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We've got the inequality negative x times the expression 2x minus 14 is greater than or equal to 24. So I encar, encourage you to pause this video now and think about what the solution set to this inequality would actually be, and actually plot the solution set on a number line. So I'm assuming you've given a go at it, so let's just try to simplify this a little bit. So on the left hand side we could distribute, we could distribute this negative x, and so if we did that we would get negative 2, negative 2x squared, negative times a negative is a positive, plus 14x is greater than or equal to 24. Now I'm gonna put the, I'm gonna subtract 24 from both sides, just so that we just have a 0 here and then we can think about factoring what we have here on the left. So we have negative 2x squared, plus 14x, I'm gonna subtract 24 from both sides, so it's minus 24 is greater than or equal to, I've subtracted 24 from the right as well, so that's going to be greater than or equal to 0. Now, I don't like, I don't like having this negative 2 out front, so what I wanna do is I wanna divide this left hand side by negative 2, but I can't just divide the left hand side only by negative 2, I have to do divide the right hand side by negative 2 as well. And anytime I multiply or divide both sides of an inequality by a negative number, it's going to flip the inequality. So if I divide both sides by negative 2, I'm going to be left with x squared, positive x squared minus, so I'm dividing by negative 2, so minus 7x, plus 12. And now since I divided by negative 2, I'm gonna flip this inequality, is less than or equal to zero divided by negative 2 is 0. So that simplified things a good bit, and now let's see if we can factor this quadratic expression. So two numbers whose product is positive 12, so that means they're gonna have the same sign, and whose, and whose sum is negative 7. So if they have the same sign, and their sum is negative 7, that tells us that they're both going to be negative. And let's see, negative 3 and negative 4 seem to fit the bill. Their product is positive 12. Their sum is negative 7. So we could write this as x minus 3 times x minus 4, is going to be less than or equal to 0. So now this is the point that we're gonna do a little bit of interesting logic. If the product of two things is less than, is less than or equal to 0, what does that tell us, tell what do we know about it? Well, that tells us that either, either one or both of them is 0, or they have different signs. The only way that you're gonna get less than 0 is if one is positive and the other is negative, or one is negative and the other is positive. So our 1 is negative, is that they have different signs. So let's write that down, let's write that down. So either, either x minus 3, x minus 3 is less than or, less than, either x minus 3 is less than or equal to 0 and, and x minus 4 is greater than or equal to 0, x minus 4 is greater than or equal to 0, so notice, note this one is non positive, this one is non negative, they're either equal to 0 or they are have different signs. So that's one situation, or, or the other way around, or x minus 3 is non negative, it's greater than or equal to 0, and x minus 4 is non positive, x minus 4 is less than or equal to 0. Once again, they're either zero or different signs, that's all I'm doing with this little with this little logic work right over here. So what are these, what is this, what is this simplified to? So x minus 3 less than or equal to 0, add 3 to both sides, you get x is less than or equal to 3. And, and x minus 4 is greater than or equal to 0. If you add 4 to both sides of this, you get x is greater than or equal to 4. So what values of x are going to be less than or equal to 3, and greater than or equal to 4? Well anything that's less than or equal to 3 is not going to be greater than or equal to 4 and anything that's greater than or equal to 4 is not going to be less than or equal to 3. So there's no x, there's no x value that can satisfy this situation right over here, there's no X value that will result in this one being negative, and this one being, or this one being non positive and this one being non negative. So let's go to this one right over here. So if we add 3 to both sides, we get x is greater than or equal to 3 and, and we get adding 4 to both sides, x is less than or equal to 4. Now does this make sense, that something could be greater than or equal to 3 and less than or equal to 4? Sure, for example, well 3 is greater than or equal to 3 and is less than or equal to 4, 4 is greater than or equal to 3 and it's less than or equal to 4 and anything in between. So we can plot the solution set here. So this is actually all that matters. Cuz this one, there's no situation which that would've been true. So this is the only thing, this is the only thing that's gonna make this or part true. This part is always going to be false. So if we wanted to make the solution set, it would look something like this. So if this is our possible values of x. So let's say that that is 0, so this is 1, 2, 3, and 4, 3 and 4, 3 and 4, x could be greater than or equal to 3, so it's greater than or equal to 3. But it's also let, it also has to be less than or equal to 4 so we can't just go all past 4. Also, less than or equal, less is since it's, less than or equal we can color in these dots, less than or equal to 4. So anything in this range, including 3 and 4, that's why we circled in the dots, this would satisfy this equation here. And if you wanted to think about it visually, hey, you know we know that this type of thing, we know that this type of thing, or this type of thing, this type of thing right over here, these are, these are parabola's. So how would that relate to this little solution set that we just thought about right over here? Well if you look at let's, let's just look at one of these. Let's say we went to, let's say we went to this form right over here. So all, everything we did, this is just another way of thinking about negative 2x squared plus 14x minus 24 is greater than or equal to 0. So this right over here, we have a negative coefficient on the x squared term, that's going to be a downward opening parabola. So when is that greater than or equal to 0? So if, if we though about the downward opening parabola, it could might look something like this, it might look something like this if we're now thinking in 2 dimensions. And this is the, this is the, if you think of this as the Y axis right over here. So when is that greater than or equal to 0? Well it's greater than or equal to 0, it's above the X axis in this range for x right over here. So that's one way of thinking about it. If we though about it from the point of view, not of that parabola, not of that parabola, but this parabola right over here, when is x squared minus 7x plus 12 less than or equal to 0? Well this is gonna be an up, upward opening parabola. So if it has a positive coefficient here, so this problem might look something like this, might look something like this, when is it less than or equal to 0? Well once again, once again it's less than or equal to 0 in that same range.