Find the solution set of the inequality negative four 𝑥 minus one is less than five in the set of real numbers. Give your answer in interval notation.
So, we’ve got negative four 𝑥 minus one is less than five. And as we said, this funny-looking capital 𝑅 is actually some notation that means a set of real numbers. So, we’re only looking for answers that are in that set. Well, to begin with, this problem is gonna be very like when we’re solving an equation. So, the first thing we want to do is try and get the 𝑥 on the left-hand side on its own. So, in order to do that, what we’re gonna do, first of all, is add one to each side of our inequality. So when we do that, what we’re gonna get negative four 𝑥 is less than six.
Well, it’s now at this stage when often there is a common error which is made. And that common error is this, and that is that both sides are now divided by negative four, so what we’re left with is 𝑥 is less than negative six over four. Well, this looks sensible. So, what’s wrong? Well, the issue is in fact with the inequality sign itself, because if you divide or multiply through by a negative, then the sign will flip. So then, we can see what we should have is 𝑥 is greater than negative six over four because we flipped the inequality sign, which is gonna give us 𝑥 is greater than negative three over two because what we’ve done is divided the numerator and denominator here by two.
So, great, have we finished the question? Well, we’ve solved our inequality. But no, we haven’t finished because the question asks for the answer in interval notation. Well, in order to use interval notation, let’s remind ourselves of what it is. Well, if we’ve got square brackets, then this is greater than or less than or equal to. However, if we’ve got parentheses, then it’s just greater than or less than.
So, therefore, in interval notation, we’ve got negative three over two to ∞. And we’ve got that inside our parentheses. And the reason that is is because if we look back at our answer, which was 𝑥 was greater than negative three over two, we can see that it’s just greater than, not greater than or equal to, which is why we have parentheses. And then, we’ve got for the upper limit ∞ because we just know that 𝑥 is greater than negative three over two. So, it could be any value up to ∞. But it is not including ∞, and that’s because we can actually have the value ∞ itself. So, that is why we use the parentheses for the right-hand side as well.