Let's solve this quadratic inequality. Algebraic Lee. Now, when we're solving algebraic Lee, we're not gonna do a graph. But we are gonna end up doing a number line once we find out. But similar to graphic and solving through graphs, we're gonna go ahead and we're gonna factor. So we'll set this up as though it were in equation. Remember, we're gonna actually subtract are six on both sides because we needed to equal zero. So when a factor, I'm gonna find two factors of negative six that will give me one. And that's gonna be X plus three and X minus two. So they're gonna equals zero. Sum will solve that out. X minus three equals zero plus three. We're going to track three on both sides. X would equal negative three. An X minus two equals zero plus two plus two. So expert equal to now, this looks like we were solving an equation and we do started off Justus that we were solving an equation. The difference is going to come in and what we do here because this point we're needing toe, look at what the inequality is and not just the solutions. So here we're gonna do is we're actually going to create a number line and on that number line, let's say we've just got negative three and positive, too. And we'll just do zero. You don't have to have a full number line. We're just gonna do this. And for this one, we're gonna divide this number line into three sections. Our first section is X is going to be less than negative three. Our second section is going to be kind of a compound inequality where we're gonna have negative three is less than X, which is less than two, and then we're gonna have X is greater than to. So what we want to do is we're gonna kind of tests, um, numbers out and we want to see if they would ring true. So if we want to find out if X is going to be less than negative three, we're gonna pick a number less than three. So in this case, that would be negative. Four. So we're going to test negative four. So basically, we're gonna put that in place of our inequality, and we're gonna go back to the original inequality so we would have negative four squared plus four. So that's going to be a 16 plus four. And that is 20 which is greater than six. So that statement is true. So this means that X is going to be less than negative three. So that part is true, so I'm not gonna have to try this middle section. So now let's try to make sure X is greater than two. So let's pick a number greater than two. Well, let's do three. So we're gonna task where X is three. So that would be three squared plus three, which would be nine plus three and 12 is greater than six. And that statement is also true. So because both of those were true, I could test the middle. Let's test just to show it. So let's just use zero, because zero is the number that comes in between. So that would be if X equals zero with that statement. Be true. Well, you have zero squared plus zero, which would be zero zero is not less than six are greater than six. So that statement falls. So that means that this is gonna be an or statement. X is less than negative three or X is greater than to. So let's try another one. Let's do X squared. Plus five X is less than six. So I'm gonna rewrite this one as an equation. So it's gonna be X squared. Plus five X plus six would be zero. So when I factored, this one that's gonna be X Plus two and X plus three would equal zero, and then we're gonna work him out. So X plus two equals zero and X plus three equals zero. Subtract two. So X would equal negative to subtract. Three X would equal negative three. Okay, so let's draw our number line. So we said X is negative tube and negative three. So we know those statements are true, So let's do negative three and then negative, too. So we're gonna divide this into three parks. One where X is less than negative. Three. One where X is greater than negative, too, and are in between where negative three is less than X, which is less than two. And we want to see where do our answers law. So for this one, let's test a number less than negative three. So let's say X is negative. Four that's less than negative. Three. Sorry equation is X squared plus five x so that would be negative. Four squared, plus five times negative four. So that would be 16. Minus 20 would be negative for and we're saying negative for is less than negative six. And that statement is false, so that can't be it. So let's go into the middle section and let's try out a number between negative two and negative three. Well, let's just do negative 2.5. So that would be negative. 2.5 squared plus five times negative 2.5. So when I do that, that's going to get me 6.25 minus 12.5. So that's gonna give me negative 6.25 And we said we wanted less than negative six. And that statement is true. So that tells me, because the middle section is true, that my solution is going to be negative. Three is less than X, which is less than negative, too

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Let's solve this quadratic inequality. Algebraic Lee. Now, when we're solving algebraic Lee, we're not gonna do a graph. But we are gonna end up doing a number line once we find out. But similar to graphic and solving through graphs, we're gonna go ahead and we're gonna factor. So we'll set this up as though it were in equation. Remember, we're gonna actually subtract are six on both sides because we needed to equal zero. So when a factor, I'm gonna find two factors of negative six that will give me one. And that's gonna be X plus three and X minus two. So they're gonna equals zero. Sum will solve that out. X minus three equals zero plus three. We're going to track three on both sides. X would equal negative three. An X minus two equals zero plus two plus two. So expert equal to now, this looks like we were solving an equation and we do started off Justus that we were solving an equation. The difference is going to come in and what we do here because this point we're needing toe, look at what the inequality is and not just the solutions. So here we're gonna do is we're actually going to create a number line and on that number line, let's say we've just got negative three and positive, too. And we'll just do zero. You don't have to have a full number line. We're just gonna do this. And for this one, we're gonna divide this number line into three sections. Our first section is X is going to be less than negative three. Our second section is going to be kind of a compound inequality where we're gonna have negative three is less than X, which is less than two, and then we're gonna have X is greater than to. So what we want to do is we're gonna kind of tests, um, numbers out and we want to see if they would ring true. So if we want to find out if X is going to be less than negative three, we're gonna pick a number less than three. So in this case, that would be negative. Four. So we're going to test negative four. So basically, we're gonna put that in place of our inequality, and we're gonna go back to the original inequality so we would have negative four squared plus four. So that's going to be a 16 plus four. And that is 20 which is greater than six. So that statement is true. So this means that X is going to be less than negative three. So that part is true, so I'm not gonna have to try this middle section. So now let's try to make sure X is greater than two. So let's pick a number greater than two. Well, let's do three. So we're gonna task where X is three. So that would be three squared plus three, which would be nine plus three and 12 is greater than six. And that statement is also true. So because both of those were true, I could test the middle. Let's test just to show it. So let's just use zero, because zero is the number that comes in between. So that would be if X equals zero with that statement. Be true. Well, you have zero squared plus zero, which would be zero zero is not less than six are greater than six. So that statement falls. So that means that this is gonna be an or statement. X is less than negative three or X is greater than to. So let's try another one. Let's do X squared. Plus five X is less than six. So I'm gonna rewrite this one as an equation. So it's gonna be X squared. Plus five X plus six would be zero. So when I factored, this one that's gonna be X Plus two and X plus three would equal zero, and then we're gonna work him out. So X plus two equals zero and X plus three equals zero. Subtract two. So X would equal negative to subtract. Three X would equal negative three. Okay, so let's draw our number line. So we said X is negative tube and negative three. So we know those statements are true, So let's do negative three and then negative, too. So we're gonna divide this into three parks. One where X is less than negative. Three. One where X is greater than negative, too, and are in between where negative three is less than X, which is less than two. And we want to see where do our answers law. So for this one, let's test a number less than negative three. So let's say X is negative. Four that's less than negative. Three. Sorry equation is X squared plus five x so that would be negative. Four squared, plus five times negative four. So that would be 16. Minus 20 would be negative for and we're saying negative for is less than negative six. And that statement is false, so that can't be it. So let's go into the middle section and let's try out a number between negative two and negative three. Well, let's just do negative 2.5. So that would be negative. 2.5 squared plus five times negative 2.5. So when I do that, that's going to get me 6.25 minus 12.5. So that's gonna give me negative 6.25 And we said we wanted less than negative six. And that statement is true. So that tells me, because the middle section is true, that my solution is going to be negative. Three is less than X, which is less than negative, too

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