 And welcome back today. What we're going to do is we're going to talk about solving quadratic inequalities by using algebra So the previous video that I did we did graphing inequalities One thing that we're going to do here today is we're going to solve quadratic inequalities using algebra So we're just going to use just our arithmetic Okay, just numbers and variables and that kind of stuff Okay, so we want to treat the first thing we want to do is we want to treat the inequality similar to an equation We want to solve for x now as we go through this This is going to be very very similar to what you already know about solving equations solving quadratic equations But this is also going to be very very different There's going to be some what we call critical points and there's some test areas that we're going to do So just try to stick with me on this So a lot of this is going to seem really really familiar and then some of this is going to be very very vague Some of this is going to be a kind of off the wall because maybe it may have been the first time you've seen this But anyway, so we want to treat the inequality similar to an equation We will solve for x so what I'm going to do is I'm just going to rewrite this I'm going to rewrite this as x squared plus 8x plus 20 is equal to 5 Okay, instead of an inequality symbol, I'm going to write an equal symbol It's going to make it easier for me to solve now what I want to do is I want to solve this like any normal equation Okay, when I want to solve quadratic equations, I want to get everything said equal to 0 So let's take this 5 subtract it over x squared plus 8x Plus 15 subtract that over equals 0 Okay, you always want to get everything on one side when you solve quadratic equations Okay, now there's a couple different ways we can solve quadratic equations We can either do factoring we can do completing the square or we can do the quadratic formula always try to do factoring first Factoring is so much easier than everything else. So I want to try to factor first Let's see if I can factor. Okay, so what I need is x is here in the front and then I need numbers So in this case, there's going to be either two positives or two negatives This tells me it's going to be two positives Okay, so 15 numbers that multiply to get 15 and then add to get 8. Oh, that's kind of easy. That's 3 and 5 Okay, 3 times 5 gives me 15 and then 3 plus 5 gives me 8. So there we go I factored pretty easily and then the solutions to this after you factor solutions to set each one of these parentheses equal to 0 x plus 3 equals 0 Equals 0 subtract 3 over x is equal to negative 3 same thing here x plus 5 Equals 0 subtract that 5 over x equals negative 5. Okay, so those those are my two solutions now When we're now with a normal quadratic equation, we would be done. Here's our answers. Here's our solutions. We're done But now with an inequality. We're looking for a range of solutions. We're looking for a lot of solutions Not just one or two numbers. We're looking for a bunch of numbers. Okay, so now these are not solutions These are not solutions. Don't don't don't get me. Don't get me wrong. They're important. They're just not solutions These are what we call critical points Critical I can spell correctly critical points Okay, now these are critical points for our inequality that we're going to use so now remember whenever we do inequalities We we usually have Our answers also include inequalities and we also graph a lot of these so we can get a better idea of what all the solutions are So that's what I'm going to do. I'm going to take I want to take a number line Okay, I'm gonna take a number line and I'm gonna take the points negative three negative five So negative five is over here. I got negative four here and then I got to negative three here So here's negative two put negative six over here Okay, here are my critical points five and negative three are my critical points Okay, now what I want to do here is I want to figure out what numbers that I plug in for x here That are going to give me an answer that's greater than or equal to five So there's a bunch of different numbers that are going to give me the ability to do that I just need to find them now what these critical points are these kind of give me the Boundaries of what points are actually going to work what x's are actually going to work Okay, so but they now the thing is that those are my boundaries now I have to figure out kind of what other numbers are going to work So negative five like it's negative six and negative seven all these numbers are to work is negative four going to work Pay negative two and negative from the one and zero are these numbers over here going to work things like that So what we're gonna what we have to do here is? After we find these critical points what we also need to do is we need to find some test points Okay, so I'm going to do a little bit of labeling here. So critical critical Points we'll abbreviating here. Okay of negative five and negative three now I want to test some certain numbers These numbers that I'm going to test are going to be around my critical points now since I have two critical points I have three areas. I have three areas. You know what? Let me put some let me put some circles here So here's there's a circle from five circle three I have three different areas that could be solutions to this inequality This right here is an area anything that's smaller than negative five. That's an area Okay, anything that's between negative five and negative three That's my second area that could work. Okay, and then this area over here any number. That's bigger than negative three That's my third area that could work and now what I have to do is I simply just have to test a Number inside of all three of those areas to see which areas are going to work That's basically what I have to do. Okay. So in this case we call these test points. So in this case test Points will abbreviating here test points. I'm gonna test out a couple of different points in this case I'm gonna test out now again when you have a choice of what numbers to choose try to choose numbers that are easy to work with So in this case, I'm gonna choose. I'm gonna choose negative six negative six is in this area So that's my first point. I'm gonna test in between here negative four is another point that I'm going to test and then now over here Now you might think I'll just test negative two now Actually, there's an easier number to work with negative one is here and zero is also over here So I'm actually gonna use zero to test to see if this area right here is a solution So actually zero is gonna be my third test point. Okay, I love working with zero You should really love working with zero. It's very easy to add subtracts multiply We can't divide by zero, but it's very very easy to do most of your operations with zero So you should always try to use zero if you have a choice Okay now the reason we call these test points is we're actually gonna take these points We're gonna take all three of these points one at a time And we're going to plug them into this inequality and we're gonna figure out if they work Yes, or no, do they work? Okay, so this takes a little bit of arithmetic just a little bit of arithmetic arithmetic Excuse me, but this is kind of the fastest most efficient way to figure this out Okay, so the first thing I'm gonna do is take in plug in negative