Video: Finding the Solution Set of Linear Inequalities with Real Numbers

Find the solution set of the inequality 1 ≥ 3𝑥 + 4 > −4 in ℝ. Give your answer in interval notation.

04:10

Video Transcript

Find the solution set of the inequality three 𝑥 plus four is less than or equal to one, but greater than negative four in the set of real numbers. Give your answer in interval notation.

Now, to solve this problem, there’re a couple of methods we could use. The first method, we’ll solve each side of our inequality first. The second method is to solve it all together. So on the left-hand side, we have one is greater than or equal to three 𝑥 plus four. And on the right-hand side, we have three 𝑥 plus four is greater than negative four.

So we’re gonna start with the left-hand side. When solving an inequality, we deal with it much the same way we would an equation. So whatever we do to one side of the inequality, we must do to the other. So the first thing we’re gonna do is subtract four from each side of the inequality. So when we do this, we’re left with negative three is greater than or equal to three 𝑥.

So now, as we’re looking for an inequality in terms of 𝑥, what we’re gonna do is divide both sides of our inequality by three because that would give us our single 𝑥 that we’re looking for. And when we do that, we’re gonna be left with negative one is greater than or equal to 𝑥. Okay, great. So that’s the first part solved. Now, we can move on to the right-hand side.

So once again, we’re gonna begin by subtracting four from each side of the inequality. And when we do that, we’re gonna have three 𝑥 is greater than negative eight. And that’s remembering that if we have negative four and we subtract four, this means it’s gonna become more negative. So let’s move left down our number line. So it takes us to negative eight. And then, once again, we divide through by three because we’re looking to find out what 𝑥 is in our inequality. So therefore, what we get is 𝑥 is greater than negative eight over three or eight-thirds. And that’s what we get if we divide both sides of the inequality by three.

Okay, well, we haven’t finished here. What we need to do now is combine our inequalities. And when we do that, we’re gonna get negative one is greater than or equal to 𝑥, which is greater than negative eight-thirds. Okay, so this is our inequality solved. However, have we solved the problem completely? Well, no, because what the question asks us is to show the answer in interval notation. But what does this mean? So if we use that in interval notation, well first, we’re gonna have a parenthesis and then negative eight-thirds comma negative one and then a square bracket. And this is because the parenthesis tells us that it does not include that value. So we know that we go from negative eight-thirds, but not including it, all the way up to negative one. But we include negative one.

And we can see from the line above that the reason that that’s the case is because we’ve got a line underneath the inequality sign. Well, that tells us that 𝑥 is less than or equal to negative one. However, there isn’t a line underneath the inequality sign when we’re going that 𝑥 is greater than negative eight-thirds. Okay, brilliant. So this is the answer. We’ve solved the problem. But I did say there was another way that we could’ve approached this.

So what we can do is use this alternative method to act as a check. So, we’ve got the same inequality. But this time, we’re gonna deal with it all together. So what we’re gonna do is subtract negative four from each term within our inequality, which will give us negative three is greater than or equal to three 𝑥, which is greater than negative eight. And then, what we’re gonna do is divide by three from each term within our inequality. And what this can do is lead us to negative one is greater than or equal to 𝑥, which is greater than negative eight-thirds, which is the same as we got with our previous method.

So therefore, we can confirm that we have the correct answer when we said that the answer in interval notation was parenthesis negative eight-thirds comma negative one square bracket, which means that the interval is in fact between negative eight-thirds, but not including negative eight-thirds, and negative one. And we include negative one.

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