What inequality involving 𝑥 is equivalent to the inequality a third 𝑥 minus four is less than five 𝑥 plus two.
Well, when we’re looking to solve an inequality, we deal with it a bit like when we’re solving equations. So if we take a look at the left-hand side, we’ve got a third 𝑥. So therefore, so that we’ve got a whole 𝑥, what we’re gonna do is multiply both sides of our inequality by three. Well, what this would mean is multiplying each and every term by three.
So what I’ve done is shown a step here just so we could completely understand what’s happening. So I’ve got three multiplied by and then we’ve got a third 𝑥 minus four inside the parentheses is less than three multiplied by and again inside the parentheses this time we have five 𝑥 plus two. Then if we distribute the three over the parentheses what we get is 𝑥 minus 12. And that’s cause three multiplied by a third 𝑥 is just 𝑥 and three multiplied by negative four is negative 12. And then this is less than 15𝑥 plus six. Now, we didn’t need that step included, but it’s just to show you exactly what was happening.
Okay, so now we’ve got this inequality. What do we want to do next. Well, we looked to see where most of our 𝑥s are. And we can see that most of the 𝑥s are on the right-hand side cause we got 15𝑥 on the right-hand side. So therefore, what we’re gonna do is subtract 𝑥 from each side of our inequality. So when I do that, you get negative 12 is less than 14𝑥 plus six. So then what we do is we subtract six from each side of the inequality. And when we do that, we get negative 18 is less than 14𝑥. So now, what we want to do is just find an inequality with single 𝑥. So what we’re gonna do is divide each side of the inequality by 14.
So when we do that, we get negative 18 over 14 is less than 𝑥. So have we finished here? Well actually, what we can do is simplify. So we’re gonna do that first. Well, once we’ve simplified, what we can do is rewrite our inequality. And that’s because if we divide both the numerator and denominator by two, we’re gonna get nine over seven. So therefore, we’ve got negative nine over seven. So we can say that 𝑥 is greater than negative nine over seven.