Find the solution set of the inequality a third 𝑥 plus one is less than negative six in the set of all real numbers. Give your answer in interval notation.
So the first thing to take a look at is some set notation we had in the question. And that’s this R-looking character. What this means is all real numbers. So as we said, it’s the set of all real numbers. Okay, so now we’ve looked at the notation, let’s get on and solve the inequality. So we’ve got a third 𝑥 plus one is less than negative six. So the first thing we’re gonna do is solve this like we’d solve any equation in fact, because we’re gonna subtract one from each side of the inequality. And when we do that, we get a third 𝑥 is less than negative seven.
So now, what’s the next step? Well, as we have a third 𝑥, if we want to find a whole 𝑥, what we do is we multiply both sides of the inequality by three. And when we do that, we get 𝑥 is less than negative 21. So great, we’ve solved our inequality. And we’ve given it in the form that we have here. However, is this correct? Is this what we’re looking for? Well, not quite because, yes, we’ve solved the inequality and we’ve got it using inequality notation, however, we want the answer to be written in interval notation.
Well, first of all, in our interval notation, what we’re gonna have is a set of parentheses. We’d have parentheses, not square brackets because if we look at the inequality sign, it’s just 𝑥 is less than negative 21. There’s no less than or equal to here. So if it was all equal to, then we’d use square brackets. Well, then, what we have are our parameters. Well, negative ∞ is the lowest possible value that 𝑥 will get to. It wouldn’t in fact ever be negative ∞. But we go right up to that point because we know that 𝑥 is less than negative 21 and the greatest value. But again, not actually being that value, hence the parentheses is going to be negative 21.
So therefore, we can say that the solution set of the inequality a third 𝑥 plus one is less than negative six is negative ∞ to negative 21. And this is inside our parentheses.