Video Transcript
Find the solution set of 𝑥 minus seven is greater than or equal to negative four but less than or equal to negative one, where 𝑥 is in the set of integers.
So the first thing we’re gonna have a quick look at is at some set notation that we’ve got here. So this Z-looking shape means that actually 𝑥 is gonna be in the set of integers. So that means that it’s gonna be a whole number, but it can be negative or positive. So what we have here is a double inequality.
Now, before we solve it, let’s remind ourselves what our inequality signs mean. Well, first of all, what we’ve got here is 𝑥 is greater than 𝑦. And we know that one thing is greater than something else when we look at the inequality sign and the open end is pointing towards that. So we got the open end pointed towards the 𝑥 so that the 𝑥 is greater than 𝑦. Then our second one shows less than. We know that 𝑥 is less than 𝑦 because the pointy end is pointing towards our 𝑥. Then, next, we’ve got 𝑥 is greater than or equal to 𝑦. And we know that it’s or equal because we have this extra line underneath our inequality sign. And then, finally, we’ve got 𝑥 is less than or equal to 𝑦. And we know it’s or equal to because again we’ve got this line underneath our inequality sign.
Okay, great, so we’ve reminded ourselves of the notation. So now let’s get on and solve our double inequality. So if we take a look at our inequality, then what we’ve got in the middle section is 𝑥 minus seven. What we want to find out is the solution set of 𝑥. So what we’re gonna do is add seven. So we’re gonna add seven to each section of our inequality. So when we’ve done this, what we’re gonna get is 𝑥 is greater than or equal to three but less than or equal to six. And that’s cause negative four add seven is three and negative one add seven is six.
Now, to help us decide what our solution set is going to be, what we’ve done is sketched our inequality on a number line. So what we’ve got here is a number line that shows that 𝑥 is greater than or equal to three but less than or equal to six. And just so you know how this works, if you’re doing a number line, the way that we show that it’s greater than or equal to or less than or equal to is by coloring in the circles. If these were open circles, then it’d just be either greater than or less than. And because it can be greater than or less than, what it means is that we can include our three and our six. So therefore, I’ve circled here each of the possible solutions of our inequality: three, four, five, and six.
So therefore, we can say that the solution set of 𝑥 minus seven is greater than or equal to negative four but less than or equal to negative one is three, four, five, and six and remembering this is where 𝑥 is in the set of integers because if this was not the case, then we’d have a lot more values because it could be anything in between these values and it’d still satisfy our inequality.