six. So take negative six squared plus eight times negative six and then plus 20 and is that going to be greater than or equal to five, okay? All right, so Negative six squared is 36 and then this is going to be negative eight times six is 48 plus 20 Is that going to be greater than or equal to five? So I got that question mark there So in this case get the positive numbers together first. So this is going to be 56 56 minus 8 is going or minus 48 is going to be 6 so 6 is greater than or equal to 5 this does work Okay, so we know that negative 6 actually does work. Okay, it actually it actually works in this equation So let's let's try the other numbers case. So let's try negative negative 4. All right So now we're going to take negative 4 plug it in negative 4 plus 20 is that going to be greater than or equal to Five is that gonna be greater than equal to five now one thing I have with this board Which I haven't used yet is the color so one thing I'm going to do is see if I can let me do it Oh, let me do it. Oh never mind So I was gonna change colors, but that's not gonna work for me this time. So let's get back to work So we're gonna do as I have 16 Plus four times eight is 32. So that's actually gonna be a negative 32. Let's just put a negative sign there Okay, so it was 16 and then the negative 32 and then plus 20 so in this case 16 and 20 make a 30 What is that 36 36 and then minus 32 is 4 Which is not less than or equal to 5 so this is this one doesn't work This one doesn't work. So this one here is just a no I know of them on the edge of my screen here, but this one doesn't work. This one's a big no Okay, that one's a big no Last but not least what I have and I'm gonna do my work kind of right here for this one I'm gonna have zero okay, so now I'm gonna plug in zero and again zero is really easy to work with So I like working with zero Is that greater than or equal to five so zero squared is zero plus eight times zero is zero plus 20 And you can see very quickly that actually 20 is greater than five. So that one actually does work Okay, now all that work. Okay, all that arithmetic to figure this out now Why do we do that? Basically what that tells us is that this area over here? This area actually works. That was the check mark this area here with the negative four all this work down here That tells us this one here is a no we can't use that area It's not gonna be part of our solution Okay, and then the zero area when we plugged in the zero that one actually worked this area over here So basically what that tells us is that our solutions are going to be this way and This way on our inequality. Okay now one thing I also have to think about with these graphs Is that do we actually include negative five and negative three that as I look back up to my work? This is a greater than or equal to science to the or equal to tells us yes that we actually do include those points Okay, so actually this right here is kind of a visual This is a visual on what the graph of this looks like what all the numbers what all my solutions are and Now what I'm gonna do is I'm actually gonna write that out okay using inequality So I'm at this part right here these X's are less than or equal to negative five. So X can be less than or equal to Negative five and then also this region over here X's can be greater than or equal to negative three X's can be greater than or equal to negative three there we go and If you want to write this depending on what your teacher wants if Sometimes teachers allow you to write this as to inequalities like I have here like this right here, or some teachers meant it that you have to write these as as a Combined inequality. So if you want to combine them together Basically what we can do is this one's in order flip this one around. So negative five is Greater than or equal to X and then combine it with this one X is greater Excuse me. Yeah greater than or equal to negative three. And so there's our combined inequality I think either one of these would work again For my students I allow them to write two inequalities because there's two regions here two regions So two inequalities that just make sense to me, but some teachers mandate that you have to they require I should say that you have to write it as a single kind of combined inequality. Okay, so that's all of our work for that Okay, now that's a lot of work. Let me go back through that real quick So first thing we want to do when we're solving quadratic inequalities by using algebra Treat the inequality similar to an equation solve for X So we take this inequality. We set it equal to zero and we solve for X now when you're solving a quadratic equation You can either solve using factoring Completing the square or the quadratic formula and there's a couple other methods, but those are the main three I would suggest always try to factor first and that actually what's what worked here If factoring doesn't work then try to use the quadratic formula quadratic formula will always work. Okay So anyway, we factored we said our parentheses equal to zero and we got these these these points here Now these are critical points critical points. Okay. These are not solutions. These are just critical points Now these critical points I put them on a number line and then what I found the after I found the critical points I found test points and the test points are points that are in certain regions So it was to the left of negative five in between negative five and negative three right here and then over here on the right side So all three of these points were in these certain areas Okay, now take these numbers and then and I know it's a lot of arithmetic But take all those numbers and then just plug them in individually back into your inequality Okay, so then we plugged in negative six found out that it did work We plugged in negative four found out it did not work and then we plugged in zero and we found out that it did work So that tells us that this region worked this region does not work This region does work and so that gives us the two regions of for our solution So the X's are less than or equal to negative five the X's are greater than or equal to negative three Okay, so those are my that's my solution here or you can also write it as a combined inequality again It depends on what your teacher is asking you to do For me for my purse for for my students what I do is I use this one right here I think that's okay because that also shows me that there's two regions For this inequality but some students some teachers will will require you to write as a combined inequality And that's how what you would have right there, okay? Little bit lengthy, but once you go through a couple of these problems You'll you'll fly through these pretty quickly because it's actually relatively straightforward. Yes There's a lot of arithmetic that you have to go through there's a lot of calculating a lot of algebra They have to go through but for the most part it is pretty straightforward Okay, solve it find the critical points take those critical points find test points around it use those test points plug them into your quadratic inequality and then that tells you what regions do work and do Not work and then the regions that do work are going to be your solution. Okay? That's kind of a very long summary of that. All right. Thank you for watching. I hope you hope you enjoyed this video And we'll see you next time