Common Core Algebra I.Unit 2.Lesson 9.Solving Linear Inequalities.by eMathInstruction
Algebra 1
Learning the Common Core Algebra I.Unit 2.Lesson 9.Solving Linear Inequalities.by eMathInstruction
Hello and welcome to another common core algebra one lesson by E math instruction. My name is Kirk Weiler, and today we're going to be doing unit two lesson number 9 on solving linear inequalities. Let me remind you that you can find the worksheet in the homework that goes along with this lesson. By clicking on the video's description. As well, don't forget about the QR codes at the top right hand corner of every worksheet. That will allow you to take your smartphone or your tablet, scan the code and come right to this video. All right, let's begin. In the last lesson, we talked a lot about whether or not inequalities were true and false. Today, what we want to do is be able to solve inequalities.
In order to do that, we have to understand the rules that inequalities live by. What their properties are, all right? Just like we understand the properties of equality, we want to understand the properties of inequality. So let's do that in exercise number one. Exercise one says, consider the true inequality for is less than 8. I think that we can all agree that four is less than 8 is a true inequality. Letter a says if we add three to both sides of the inequality, what is the resulting inequality? Is it true? All right, so let's do that. Let's add three to both sides of this inequality. And we'll get 7 is less than 11. So that's the resulting inequality. Well, and that's definitely true. All right. Maybe it won't work though if we subtract, right? So let it be says if we subtract four from both sides of the inequality, what is the resulting inequality? And is it true?
All right, so we start with this true inequality. We subtract four from both sides and we get zero is less than four. Ah, that's also true. Great. I like it. What if we multiply both sides of the inequality by two? So let's do times two over here. Let's do two over here, and we'll get 8 is less than 16. Well, that's true. And then it says, what happens if we divide both sides by two? Let's divide both sides by two, and we get two is less than four. That's also true. Wow. So this almost seems it almost seems like inequalities have the exactly the exact same two properties that equalities had. In other words, equalities have the additive property of equality, which said we could add or subtract anything we wanted from both sides and it would retain its true nature.
If it started off being true. And we could divide or multiply both sides, and it would still remain true, so it appears that that's the case, right? A and B, we added and subtracted, and it stayed true. C and D, we multiplied and divided, and it stayed true. All right. But there is a catch with inequalities. I'm going to clear this out, so write it down first, and then we're going to see what that catch is. All right. Here we go. Clearing out the text. It would be so, so lovely, if inequalities worked exactly like equations. But let's take a look at four is less than 8. Letter aces, if we multiply both sides of the inequality by negative two, what is the resulting inequality? And is it true? So now I'm going to multiply the left by negative two and the right by negative two, and I'm going to get negative 8 is less than negative 16. Oh, that's false. Right? That's definitely false. Negative 8 is not less than negative 16. In fact, negative 16 is less than negative 8. What if we divide both sides by negative two? And we get negative two is less than negative four. That's also false. Um. Wow, that stinks.
You know, when we added or subtracted from both sides, we were fine. When we multiplied or divided both sides by a positive number, we were fine. But when we multiplied or divided both sides by a negative, it changed our inequality from being true to being false. All right? So this really leads us into two properties of inequality that we need to look at right now. I'm going to clear out this text, so really I want you to really process what we just did. Okay? Then I'm going to clear it out. All right, let's take a look at the properties of inequality. All right? We've got the addition and subtraction property of inequality. Now here's the great thing about the addition subtraction property of inequality, it is exactly the same as the addition subtraction property of equality. In other words, if a is greater than B, then a plus C will be greater than B plus C in other words, you can add or subtract anything you want from the two sides of the inequality and it will remain true. All right? But the multiplication and division, that's a little bit worse. So if a is greater than B, then C times a will be greater than C times B if what we're multiplying by, we're multiplying by C on both sides, is positive.
So if C is positive, we're all good. Unfortunately, if a is greater than B, then C times a will be less than, look at this, look at how it's greater than here. And how it switches into less than here, if C is negative. All right, a lot of teachers, including me, sorta describe it this way. Solving a linear inequality works exactly like solving a linear equation with one exception. If you either multiply or divide both sides of the inequality by a negative. Then you must flip the inequality symbol. All right? Because if you don't flip the inequality symbol, the inequality will be go from true to false. Or if for some reason you started with a false inequality, it would go from false to true. But let's not go there. In fact, let me clear this text out, and let's do one more problem before we solve inequalities. Number three says write a true inequality and show that it becomes false when multiplying or dividing your choice. Each side by a negative. Pause the video now and try to do what the problem suggests. All right, let's do it.
