Solving a quadratic inequality. Basically all we have to do to solve such a problem is to sketch a graph of our parabola, choose a test point and see if our test point gave us a true of false statement.
So what I have here is a parabola in vertex form. So I know the vertex, I know the vertex is going to be down 4 units and back 1 and then this 2 tells me that the graph is going to be facing upwards and it's going to be a little bit steeper than normal. So we know how to make this graph precise if we wanted to, but I just want to get a rough sort of graph. It's pretty rough it's not very pretty, but hopefully you get the idea.
Now what we want to do is choose a test point and the test point that we choose has to be clearly on the one side of the region or the other and before I do such test point, I do want to point out a mistake I just made. This is a strictly greater than equality which means that our line is not included, so this actually should be a dotted line, what I'm going to do is just take out some pieces of it to make sure we have that dotted form. If it was equal to, it would be a solid line, so that fix back to our test point.
We need to choose a test point that is clearly in one region outside the parabola or inside the parabola. It doesn't matter which, it just doesn't what to be on that line. So the origin is typically a great point to choose, but I'm not entirely sure which side of the line this origin is on.
It's probably on the inside because I probably drew this a little bit steeper than I should have, but I don't want to make a assumption on what should be the case, so let's choose another point and in general I try to choose points that have one coordinate is 0 because that is going to make my life a little bit easier something is going to go away.
So let's choose the point say 3, 0. Now that's clearly outside of our parabola and so that will give us a good idea of what is going on. We plug 3, what we end up with is 2 times 4² minus 4. 4² 16 32 minus 4 28 and 28 is obviously not less than 0, so this gave us a false statement which means that out here is not true, so we have to shade the other region which is inside the parabola.
So basically all we want to do is graph our parabola, choose a test point and interpret our results.