 So once you can solve quadratic equations, you can also solve quadratic inequalities. And in fact, this will establish a general trend. Once we can solve equations, we can solve inequalities. And we'll do so using the following method. First, we'll find the critical values, which will solve the equation. Next, we'll determine whether to include the critical values in the solution. Then, we'll test points in each interval defined by the critical values. So let's start with a simple inequality x squared less than 50, and we'll graph and express our answer in interval notation. So first, we'll ignore the inequality, and we'll simply assume that x squared is equal to 50. And this gives us two solutions. x equals plus or minus square root of 50. Now, we do want to graph it. So here's an important idea. We'll plot first, then label. In other words, don't spend a lot of time trying to figure out exactly where square root of 50 is on the number line. And that's because our number line is meant to organize our answer, but is not the answer itself. So we'll just set down a number line. So we know that positive square root of 50 is a positive number, so it's someplace to the right of 0, about here. So we'll plot it first, but then we'll be sure to label it square root 50. And likewise, negative square root of 50 is a negative number, and it's somewhere to the left of 0. So maybe around here. Now, remember, x equals plus or minus square root of 50 solves the equality. But because we're good math students and good human beings, we remember that the inequality exists. And our inequality is we want x squared to be strictly less than 50. And so that means, since the inequality is strict, we should have open circles at x equals square root 50 and at x equals minus square root of 50. And this gives us three intervals to check. In the middle interval, we can test x equals 0, substituting that into our inequality. We get a true statement, so we include the middle interval. On the right, we can go big or go home. We can test x equals 1 million. And this is false, so we exclude the right interval. On the left-hand side, we can test x equals minus 1 million. So we'll substitute that into our inequality. And we get a false statement, so we exclude the left section. Since we have the graph of our solution, it's easy enough to convert that into our solution in interval notation. So the solution interval goes from negative root 50 to positive root 50. And we can express it in interval notation as we can also solve a more complicated quadratic inequality, x squared minus 8x greater than or equal to 84. So we'll solve the corresponding equality. So solve this any way you feel like. I feel like completing the square. So let's put down a number line and graph the critical values. So x equals 14 is someplace to the right of 0. x equals negative 6 is someplace to the left. Since the inequality is greater than or equal to, the critical values, which will solve the equation, will be included. So we should mark them with closed circles. And the critical values split the number line into three parts, so we'll test to point in each part. On the left-hand side, we'll try x equals minus 1 million, substituting that into our inequality. And this is a true statement, so we do include this interval. In the middle, we'll try x equals 0 seems good, substituting that into our inequality. This is false, so we exclude the middle. And on the right, we'll try x equals 1 million, substituting that into our inequality. We get a true statement, so we include this interval. And because we've graphed the solution, it's easy to convert that into a solution in interval notation. And so our solution is going to be.