Video Transcript
Find the solution set of the
inequality negative 14𝑥 minus 52 is less than or equal to negative 18𝑥 in the set
of real numbers. Give your answer in interval
notation.
Well, the first strange thing we
want to see in this question is this R-looking character. And what this means is all real
numbers. So we’ll know that the results of
our inequality is gonna be a real number. Well, you might think, “Well,
aren’t all numbers real?”
Well, in fact, no, there are
imaginary numbers that we deal with in maths. But we deal with them later on in
the maths course. So now when we’re gonna solve our
inequality, we’ll do it in the same way that we’d solve an equation. So we’ve got negative 14𝑥 minus 52
is less than or equal to negative 18𝑥. So what we’re gonna look to do now
is add 18𝑥 to each side of the inequality and also add 52. And when we do that, what we’re
gonna get is four 𝑥 is less than or equal to 52.
And you can see here I’ve completed
two steps. Well, this is in fact a multistep
inequality because we’ve done two steps. But we haven’t quite finished yet
because we’ve got another step that we need to do to solve our inequality.
So the final step is to divide
through by four. And that’s because we want to find
out what one 𝑥 is. Well, when we do that, we’re gonna
get 𝑥 is less than or equal to 13. So great, we’ve solved the
inequality. So have we solved the problem?
Well, no, not quite, because the
question asked us to give our answer in interval notation. So what will this be, in interval
notation? Well, when we write this in
interval notation, we have this here. We’ve got parentheses, then
negative ∞ comma 13, and then a square bracket. So let’s think what all of this
means.
Well, the first of all, on the
left-hand side, we have a parenthesis. And that’s because we know that if
our 𝑥 is less than or equal to 13, then it can go all the way down to negative
∞. But it would not include negative
∞. Then, we can see on the right-hand
side, we’ve got a square bracket. And this means that it does include
13 because we were told that 𝑥 is less than or equal to 13. So we know that our values can take
any value from negative ∞, but not including negative ∞, all the way up to 13, but
also including 13.
