Video: Finding the Solution Set of a Linear Inequality

Find the solution set of the inequality 2 ≤ (−𝑥 + 6)/2 ≤ 3 in ℝ. Give your answer in interval notation.

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Video Transcript

Find the solution set of the inequality negative 𝑥 plus six over two is greater than or equal to two but less than or equal to three in the set of all real numbers. Give your answer in interval notation.

So now, if we’re looking to solve this inequality, there’s a couple of methods we can use. We can first deal with it in two parts. Or, there’s another method we can use where we deal with it altogether. I’m gonna go through the dealing with “in parts” method first. So if we deal with the left-hand side first, what we’re gonna have is two is less than or equal to negative 𝑥 plus six over two. So then, what we do, first of all, is multiply each side by two. And that’s because we’ve got a two as the denominator of the right-hand side. So we don’t want a fraction. So what we do is we have four is less than or equal to negative 𝑥 plus six.

So then, in order to have a positive 𝑥 term, what I’m gonna do is add 𝑥 to each side of our inequality. So we have 𝑥 plus four is less than or equal to six. And then finally, what I’m gonna do is subtract four and what I’m left with is 𝑥 is less than or equal to two. So great, we’ve solved the left-hand side of our inequality. Now, let’s move on to the right-hand side. Now, for the right-hand side, we have negative 𝑥 plus six over two is less than or equals to three. So once again, what we’re gonna do is multiply each side of the inequality by two. When we get that, we get negative 𝑥 plus six is less than or equal to six. So then, again, we’re gonna add 𝑥 to each side of the inequality so that we’ve got positive 𝑥 term. So we’ve got six is less than or equal to six plus 𝑥. And then, finally, what we do is we subtract six from each side of the inequality and we get zero is less than or equal to 𝑥.

So we’ve now solved the inequality. So let’s put it back together. So now, what we’ve got is 𝑥 is greater than or equal to zero or less than or equal to two. So we’ve solved our inequality and it’s in an inequality notation. But if we look at the question, the question wants the answer in interval notation. And with interval notation, we have a couple of different types that we can use. So if we use, like, the top version here, parentheses, then we know it’s gonna be greater than or less than. So it can’t be the numbers included. Whereas if it’s greater than or equal to or less than or equal to, so includes the numbers within our interval, then we use brackets. So therefore, for our answer using interval notation, we’ll have bracket, zero, comma, two, bracket. And that shows that our values lie between zero and two inclusive.

Well, at the beginning, I did say there was a second method we could use. So I’m just gonna quickly run through this to show you another method you could use if you’re confident of doing everything at once. Now, with this second method, what we do, like I said, is deal with it as one. So the first thing we do is multiply through the whole inequality by two. So as you can see, what I’ve got here is four is less than or equal to negative 𝑥 plus six, which is less than or equal to six. So then, what I do is subtract six for every part throughout our inequality.

So what I’m left with now is negative two. And that’s cause four minus six is negative two is less than or equal to negative 𝑥. And that’s because we subtracted six from six which is just zero. And then, we’ve got is less than or equal to zero, again, because we subtracted six from six which is just zero. Now, have we finished? Well, no, because what we have here is negative 𝑥 and we want positive 𝑥. So what I’m gonna do is multiply through by negative one or divide through by negative one. It would have the same result.

So when I do this, I’ve got two is greater than or equal to 𝑥 is greater than or equal to zero. But it’s worth noting, as you can see, the inequality signs have changed the other direction. And that’s because if we divide or multiply through by a negative number, then we must change the direction of our inequality signs. So, as you can see, we now have the same as we got with the first method. Because you can see that 𝑥 is greater than or equal to zero or less than or equal to two. And we can show it using the same interval notation. So we’ve confirmed that the solution of the inequality in interval notation is bracket, zero, comma, two, bracket.

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