Find all the possible values of 𝑥 that satisfy nine over 𝑥 is greater than nine over four, given that 𝑥 belongs to the natural numbers.
Okay, so we’re looking at an inequality problem here. I’m gonna go ahead and have a go at solving that. But before we do, I just wanna have a quick look at some of the notation, that you might not be so familiar with. We’re looking at the notation here. We actually have 𝑥. And then we have this giant ∈. But what does this giant ∈ mean? What it actually means is that this means “belongs to”. So it’s telling us that 𝑥 is actually gonna belong to something. And then we have this kind of capital ℕ. And this capital ℕ means the natural numbers. But what do we mean by the natural numbers? Well, the natural numbers are actually any positive integer. For this question then what you’ll be doing, what this actually means in practice is any number from one upwards, that is a positive integer. Sometimes people do actually include zero, but we won’t be including zero in this set of numbers.
Great! So now we know what the parameters are, that 𝑥 needs to be between, to satisfy our inequality. We’re now gonna start to solve our inequality. And to do this, we’re actually gonna treat it a bit like a normal equation. So we’re gonna use similar methods and steps to actually solve the inequality. So first of all, what we’re actually gonna do is multiply by 𝑥. And that’s actually okay in this question because we know that 𝑥 has to be positive. However, usually in inequalities, we wouldn’t actually know whether 𝑥 is positive or negative. So this could actually be prob- problematic. But for this question, it will work.
So we get nine is greater than nine 𝑥 over four. So for our next step, we’re actually gonna multiply both sides by four which gives us 36 is greater than nine 𝑥. So at this point, we’ve now multiplied by 𝑥 and multiplied by four. At this point, I just wanna draw your attention to something which is quite important with other inequalities. In the last step, we actually multiplied by four. Well, multiplied by a positive, so it’s positive four. However, if we multiplied the inequality by negative four, we would have to actually change the direction of our inequality sign. So that’s something really key to remember.
So therefore in this situation, instead of being greater than, it would’ve been less than. And I’ve made a note of this tip on the right-hand side. So now we’re gonna complete the final step of solving this inequality. And I’m actually gonna divide both sides through by nine. Having divided both sides by nine, what I’m left with now is four is greater than 𝑥. Sometimes, what you’ll actually find is that people stop at this point and think, “Great! I’ve solved the inequality. That’s it. I’ve finished.” But that’s not the case. You’ll not receive all the marks unless you’re actually answering the question in front of you. And the question says: find all the possible values of 𝑥 that actually satisfy this inequality.
So we need to go one stage further. We need to decide what is going to satisfy our inequality. If we represent this inequality on a number line, we can see that we want all the values that are less than four. So this number line actually represents the inequality really well and gives a good visual depiction of what we will have. However, what makes this different is that we’re actually given the set of numbers that 𝑥 has to be within. So we go back to what we started with. And we actually remember that 𝑥 must belong to the natural numbers. So therefore, 𝑥 must be a positive integer. So with that in mind, we can now decide which numbers are gonna be possible values of 𝑥. And I’ve circled the possible values of 𝑥 that are actually natural numbers. So we have three, two, and one. Therefore, that gives us our final answer which is that the possible value of 𝑥 that satisfies the inequality nine over 𝑥 is greater than nine over four, given that 𝑥 belongs to the natural numbers, is: 𝑥 is equal to one, two, or three.
So just recapping on what we’ve done, first of all, solve your inequality like you’d solve an equation, using the same methods. So then we have to look at the value of our inequality. So in this case, it’d be four is greater than 𝑥. Then we look at any restrictions for 𝑥 in the question. There may not be any restrictions. But in this case, we actually did have some. And we knew that 𝑥 had to belong to the natural numbers. So then therefore, we knew that 𝑥 had to be a positive integer. So that’s how we arrived finally at our answer of 𝑥 is equal to one, two, or three.