Video Transcript
Find the solution set of the inequality negative five is less than or equal to a fifth 𝑥 minus two, which is less than two, in a set of real numbers. Give your answer in interval notation.
So when we have a double inequality like this, there’s a couple of ways in which we could solve it. The first way would be to deal with it altogether as one. The second way would be to take it into two parts. So, for our first method, what we’re gonna do is deal with it all as one. So in this method, the first thing we’re gonna do is add two to each section of our inequality. So when we do this, what we’re gonna get is a fifth 𝑥 is greater than or equal to negative three but less than four.
And now, if we’ve got a fifth 𝑥, then we want to find out what 𝑥 is. So we’re trying to solve our inequality for 𝑥. So what we’re gonna do now is multiply each section by five. So what this is gonna leave us with is 𝑥 is greater than or equal to negative 15 but less than 20. So, using this method, great, we’ve solved our inequality. So we’ll have a quick look at our second method.
Well, for our second method, what we’re gonna do is split up our inequality into two parts. So we’re gonna have a left-hand part and a right-hand part. So if we deal with the left-hand part first, what we’ve got is a fifth 𝑥 minus two is greater than or equal to negative five. Now what we’re gonna do is add two to each side of the inequality. And when we do this, we have a fifth 𝑥 is greater than or equal to negative three. And then all we need to do is multiply through by five, which is gonna give us 𝑥 is greater than or equal to negative 15. So that’s the left-hand part dealt with. So now we can deal with the right-hand part of our inequality.
Well, for the right-hand part of our inequality, we’ve got a fifth 𝑥 minus two is less than two. So once again, if we add two to each side of the inequality first, we’re gonna get a fifth 𝑥 is less than four. And then, if we multiply through by five, we’re gonna get 𝑥 is less than 20. Okay, great. So now we’re gonna just pull both parts together. And when we do that, we’re gonna get the same answer we got with our first method. And that is 𝑥 is greater than or equal to negative 15 but less than 20.
So great, have we finished the question? Well, no, and that’s because the question wants our answer in interval notation. So in order to write an interval notation, we can remind ourselves what that is. Well, when we’re writing an interval notation, if we use a square bracket, this means or equal to. So it could be greater than or equal to or less than or equal to. And if we use a parenthesis, so a rounded parenthesis or bracket, then we can see this will be greater than or less than. So therefore, in our interval notation, our answer is square bracket negative 15 comma 20 and then parenthesis, which means that 𝑥 is greater than or equal to negative 15 but less than 20.
