 In this video I'm going to talk about solving inequalities. This is actually very similar to solving equations except for we have the inequality symbols which are less than, less than or equal to, greater than and greater than or equal to. Those are our four different symbols that we have for inequalities. Okay, so not only are we going to solve these but we're also going to graph them on a number line so we can see kind of what the solutions look like. Okay, so now the rules for solving these inequalities is actually the same rules that we use for solving equations. We treat this inequality, we treat it just like an equal sign. For the most part we treat it just like an equal sign. So I'm going to add some tracks, multiply, divide on the left and on the right side of this inequality so that I can solve for this variable. So basically what I'm looking for is I'm looking for when, what numbers here, if I plug in some numbers here, what numbers are going to be less than 28? So if I plug in the number 1, 1 times 3 plus 7, so that's 3 plus 7 is 10, 10 is less than 28 so 1 is an answer here. But what other solutions are there? What other numbers can I use? That's what I'm going to find out. What I'm going to attempt to do is find out all of them, not just a few of them but just all of them. Alright, so again when I'm solving this I want to do this just like I do any other equations. So as I see here what I want to do is I want to get rid of this 7 first. So subtract 7 from both sides. So 3x is, excuse me, 3x is less than 21 and this 3 is multiplying times x so I'm going to divide by 3. So x in this case is less than 7. x is less than 7, let's put a little, let's put a little extra there for the 7th so we know the difference between our data qualities and our 7th. Anyway, now what does that mean? x is less than 7 which means all of my solutions, all of the numbers that work are going to be smaller than 7. All the numbers that work are smaller than 7. Okay, so if I look, if I want to check that, so what's a number smaller than 7? Well 6, 6 is smaller than 7. So if I plug in 6 here, 3 times 6 is 18, 18 plus 7 is going to be 25, 25 in fact is smaller than 28. Okay, that doesn't work. Okay, but what about 7 itself? What about 7? 7 is not smaller than 7 but let's see what happens right at 7. So if I plug 7 in, 3 times 7 is 21, 21 plus 7 is going to be 28, 28 is not smaller than 28. So that actually doesn't work. So 7 doesn't work in this case. But anything smaller than that, 6, 5, 4, 3, 2, 1, anything smaller than that works. Okay, so let's actually look at this, let's actually graph this. I'm going to make myself a number line. Now since I only have one number in my solution, 7 is the only number in my solution, I'm actually going to put that right in the middle. 7 is my number that I'm going to put right in the middle. And from there I want to illustrate all of the possible solutions that I have. All of my solutions are smaller than 7, so that means all of my solutions are going to be this way on my number line. All of my solutions, all the x's are going to be right at 7 and then anything that's smaller than that. Now look here, I do have a circle. Now when I use, when I graph my inequalities, I either use an open circle or a closed circle. Now the difference between them, actually I'm going to illustrate that up here, is if I have a less than or greater than, I use an open circle. If I have a less than or a greater than, keep messing that one up, if I have a less than or a greater than, I'm going to use an open circle. If I have a less than or equal to, or a greater than or equal to, I'm going to use a closed circle. Okay, now what that means is the open circle means I do not include that number. The closed circle means I do include that number. So in this case, we do not include 7. As we saw earlier, if I plug in 7, I actually get 28, but I don't want 28, I want something that's smaller than 28. Okay, so 7 doesn't work, open circle to show that 7 doesn't work, but anything smaller than that is going to work. So 9, 6.8, 6.7, and then all the way down, 6, 5, 4, 3, 2, 1, 0, negative 1, negative 2, all those numbers down this way on the number line, all the numbers down this way, all the numbers down this way are going to work. Okay, so that's what this, that's what it means when we solve inequalities. We're finding all of the solutions that work, not just one number, but we're finding a lot of numbers that work. Okay. All right, let's try a different example. Let's show you a different example of solving and graphing inequalities. Here we go. How about 12y minus 3 is less than or equal to 57. So now we got less than or equal to, so as I kind of mentioned earlier, when we graph it, we're going to use that closed circle. Okay. So again, when I solve these, I solve these just like normal equations. All right, what I'm going to do is I'm going to add 3 to both sides, so 12y is less than or equal to 60. And now I'm going to divide by 12, I'm not going to stuff over here, I'm going to divide by 12, which means y is less than or equal to 5. Y is less than or equal to 5. So what does that mean? All the solutions that I want, all the solutions that work here are going to be less than or equal to 5. So all the numbers smaller than or equal to 5. So 1 is going to work, 2 is going to work, 3 is going to work, 4 is going to work, and 5 is also going to work. 6 on the other hand is too big, that's not going to work. Okay, so let's test this, let's see what happens. So if I take, let's say I take 5. Since my solutions are less than or equal to 5, let's actually use 5. So let's take 5, plug it in. So if I take 12 times 5, I get 60, 60 minus 3 is 57. 