Hi, and welcome. My name is Anthony Varela, and today we're going to solve quadratic inequalities that represent motion. So we'll be solving quadratic inequalities. We'll outline the steps for doing that. We're going to be solving algebraically, but we'll also connect back to the graph of the situation, interpret our solution graphically, and we're going to go through an example of an inequality that has no solution.
So first, I'd like to talk a bit about our context, an object in motion. So here we have a picture of our situation. A ball or some object is going to be thrown into the air, and it's going to go up and up and up, eventually reach a maximum height before due to the force of gravity, it starts coming back down towards the ground. And so we're going to be labeling our x-axis as time and our y-axis as height.
Now, some other motion problems we'll have the x-axis also be distance. It would be the distance from the starting point at which our object, it was launched into the air. But in our examples, it's going to represent time. And we could look at this point right here. This is the y-intercept. This is when time equals 0.
So this represents the starting point of our object, and looking over here at this x-intercept, this is when height is 0. So this would then be the point in time where our object hits the ground.
Now, due to this being a real-world situation, we have a couple of natural restrictions. One, is that we're only going to be dealing with positive x values because this represents time, and it doesn't make sense to consider negative seconds.
Now, a similar idea exists with our y-axis. We're only going to be considering positive values of y because it doesn't make sense that when a ball hits the ground, it's going to somehow travel negative feet. So we have a couple of natural boundaries to our situation.
Well, here's my first question and our first example. So a ball can be represented by the equation, y equals negative 4x squared plus 20x plus 8, where y is the height of the ball and x is the time that the ball has been in the air. And we want to know when is the ball above 20 feet.
So we can represent this using a quadratic inequality, so we want to know at what times is the ball above 20 feet, and this is how I'm going to represent it as an inequality. I have 20 representing y, because that's the height, and then I have an inequality statement, and I need this quantity to be above 20, so I'm using this less than sign here. So 20 is less than negative 4x squared plus 20x plus 8.
So how do we then solve quadratic inequalities? Well, our first step is we need to solve this as an equation, an equation that is set equal to 0. So here I made this an equation, and in order to set this equal to 0, I had to subtract 20 from both sides of my equation. So that affected this constant term here, 8 minus 20 is negative 12.
Well, we can solve then this equation by factoring or using the quadratic formula. I prefer to use the quadratic formula. And one thing that I also like to do is, if I can, factor out a common factor between all three of these terms. So I can factor our a 4, so I'll negative x squared plus 5x minus 3, and I'm going to use these values for a, b, and c in my quadratic formula. I'll still get the same results because this is going to give me x values that make this expression here equal to 0. When you multiply that by 4, we'll still get a 0.
So I'm going to be using a equals negative 1, b equals 5, and c equals negative 3 in my quadratic formula. So constructing the quadratic formula, I have x equals negative 5 plus or minus the square root of b squared-- so that would be 25-- minus 4 times negative 1 times negative 3. And this is all over 2a, so that would be 2 times negative 1 all over negative 2.
Let's go ahead and simplify what we have underneath our radical sign, so 25 minus 4 times negative 1 times negative 3. Well, 4 times negative 1 times negative 3 is a positive 12, so 25 minus 12 is 13. So we have negative 5 plus or minus the square root of 13 all over negative 2.
So I have two solutions, one would be negative 5 plus root 13 all over negative 2 and negative 5 minus square root of 13 all over negative 2. So plugging that into my calculator, I get two x values, one is 0.697 when we round, and another is 4.303 when we round.
So what do we do then with these two solutions to our equation? Well, we're going to create intervals on a number line. So here's what our number line looks like. I'm using open circles due to our inequality symbol not allowing the exact value. So I have 0.697 here and 4.303 here. This creates three intervals to our number line.
So what we do now is choose a point that exists within each interval. So I'm choosing 0, 3, and 5. And what we do with these test points is we plug them into our inequality, and we'll see if we get a true or false statements that will guide us to our solution. So I'm going to be plugging in 0, 3, and 5 in for x and seeing what we get in this inequality statement.
So when x equals 0, my inequality would be 0 is less than 0 plus 0 minus 12. And we see that this is a false statement. 0 is not less than negative 12, so we'll get to that in a minute. But now let's substitute 3 in for x. So this would be 0 is less than negative 36 plus 60 minus 12. And simplifying that, we see that 0 is less than 12 is a true statement.
When x equals 5, we have 0 is less than negative 100 plus 100 minus 12. So this combines to 0, so then we just have a minus 12, and we know that 0 is less than negative 12 is a false statement. So false statements are not part of our solution to the inequality, but true statements are. So that will help us then determine solutions to our overall quadratic inequality. So that would be this interval right here in between but not including 0.697 and 4.303.
So what does this then mean for the ball in motion? Well, I'm going to plot 20 feet on our sketch here, and we can see that these then would be x values where the ball is underneath 20 feet, this is the range of values where the ball is above 20 feet, and then once again below. And that occurs in between these two x values that we solved just a few minutes ago.
Well, let's go through example number 2 here. We're going to be using the same situation, the same equation, but we want to know when the ball will be above 40 feet instead of 20 feet this time. So we're going to go through the same steps. First, we need to solve as an equation set equal to 0. So we have this inequality 40 is less than negative 4x squared plus 20x plus 8. To solve as an equation, set equal to 0, I have an equation now, and I've subtracted 40, so that effects this term here. 8 minus 40 is negative 32.
Now, I'm going to factor out the 4, so I can get a, b, and c values to plug into my quadratic formula. So my quadratic formula is x equals negative 5 plus or minus 5 squared minus 4 times a times c-- so really the only thing that changed so far from our last example is our c value-- all over 2a. So that would be negative 2.
Well now, let's simplify what's underneath our square roots. We have a negative number. 25 minus 4 times negative 1 times negative 8 equals negative 7. And if we have a negative number underneath this square root, that means that we have no solution to this quadratic inequality because we can't evaluate the square root of a negative number. So this means that at no point does our ball travel above 40 feet.
So when is the ball above 40 feet? Well, here's our picture. This is where 40 feet would be. And you can see that our ball never reaches a height of 40 feet, so our inequality has no solution. So let's review solving quadratic inequalities that represent motion.
When our object in motion problems are x-axis represented time and our y-axis represented height, here are steps to solving a quadratic inequality, so you can write these down. And remember, if this b squared minus 4ac value from our quadratic formula is a negative number, we have no solution to the quadratic inequality.
So thanks for watching this tutorial on solving quadratic inequalities representing motion. Hope to see you next time.
