 This video is going to talk about solving inequalities by using a graph, by using paper, and by using a table. This first example, what we're going to remember is that when we multiply or divide by a negative number, then we have to switch the inequality. To solve this one, I'm going to subtract 10 from both sides. So negative 6b is going to be greater than or equal to 30. Now I'm okay so far. I don't have to switch the inequality until I do this next step. When I divide by negative 6, then I have to watch my inequality. And I'm going to move it right away so that I don't forget. So this is b, and that was a negative 6 that it was being divided by, so that would give me negative 5. So b has to be less than or equal to negative 5. And let's try negative 6. Negative 6 times negative 6 would be 36 plus 10 would be 46, and 46 is greater than 40, so it works. The only tricky part is that we have x's on both sides, so we want to be careful. So the tricky part about this one is that we have x's on both sides. But if we're careful, we can make sure that it's a positive x so that we don't have to worry about if we have to change the inequality or not. Now when I look at these 2x values, this is 8 and this is 2. 8 is larger and it's already positive. So it would be better to bring the 2 to the other side. So I'm going to add 2 to both sides, 2x to both sides. So now I have 10x plus 1 greater than 51. And if I subtract my 1, 10x is going to be greater than 50, and I'm dividing by a positive number, so x is going to be greater than 5. Now when you look at this one, you have negative 3x and negative 5x on both sides, but if I compare just the 3 and the 5, the 5 is larger. So it would be better to move the 5 to the other side because it will become positive and be the bigger number. So I'm going to add 5 to both sides. So 5x plus a negative 3x will give me 2x plus 5, less than or equal to negative 7, and then I'm going to subtract my 5 and 2x would be less than or equal to negative 12, and when I divide by 2, I'm again dividing by a positive so my inequality stays the same and this x is less than or equal to negative 6. Let's look at some examples that are from a table. So in this example, I'm going to kind of lead you to where it is. So where is f of x equal to g of x? Well, if you look at the table when the y values are the same, then I know that it's equal. Okay, so we would say that they're equal at x equal 5. Now it says compare f of 3. Well, at f of 3 and g of 3, we have 25 for f of 3 and 45 for g of 3. So f of 3 is less than g of 3. So there's a less than in here. Now it asked me to go on the other side of my equal sign and look at the x equal 6. So f of 6 is 49, g of 6 is 29. So now f of 6 is bigger than g of 6. So it's greater than. So what intervals of value satisfy the inequality where f of x, 8x plus 1, is greater than negative 2x plus 51, that g of x. That's going to be down here below my 5 because if I look at 7, 57 is bigger than 35. It's also greater than. So it's going to start where? Hmm. Well, it starts at 5. Remember, we're just going to put a parenthesis there because it means really close to 5 but not exactly 5. And then it's going to go like that forever. So it would be infinity. And infinities always have a parenthesis. So how about we're going to look at the graph. It's the same inequality. It's just the graph of it. So now it's asking me to compare f of negative 5 and g of negative 5. Well, negative 5 is right here. So here's g of negative 5 and here is f of negative 5. And it looks like these values up here, this is a positive y value and this would give me a positive y value. So f of negative 5 is actually going to be less than g of negative 5. Another way to look at it is we know the higher up on a graph we go, the bigger the numbers. G of x value is on top of or above the f of x value. So it's got to be bigger. Alright, now it asks us to do f of 10 and g of 10. Here's 10. Now this is g of 10 and then over here I have f of 10. See it's on the f function. And when I compare those, f of 10 now is higher or its value looks like 80. It looks like it's maybe 30. So f of 10 is going to be greater than g of 10. So again it asks us for the values that would make it true and it would be the same thing. If we think about the intersection, remember we know that at intersections of two graphs that's where they're the same. And over here on this side we have f is greater than g. And over here on this side we have f is less than g. So we want to go from here over. So it's going to be from 5 but not including 5 to infinity just like we found in the case before. So let's talk about a model using inequalities. So we have these two equations here that are, one is for the cost of printing flyers by traditional methods and one is for cost of printing flyers using digital printing. And we want to know when traditional will be cheaper than digital. In other words, when t of n is going to be less than d of n. That's where we're headed. So for a table I would need to have my equation in here and here's my two equations. And I've already actually set up the table. So we come back in here to table and you can see I'm just trying to find, narrow it down. And I'm kind of close but not real close so I'm going to jump around. I'm going to come in here to table set and I'm going to stay at a thousand but I'm going to try to go maybe every thousand. See if I can get in the right set of thousands and then I can narrow it down if I need to after that. So second graph again and I can see that they're getting closer and closer and oh look right there they're the same. So that tells me that they are the same for x equal or in this case n equal four thousand. So in our table the way we would look at it that way is we would say okay well then we have three thousand and four thousand and five thousand. Have to have a couple pieces of information in there. And then for five three thousand this is seven hundred this is t of n and d here the three thousand would be six fifty. And if I do four thousand that's whether the same it's eight fifty for both and the five thousand gives me one thousand and one thousand fifty. And again we're just putting more than one piece of data if you were to putting this on a test for me this is what you would want to do. You would want to show me a table like this and then from here interpret your answer. So when t of n is cheaper at three thousand t of n is bigger but at so we could say that it's greater here it's equal here and it's less than so when it's four thousand it's the same but anything after four thousand. So we would say that it would be four thousand up into you know infinity for the situation infinity may not make sense but more than four thousand maybe we should put it that way. More than four thousand flyers. Well now I know what the answer is so let's see if we can get that same answer by graphing in by symbols. So in a graph I'm going to set my window and now since I've looked at the table I know what my values need to be. So I went to five thousand in the max and since I was in the thousands I went every hundred and I went to a thousand because eight fifty I usually go to the nearest multiple ten. And when it's that close I would go to a thousand and again every one hundred and if I look at my graph hopefully I can see the intersection. And it's on my it's on my thing right here so I need to do second trace five for intersect and then remember you just press enter three times you don't even have to be close. And we see that x is equal to four thousand and y is equal to eight fifty and we would put our two equations in here kind of like we see them. And so it's a way you would get your answer show your answer this is your work. This is a number of flyers and this is cost and then you got your order pair. And then finally for symbols we come back up here to T of n is less than D of n we can literally put that in. So I'm going to do my symbols over here give me a little bit more space. So two fifty plus one point one five n that's T of n is going to be less than D of n which was fifty plus point two zero in. So gathering my like terms I would want to subtract my point one five because it's smaller than my point two. So two fifty is less than fifty plus point zero five that's the difference in. And then I would want to subtract fifty from both sides so two hundred is less than point zero five n. And since I have my calculator here we'll just verify that two hundred divided by my point zero five will give us n is greater than four thousand. There's that number we expected and that's exactly what we said more than four thousand flyers and my inequality says n has to be greater than four thousand. So you would of course want to write a sentence here so you say that four thousand and one flyers or more must be made printed. I guess if we want to be real technical here for the T of n you don't know it by any other name to be cheaper.