57 is less than or equal to 57. That actually works because of this or equal to part. Okay, so 57 is actually okay to get as opposed to last time when we had a different symbol. Okay, if I put in the number 4, so instead of 5, a number less than 5 would be 4. So 12 times 4 is going to be 48, 48 minus 3 is 45, 45 is smaller, is less than 57. So that does work. Okay, so let's graph this now. Let's see all the different solutions. I'm going to make my number line. And now the only number I'm really concerned with is 5, so I'm going to put that right in the middle. We'll put a 4 down here, we'll put a 6 out here to show numbers get smaller as you go left, numbers get bigger as you go to the right. I'm going to use a closed circle. The reason I use a closed circle is because 5 is included, it does work. It does satisfy my inequality up here. I can see that by the or equal to sign here. And then the y's are smaller than 5. My solutions are smaller than 5, so the smaller numbers are that way. Smaller numbers are that way. Okay, so all my solutions, 4.9, 4.8, 4.7, 4.5, 4, 3, 2, 1, all the decimals, all the fractions, anything that is less than 5, or 5 itself actually works. Okay, so there's another example. Let's do one more to give us a better understanding of this. Let's do one more. 2 times the quantity 4 minus x is greater, excuse me, less than, is less than 10. Okay, so it looks like I got to do a little bit of distribution first. Got to do a little distributing first before I can solve this. No trouble, let's just take 2 times 4 and 2 times x to get 8 minus 2x is less than 10. Okay, I'm just following my normal rules for solving equations. So I'm going to get rid of this 8 first. I'm going to subtract 8 from both sides. Negative 2x is less than negative 8. Okay, now to get x by itself, I got to divide by negative 2, divide by negative 2. Now, this is where I need to stop and explain something. This is where inequalities is different from solving regular equations. If I multiply or divide by a negative number, I need to flip my inequality symbol. So, that's why I stopped here because right here I'm dividing by a negative 2. Dividing by a negative 2, which means this cancels, that's all the same. And then on this right side, negative 8 divided by negative 2 is 8 positive 4, but my inequality symbol is going to flip from a less than to a greater than. Okay, that's one thing you got to remember when solving inequalities is if you multiply or divide by a negative number, you need to make sure and flip your inequality symbol. Okay, so now my solution says, all my solutions, all of my numbers that are correct, all my solutions, all my answers are bigger than 4. All of my answers are bigger than 4. Okay, so what's the number bigger than 4? 5 is the number that's bigger than 4, so let's use that number. So, I'm going to plug this back up here. 4 minus 5 is going to be negative 1. Negative 1 times 2 is a negative 2. Negative 2 is in fact smaller than 10. Hey, it doesn't work. Okay, and I can keep going. I can keep choosing different numbers. I can choose any number that's bigger than 4, that would be 5, or 6, or 7, or 8, or 9, or 10. What about 10? 10 is bigger than 4. What about that one? Well, 4 minus 10 is negative 6. Negative 6 times 2 is negative 12. Negative 12 is smaller than 10. That does work. So again, with these inequalities, we're not just getting one answer, we're getting many, many answers. All right, so let's graph this. Let's see what all the answers look like. Graph my number line. The one important number I'm looking at is 4. I'm going to put some numbers around it. We'll put 3 over here. We'll put 5 here. Numbers get smaller. Numbers get bigger. And now I'm going to use an open circle at 4. And all of my solutions are bigger than 4. All the numbers that I want are bigger than 4. So where are the bigger numbers? They're that way. Okay, so that's what the graph of this solution would look like. All right, that is solving inequalities, solving and graphing inequalities. And kind of a short explanation of all the solutions that we're getting. The couple of things to remember, you solve these equations, you solve inequalities just like you do normal equations. The couple of things you've got to remember is if you multiply or divide by a negative number, you must flip your inequality symbol. That's one thing you need to remember. Also, when you are graphing your inequalities, make sure you use the right circle. You use an open circle with less than or greater than, and you use a closed circle with less than or equal to and greater than or equal to. Just a couple of things to remember.