal to three 𝑥 plus two. So we’re gonna solve this inequality, but remembering the same way that we’d solve an equation. So the first thing we’re gonna do is actually we’re gonna subtract two from each side, which gives us negative 22 minus 𝑥 is less than or equal to three 𝑥.

Okay, now our next step, we’re actually gonna add 𝑥 to each side, which gives us negative 22 is less than or equal to four 𝑥. Okay, now we can solve this inequality because what we’re gonna do is divide three by four, which gives us negative 22 over four is less than or equal to 𝑥. Just to tidy this up, we’re gonna simplify this fraction on the left-hand side. So we can say that negative 11 over two is less than or equal to 𝑥. Great! So that’s our first inequality solved.

Now, we’re gonna go on and solve the right-hand side, so our second inequality. And this second inequality tells us that three 𝑥 plus two is less than 18 minus two 𝑥. Okay, again solving it in the way that we’d solve an equation, so the first thing we’re gonna do here is actually we’re going to subtract two from each side, which gives us three 𝑥 is less than 16 minus two 𝑥. So now, we’re gonna add two 𝑥 to each side. So we get five 𝑥 is less than 16.

And then finally, to solve our inequality, we divide both sides by five, which gives us that 𝑥 is less than 16 over five. So great! We’ve solved both inequalities. Okay, so let’s now bring them together to see what our possible values of 𝑥 could be.

If we combine our solutions, we can see that 𝑥 is greater than or equal to negative 11 over two and less than 16 over five. Okay, so let’s show what this actually means. So if we looked at that on a number line, we could see it like this, where we actually got a filled-in dot above our negative 11 over two. And that’s because it is less than or equal to. And then, we’ve actually got an open circle over our 16 over five. And that’s because it’s just saying that this is greater than 𝑥.

Okay, so we’ve now solved it. And we know where our values would lie. But have we finished? Well, no, because actually our question asks us to write our answer as an interval. So in interval notation, this would be our solution with a bracket on the left-hand side. And this is because it is less than or equal to. And again it’s the same as when we’re showing on the number line. We have a different notation there and we have a coloured in circle.

But when we’re using the interval notation, we actually use a bracket for that. And on the right-hand side, we have the parenthesis. And we use this parenthesis on the right-hand side because as we can see here it is just 𝑥 is less than 16 over five, not less than or equal to. And again, we got the notation there of the open circle as well. So we’ve now arrived at our final answer. And it’s written as an interval.

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