 So here's a few more examples of solving quadratic inequalities. So let's say I want to solve the inequality x squared minus 4x plus 7 greater than or equal to 5. So again, we'll start by ignoring the inequality and solving the corresponding equation. And this time for a variety sake, let's solve this using the quadratic formula. And this gives us our critical values. So let's graph this inequality. We'll put down our number line. And ordinarily, our next step would be plotting the critical values. But where are they? To figure out where these are, it helps to think about our critical values as having two solutions, which we can rewrite through a little bit of arithmetic. In this case, we can split up the fractions and do a little bit of simplification. The reason this is useful is that we can see one of our solutions is 2 plus a little bit. And that means we're someplace to the right of 2. So plot first, then label, we're someplace to the right of 2. Likewise, our other solution is 2 minus something, so we're someplace to the left of 2. So a good math student and a good human being acknowledges the existence of the inequality. And since this is a greater than or equal to, then we want to include both of our solutions. So that means we'll use closed circles of both of these boundary points. And these boundaries separate the number line into three parts. We'll try a test point in each part. Now over on the way, way, way, way left, we can try x equals negative 1 million. Substituting that into our inequality, we get a true statement. So we want to shade this portion. How about the middle? Remember these two solutions, these two closed dots, are at 2 plus a little bit and at 2 minus a little bit. So that means x equals 2 is right in the middle. So we can use x equals 2 as our test point. Substituting that in. Our inequality is false, so we exclude the middle. And finally, on the way, way, right, we try x equals oh, I don't know, 1 million. Substituting that in. We get a true statement, and so we include the right-hand side. And since we have the graph, we can now write our answer in interval notation. Our solution goes from minus infinity up to and including 2 minus root 8 over 2. But it also includes union, the interval from 2 plus root 8 over 2 onto infinity. How about this inequality? The corresponding equality is going to give us the quadratic equation, and we can solve this any way we want. I don't know about you, but I want to solve this using the quadratic formula. But this equation has no real solutions. That shouldn't bother us, we can still plot our number line. And since there are no boundaries, the real number line is one big interval, so we can test any point. So we'll test x equals... Well, how about x equals 0? We see that the inequality is true, so we include 0 and the entire interval that contains it, which means that we include everything. So our solution is going to be x in the interval minus infinity to positive infinity. Or how about this inequality? We'll solve the corresponding equation. And this has no real solutions, so the entire real number line is one big interval. Test any value of x that you want, how about x equals 0? Substitute that in. And this statement is true. I mean false. So we don't include this interval that includes x equals 0, but since there's no other interval, this means we don't include any interval. And so we say that this inequality has no solution.