Everyone. Welcome back. So in previous videos, we saw how to graph linear inequalities, something like y>2x-1. But now you might start to see problems in which you have nonlinear equations, things like quadratics or exponentials, radicals, and rational functions, things like that. I might think that it's an entirely new process, but I'm actually going to show you here that graphing nonlinear inequalities is exactly like graphing linear inequalities. We're actually going to follow the exact same list of steps here. So let's just jump right into this problem and see how it works. We've got y≥x2-1. This is clearly going to be a quadratic equation. Right? So we have something squared. So let's just go ahead and stick to the steps here. The first thing we do is figure out if we're dealing with a solid or dashed line. Let's look at the symbol. So we see a greater than or equal to symbol. If I see a solid line underneath, it's going to be a solid line. That's going to be my equation. Alright? So what does x2-1 look like? Well, first, you're going to have to sort of pretend that there's, like, an equal sign there. So y=x2-1 is going to be a parabola in which the vertex is going to be at 0,-1, and then it opens up like this. So what's going to happen is you're going to look something like this. I'm just going to sketch it out. It doesn't actually have to be perfect unless your professor, really wants this to be perfect. But this is going to be what our parabola looks like. Right? So that's y=x2-1. Let's take a look at the second step. The second step says we're going to test the points on either side by plugging and, actually, it doesn't need to be a line. It could be a curve or something like that. By plugging x,y, the values, into the inequality. Alright? So we can use something on the x or y axis. So we could just pick a point at random. What I'm going to do is I'm just going to pick this point over here, 0,2. That's pretty much, like, well within, sort of the upper portion of the parabola. So let's go ahead and test this out, 0,2. Remember, this is just my x and y values. So what this means here is that y≥x2-1. So I'm just going to replace them now. 2≥02-1. So does this make a true statement? Well, what you'll see here is that you get 2≥02=0, and is 2≥-1? This actually is a true statement. 2>-1. So because this is a true statement now, we can move on to step number 3. We're just going to shade the side that includes that point. If our statement is true, we include that point. So really all this is here is, remember, the curve or this equation here is kind of like a barrier. We can't shade anything that's on the opposite side of that barrier. So you might think, oh, well, I just shade everything that's above this line over here, but that's not how it works. Because if you were to pick a point, let's say, that's out here, above the equation or above the graph but still on the opposite side of this line, you'll find that this actually won't be true. So what you have to do is you're sort of, like, bounded by the shape of the graph itself. So, really, the points that make this inequality true are actually going to be everything that's just above the parabola, but still kind of, like, on the inside of it. So it's kind of, like, on the inside of the bowl. Alright? And that's really all there is to it. So that would be how to graph this inequality and shade the appropriate areas. Alright. So that's how to graph nonlinear inequalities. Let's go ahead and take a look at some practice.
Table of contents
- 0. Fundamental Concepts of Algebra3h 29m
- 1. Equations and Inequalities3h 27m
- 2. Graphs1h 43m
- 3. Functions & Graphs2h 17m
- 4. Polynomial Functions1h 54m
- 5. Rational Functions1h 23m
- 6. Exponential and Logarithmic Functions2h 28m
- 7. Measuring Angles39m
- 8. Trigonometric Functions on Right Triangles2h 5m
- 9. Unit Circle1h 19m
- 10. Graphing Trigonometric Functions1h 19m
- 11. Inverse Trigonometric Functions and Basic Trig Equations1h 41m
- 12. Trigonometric Identities 2h 34m
- 13. Non-Right Triangles1h 38m
- 14. Vectors2h 25m
- 15. Polar Equations2h 5m
- 16. Parametric Equations1h 6m
- 17. Graphing Complex Numbers1h 7m
- 18. Systems of Equations and Matrices1h 6m
- 19. Conic Sections2h 36m
- 20. Sequences, Series & Induction1h 15m
- 21. Combinatorics and Probability1h 45m
- 22. Limits & Continuity1h 49m
- 23. Intro to Derivatives & Area Under the Curve2h 9m
18. Systems of Equations and Matrices
Graphing Systems of Inequalities
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