Hey, look, you know, I doubt I'm going to get the one that you did. Let me do something like this. Let me do ten is greater than four. Now watch this. Let me multiply both sides by negative three. Right? So this is definitely true. But when I multiply both sides by negative three, I get negative 30 is greater than negative 12, and that's false. So really what inequality theory tells me is that as soon as I multiply both sides by negative three, I have to switch that into a less than symbol. All right? Let me show you one with division. Let's say I've got, let's go with this. Let's start with 8 is less than 20. And let me now divide right. So that's definitely true. Let me divide both sides by negative four. And I'd get negative two is less than negative 5, and that's now false. So really, as soon as I divide by that negative four, I want to switch that into a greater than. It even works with negatives and positives. Watch this. Let's say that we have one is greater than negative three.
Let's say I multiply both sides of that equation by negative two. It's definitely true. Weird but true. And then I get negative two is greater than 6. And that's definitely false. All right, so it's very important. If we start off with the premise that the inequality that we're trying to solve is true. Then as soon as we multiply or divide both sides by a negative number, then we have to switch the inequality symbol to whatever is the opposite of what it started with. That sounded weird. Let's do some examples. I'm going to clear out the text so write down anything you need to. All right, here we go. All right. Given the linear inequality, do the following. Solve the inequality by applying the properties of inequalities we found earlier. All right. So one property of inequality that we had was the additive property of inequality, and it's said that if we add or subtract, then the inequality as written continues to be true. We don't have to switch anything. Now we have the multiplicative property of inequality that says if we divide both sides by a positive integer, the inequality remains true.
So X is greater than or equal to two is our solution to this inequality. Now that our base is right 5 numbers that make the final solution true and plot them on the number line below and see. So what are 5 numbers that are either greater than or equal to two? Now there's many that are greater than or equal to two, but here are some of them. Well, two is greater than or equal to two. Three is 4.5 is. I am one of that throw in some that are fractional. 7.1 is, how many do I need to do 5, one, two, three, four, ten is. So for instance, this is this is good four and a half is good, 7.1 is good and be right about there. Ten is good. It's this graph all solutions to the number line below. This is called the solution set. Well, our solution set starts at two and every number greater than two should get shaded. Some students don't like that, so they'll kind of put the shading down here. I don't have any issue with that, but you never know so you might want to ask your teacher if it's okay. All numbers two and above solve this. Notice that nothing down here is colored, right? Because those would all be numbers that were less than two. Simple enough. Okay? So I am going to clear this out, and then we'll go on and do some more.
All right, all gone. Exercise 5. Given the linear inequality, 8 minus two X is greater than 16, do the following. Rewrite the left hand expression as an equivalent expression using addition. All right. Now, I want you to keep in mind that any time you are operating just on one side of an inequality or an equation, right? Then the properties of equality or the properties of inequality have nothing to do with it. Here what we're going to do is we're going to use the property of rewriting subtraction into adding of opposites, and then we'll use the commutative property to change this into negative two X plus 8 is greater than 6. All right. I'd just like to have it in the same format I did before. Now I'm going to solve in letter B, this inequality using the properties of inequality. Now here we're going to be careful. We're going to use the additive property of inequality to add negative 8 to both sides. Remember that does nothing in terms of changing the inequality symbol, but here's where we have some issues. As soon as I do this division by negative two, then this inequality would suddenly become false to keep it true, I have to flip the inequality. Why? Because I divided by a negative any time you divide by a negative.
We get something that changes the truth value of our inequality, so to keep it true, we have to flip flop that inequality from greater than or less than to less than. At least in this case. Now let her see, pick a number that is true based on your solution to B and show that it makes the original inequality true. So I need to pick any number that's less than one. Any number, don't pick one. One is not less than one. I pick any number less than one, and check it. So, what do you think? Maybe we should pick, I don't know. Oh, wait a second. Huh. That's funny. I wrote the problem down wrong. Do you notice that? I wrote down a 6 instead of a 16. So I was going to pick X equals zero and test it in here. Then I started to do the test in my head, and I was getting 8 is greater than 16, which I knew was wrong. So I looked back at the original problem, and I realized that I had made a mistake. So let's correct that mistake.
Let's put a one in here and a one in here and a one in here to make that 16. Sorry about that. Make that 16. Now I need some eraser going on here. Whoops. My eraser is way too small. Much better. Great. So now we have 16 -8. And now we will have X is less than negative four. Okay. There we go. Back up to speed. Sorry about that. Now what I want to do is I want to pick a number that lies in this solution and test it. So think about a number that would make that true. And an easy one is X equals negative 5, right? Negative 5 is less than negative four, lies to the left of it on a number line. So let's test it in the original inequality. So 8 minus two times negative 5 is greater than 16. So a negative times a negative is a positive, so 8 plus ten is greater than 16. And 18 is greater than 16. And that is most certainly true. You know, I thought even for a moment when I had noticed that I'd made a mistake about just stopping the video, starting it all over. But I think it's important to always note everyone makes mistakes in math. I make mistakes in math, lots of mistakes. I make mistakes in math. Everyone makes mistakes in math. The good thing was, I caught my mistake by actually doing letter C, right? I didn't make the mistake on purpose, trust me.
