Common Core Algebra I.Unit 2.Lesson 8.Inequalities
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Learning the Common Core Algebra I.Unit 2.Lesson 8.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 8 inequalities. Just another true false question. Before we get into that true false question, keep in mind that the worksheet and a homework that go along with this video can be found by clicking on the video's description. Or by visiting our website at WWW dot E math instruction dot com. Also, don't forget about the QR codes that are at the top right hand corner of every worksheet. Use your smartphone or a tablet to scan that code and take your right to this and other videos. Great. Let's begin.
One of the things that we're going to work a lot with in mathematics throughout the years are inequalities. Now, just like equations, inequalities can be either true, false, or open. You can't determine them. That's, of course, if we have variables that we don't know. But before we get into anything more complicated, what we really have to make sure of is that you can interpret inequality symbols. So here's what I'd like you to do is a little bit of a warm up. I would like you to pause the video and take not too much time, hopefully about two minutes to tell me whether each of the following is either true or false. Do that now. All right.
Let's go through them. Letter a 7 is greater than three. Well, that's absolutely true. Letter B zero is less than ten. That's certainly true. Letter C 9 is greater than 12. Oh, no, that's definitely false. Um, letter D, four is less than or equal to four. Well, it's equal to four, so yeah, that's true. Let's see, they're less than four. It's equal to four, and it's equal to four, so yeah, that's true. Two is greater than, or equal to 7. No, it's neither of those things, so that's false. 3.5 is less than or equal to 4.2. Sure it is because it's less than 4.2. So that's true. 256 is greater than 312. No, it's not. That's false. 1978 is less than or equal to 2042. Yeah, that's true, because it's less than. All right. Hopefully, you got all of those right. It's really important that you have that down. That's probably a skill barring the decimals, barring letter F that's a skill that you really started to work with as early as second grade. Second grade.
So you've been working with the, I don't know, the alligator mouth for a long time. So make sure that you can really read those symbols properly. Okay? Pause the video now if you need to because I'm clear now the text. Okay, here it goes. Let's move on. Now, this is going to be a little bit trickier. Once negative numbers come in, it's tough. So exercise two says, consider the statement negative 8 is less than four. Do you think this is true or false? Why? And then which is the correct truth value in Y so pause the video now and really think about this. Don't look it up on the web. Don't do anything like that. Just think about what you remember, what your gut tells you. Is negative 8 less than four. All right.
Well, I can't really answer the first question because it was an opinion. It was, do you think this is true or false? And why? The correct, the correct reasoning is that yes, it is true. All right, it is true. I like to use temperature a lot of times when thinking about negative numbers because it's one of the instances where people have some real world experience. And if I asked you, is negative 8, a lower temperature than positive four, you would say yes, yes it is, right? So try to put things in context if you can. Now, we're going to discuss technically why it is a true statement in just a moment, okay? But negative 8 is less than four is true, and if you set it, good. One of the reasons it's confusing is that the number 8, the number 8 is bigger than the number four. But the number negative 8 is considered to be less than the number positive for.
Anyhow, I'm clearing out the small amount of text that's on the screen. And then we're going to talk about how you can truly have a little bit of a problem clearing the one word. Then we're going to talk about how you can truly think about comparing numbers. All right, it's this symbol. If we compare two numbers, a and B, I know this is way down here. We will say that a is greater than B if a lies to the right of B on a standard horizontal number line. In other words, if I have a standard number line, I'm not going to even mark it. I'm not going to even put stuff on here, right? But if I've got B sitting here, I've got a sitting here, then we say a is greater than B or we say that B is less than a on the other hand, if we orient the number line vertically and a is above B, then we say that a is greater than B or we say that B is less than a it's only when the numbers are in the same place on the number line that we would use the equal sign. Okay? But that's how we judge it. It's really that simple. So I'll be using number lines a little bit here and there just to make sure that you understand this idea.
There are really especially helpful when you're comparing a negative to a positive and they're really helpful when you compare two negatives to one another. That's probably the trickiest thing, right? So, I mean, if I'm looking at something like negative three and negative 5, let's say, right? If you think about this on a number line, you have zero, one, two, there's my negative three. Four, 5. There's my negative 5, right? Then we would say that negative three is greater than negative 5, because it lies to the right of it. Okay? So it's a very, very important. Let me clear this out. Copy down anything you need to. Okay. Let's do some problems. Get the truth values for each of the following statements draw a number line to support your work. Okay, let's do the first one together. The first one says three is greater than negative four. Well, if I had a little number line, this would be zero. Three would be right here. Put a big dot there. And negative four would be right there. All right. So is three greater than negative four? Well, that's absolutely true. Why? Because three lies to the right of negative four.
Three lies to the right of negative four. Okay? So I'd like you to pause the video right now and think about letter B and letter C, okay? All right, let's take a look. Letter B says negative 5 is greater than negative three. Well, let me put the zero here. One, two, three. There's my negative three, four, 5. There's my negative 5. Why did I choose these two before? Well, if negative 5 was greater than negative three, it would lie to the right of it. It doesn't, it rise to the left of it, and therefore that is a false statement. Zero is greater than negative 6. Well, if I draw a little number line, then I put, let's say, zero here on to three, four, 5, 6, negative 6 here, where's my zero? There's my negative 6. Since zero lies to the right of negative 6, it is larger. It is greater than it. So that is also true.
All right, that's one that really throws people off. People think, oh, zero, zero is the smallest number that there is. Zero is not bigger than anything. But that's not true. Zero is in fact greater than all of the negative numbers, right? All of the negative numbers, and likewise, it's smaller than all of the positive numbers. Okay? So always go back to those number lines if you need to. Numbers that lie to the right of other numbers are greater than them. And it's that simple. Okay, let me clear this out. I also think that the temperature just works great. So if you think about 0° versus negative 6°, negative 6 is definitely lower than zero. All right? Moving on, now.
