Determine the solution set of negative seven 𝑥 plus five is greater than negative nine given that 𝑥 is in the set of integers.
The first thing I wanted to take a quick look at was the set notation that we had, that is, this Z-looking character. And what this means is integers, so the set of integers. And then, the E character next to it means that it’s in the set of integers. So we’ve got 𝑥 is in the set of integers. Okay, great. So now we know what the notation means, let’s get on and solve our inequality.
So what we have is negative seven 𝑥 plus five is greater than negative nine. So then, to solve this inequality, we do it in the same way that we’d solve an equation. So the first thing we do is subtract five from each side. So when we do that, we get negative seven 𝑥 is greater than negative 14. And then, what we do is we divide both sides of our inequality by negative seven. So this seems fairly straightforward, doesn’t it? Because what we’re gonna have on the right-hand side is two. And on the left-hand side is 𝑥.
And we get two on the right-side because negative 14 divided by negative seven is two. And negative seven 𝑥 divided by negative seven is just 𝑥. But what goes in the middle? This might seem straightforward because we just bring down the inequality sign. However, this is not the case. This is where we have to be very careful when we’re dealing with inequalities and negative numbers because if we divide or multiply by a negative number, our inequality sign changes direction.
So what our final answer is in fact is that 𝑥 is less than two. Not 𝑥 is greater than two, but 𝑥 is less than two. So be very careful with this kind of question. Okay, well, we’ve solved our inequality. So have we solved the problem? Well, not quite, because what we’re looking for is not just the solution to the inequality. But we’re looking for the solution set. And what the solution set is is all the values that 𝑥 could take. Well, we know that 𝑥 could take any value that’s less than two. So the solution set is gonna be one, zero, negative one, et cetera.