Sometimes I do, but not this time. All right, letter D says graph the solution to the inequality on the number line below. Now this is a little bit tricky. Clearly, negative 5 is part of the solution. So are things like negative 6 negative 7, et cetera. But is negative four part of the solution should I color that in? And the answer is no. Right? I need all numbers that are less than negative four. So what I'm going to do is I'm going to circle negative four. I'm not going to color it in, and then I'm going to color in everything to the left. Again, if you like to color above the number line, it would look something like this. All right, the circle indicates that everything up to negative four, everything up to negative four is colored, but not negative four itself. A lot of students end up doing some lights or a rote memorization. Okay, so if there's an equal sign, a colored in, and if there's no equal sign, I know color it in. Fair enough, it works. But at the end of the day, this is what you should know. If a number solves the inequality, if it makes it true, it should be colored in. And if it doesn't make it true, it shouldn't be. And negative four doesn't make this inequality true. So it shouldn't be shaded in. All right? So I'm going to clear this out, pause the video if you need to. There it goes. All right. Next problem.
Now, wow. This is as bad as it gets. Okay? It doesn't get any worse than this. And the thing is, it's not that bad, right? We've solved equations that look exactly like this, but now we're going to solve in inequality. Normally, we wouldn't necessarily break it down into part a part B but I really want you to understand something as we solve this inequality. All right? As we solve this inequality, when we're manipulating the expressions, not the equation, not the inequality. When we're manipulating the expression on the left hand side, expression. Number one, that seems like exercise number one. And we're thinking about expression number two, right? When we manipulate these two things, independently, that has nothing to do with the inequality. Nothing at all, and if there was an equation here, it would have nothing to do with the equation. We're simply going to be using the associative commutative and distributive properties of numbers. To rewrite these things. All right, so the first things first, let's use the distributive property. Let's distribute the 8 through the parentheses. Let's distribute the negative three, be careful there.
All right. Now you might say, wow, we multiplied by a negative. We got to switch to inequality. No. That's when we multiply both sides of the inequality by a negative, or divide both sides by a negative. We didn't do that here. We're just manipulating those expressions. So on the right hand side, nothing with the 7 X minus four or sorry 7 X plus four, but then we're going to distribute that negative three. And get negative three X minus three. So we distributed innate. We distributed in a negative three. We distributed another negative three. Now we're going to use the commutative properties and associative properties here to flip flops and things so that we can combine them. So what we've done here is use the commutative property to flip flop this and flip flop that. Now we're going to do a little bit of associative properties. Have a little plus in between that. All right, 8 X -6 X is two X, negative 16 negative three is negative 19. Here we have 7 X minus three X, which is four X, four minus three is one.
All right, that's really all we wanted to do in part a wow. Here comes solving the inequality. All right, so I really want you to keep in mind that what we were doing before in part a really wasn't solving the inequality, it was just manipulating expressions to get equivalent expressions that were simpler. Now what we'll do is we'll subtract four X from both sides using the additive property of inequalities. So we'll get negative two X -19 is less than or equal to one. Well, again, use the additive property of inequalities to add 19 to both sides, negative two X is less than or equal to 20. And now we will use the multiplicative property of inequality, but we have to watch out here because we divided by negative. We'll get X is greater than or equal to negative ten. There's our solution to graph our solution set. We would draw a little number line, maybe orient zero on there, orient to negative ten on there. And with it all in blue, you can't see it very much. That's where students like to do this. And there's our solution.
All right. That's it. So I'm going to clear this out. Pause the video if you need to. Okay. Let's wrap this lesson up. So today, we looked at how to solve linear inequalities in order to do that. We needed to make sure we understood the rules that inequalities play by. The first rule of the additive property of inequalities was exactly the same with equations. Add or subtract anything you want from both sides as long as you do it to the same size, the same thing to this. To each side of the inequality. The multiplicative property though of inequalities was sort of twofold. If you divide or multiply by a positive on both sides, the inequality stays the same. If you multiply or divide by a negative though, the truth value of the inequality will switch. So in order to make it remain true, we have to flip the inequality symbol whenever we multiply or more commonly when we divide by a negative. All right. Thank you for joining me for another common core algebra one lesson by email instruction. My name is Kirk Weiler. And until next time, keep thinking and keep solving problems.