Now that we can compare two numbers, we can now determine whether or not an inequality with algebraic expressions in it is true or false for given values of X so this is this is no different than when we determine whether equations were true or false based on various values of X so let me do the first one with you and then you can take it away from there. Let's see if this inequality is true or false for the given value of X so we have done something like this before. What we're going to do is we're going to put that ten in for X wherever there's an X, all right? We want to now evaluate the left hand expression and the right hand expression. That'll be three times 8. Greater than or equal to 20 plus one, three times 8 is 24, 20 plus one is obviously 21, is 24 greater than or equal to 21. Absolutely. That's true. All right? So what I'd like you to do now is pause the video and try the other three values and see if they make the inequality true or false.
Please go ahead and pause the video now. All right. Let's go through them. Let's see. Speed it up a little bit now. Since we have hopefully the general idea, I'm putting in 5 everywhere I see an X I'm evaluating 9 is greater than or equal to 11. That is most certainly false. It's neither greater than 11, nor equal to 11. Let's try one. Three times one minus two. Greater than or equal to two times one plus one. Be careful here, one minus two is negative one. Three times negative one is negative three. Three is greater than or equal to negative three. Oh no, right? Negative three lies to the left of three on the number line. It does not lie to the right of it, so that's going to be a big fat false. And let's do 7. Three times 7 minus two. Greater than or equal to two times 7 plus one. Three times 5, greater than or equal to 14 plus one. 15 is greater than or equal to 15.
Now let's pause for a second here. So greater than or equal to and less than or equal to fall into the category of what are known as or statements. Also known as disjunctions, but that's a fancy word we don't really need to worry about right now. So because 15 is equal to 15. This actually makes this a true statement, right? Because one of the two things had to be true. Either 15 had to be greater than 15, which it's not, because it doesn't lie right of itself. Or it has to be equal to 15, meaning it lies on top of itself, which it does. So we can say that that is a true statement. Now this is really cool, right? I want you to understand. We can now very easily as long as we understand arithmetic and order of operations. We can easily determine if an inequality is true or false for a particular value of X by simply substituting the value of X in, and evaluating the two expressions on the left and right hand sides. All right, pause the video now if you need to, and then I'm going to clear out the text.
Okay, here it goes. Next page. Oh, okay. So here we've got some more complicated expressions, and they do look a little bit ugly. But the same idea is true. All we're going to be doing is taking those various values of X substituting them into the inequality and determining if they're true or false. Since you already understand hopefully what to do, I'm going to ask you to pause the video now. I would say for upwards of ten minutes because some of them, especially let her C and D take a little bit more work to actually just work through the arithmetic. But take up to ten minutes and determine whether these are true or false. All right? Pause the video now, please. All right. Let's go through them. So the first one is not too bad. Nice linear. We're going to put one in for X, just have to be careful about our order of operations.
Remember the multiplication here comes first, so we'll get two plus four is greater than four minus one. We'll get 6 is greater than three. And that is absolutely true. All right. Let's do letter B, we have negative three times negative three plus 5. Greater than or equal to negative three plus 7 divided by two. Let's take a look at this negative three here. We'll have two. Greater than or equal to four divided by two. Then we'll have negative 6 is greater than or equal to two. Now again, pause for a second on this. It would be easy to say true here, because 6 is certainly 6 is certainly greater than or equal to two. But negative 6 is not greater than or equal to two. So that's false. Negative 6 lies to the left of two on the number line. Or if you want to use temperature, negative 6 is colder than positive two. Okay. Oh, yeah, let's watch out here. Here we've got X squared.
Real good idea to put that negative two in parentheses, especially if you're relying on your calculator to do some of the arithmetic. Plus just correct. Never hurts wherever there is an X to put parentheses around the input. Don't forget that a negative times a negative is a positive, so negative two squared is positive four. Watch this. I have a negative times a negative is again a positive, be careful, be easy to say 12 there it's positive 20. And then I get positive one. Here, we're going to have 5 times negative two, which is negative ten. All right, 20 plus four plus one is 25, and it says 25 is less than ten. Now it's not. That's false. All right. Last one. Two times 5 -5. Plus one divided by three. Is less than or equal to 5 minus two divided by 9.
Let's just be careful. 5 -5 is zero. Less than or equal to three divided by 9, a little more work, two times zero is zero. Um, maybe we should reduce this. Divide both sides by three, one third, so we get one third is less than or equal to one third. And there, that is a true statement because one third is equal to one third. That's it. That's the whole story, right? To evaluate whether an inequality is true or false, we do exactly the same thing we do with an equation. This is going to eventually lead, I'm sure you expect to understanding what it means to solve inequality. IE to find all the values of X that make the inequality true. Just like when we solved an equation we were looking for all the values of X that made the equation true. All right. Well, I'm going to clear out the text so pause the video now if you need to.
All right, here it goes. Let's wrap this lesson up. So today, we learned how to compare various numbers, especially using number lines and the idea of temperature. And we also determined ways of figuring out whether or not an inequality involving algebraic expressions is true or false. By simply substituting in a value of X and determining the inequality truth value of the overall inequality. But that was not very articulate there at the end. It's kind of hard to say it any other way. You know, we're just trying to figure out whether or not a statement is true or false based on a value or values of X all right, thank you for joining me for another common core algebra one lesson by E math instruction. My name is Kirk Weiler. And until next time, keep thinking and keep solving problems.