 So you know a little bit about solving equations, how do you solve inequalities? There are two approaches to solving inequalities. First, we can utilize special rules involving inequalities. Alternatively, what we can do is solve the equality and identify the solution intervals using test points. We'll take a look at both methods. If we want to use the special rules, then we can proceed as follows. We can work with inequalities in the same way we can work with equalities with two important differences. Never multiply or divide by a variable expression. And if you multiply or divide by a negative number, the direction of the inequality must change. So let's see if we can solve the inequality 3x plus 5 less than or equal to 15 plus 5x, and we'll keep in mind those special rules. So if this was an equation, one of the things we would try to do is to get all of our x terms to one side. So our first step might be to subtract 5x from both sides. So let's do that. Our next step, we'll subtract 5. And if this was an equation, my next step at this point is to divide by negative 2. Well, we can do that as long as we keep in mind that when we multiply or divide by a negative number, the direction of our inequality must change. So when we divide by negative 2, this less than or equal to becomes greater than or equal to. And cleaning things up, we get our solution. Unfortunately, not every inequality can be solved even with these special rules for handling inequalities. So another approach involves finding critical values. And the idea is the following. Rather than trying to keep track of when to chase a direction of the inequality, we can solve the corresponding equality. The solution is called a critical value, and it will separate the number line into two parts. The number is less than the critical value, and, wait for it, the number is more than the critical value. We can then use a test point in each interval to decide whether to include it. It's also important to remember that we also need to test the critical value itself to decide whether to include it. So let's go back to that inequality 3x plus 5 less than or equal to 15 plus 5x. So first, we'll ignore the inequality and we'll solve the corresponding equality. Now we started out by ignoring the inequality, but if you're a good math student or a good human being, you'll recognize that the inequality exists. So our critical value will separate the real number line into two parts, and we'll test a point in each part as well as the critical value to see what we should include in our solution. So first, let's set down our number line, and let's test that critical value. The important thing to remember here is that our critical value, x equals negative 5, makes 3x plus 5 equal to 15 minus 5x. Since our inequality was less than or equal to, equality is allowed, so we should include x equal to negative 5 in our solution, and so when we graph it, there's going to be a closed circle there. Now that we have the critical value, we want to test a point in every interval. And here, notice that the critical value splits the number line into two intervals, x greater than negative 5 and x less than negative 5. We could use any number in an interval to be a test point. So in this first interval, we can pick, well, how about zero? That's an easy number to work with. So we'll see if zero satisfies our inequality. So we'll substitute zero into our inequality relationship, and since this is a true statement, we want to include the interval that contains our test point. To keep track of this, we'll shade that interval. Now we have two intervals, so we do need to pick a test point in the other interval. And while we might mess around with numbers like negative 6, negative 7, a useful idea when dealing with inequalities is the following. Go big or go home. Well, actually if you're watching this, there's a reasonable chance that you're already home, so go big. And so let's pick x equals, oh, I don't know, minus one million. And the reason this is useful is that we're only interested in whether or not our test point solves the inequality or not. And if it's a large number, we don't have to worry about the small bits of arithmetic we can focus on relative magnitude. So I'll drop in x equals minus one million into our inequality. And so over on the left-hand side, we have minus three million plus five. Well, over on the right-hand side, we have 15 minus five million. And we want to know whether the inequality is satisfied. The thing that's useful is that minus three million plus five is pretty close to minus three million. And it's as if we could ignore this plus five as not being that important. Similarly, 15 minus five million, well, that's almost like minus five million. So again, this 15 is sort of unimportant and we can kind of fuzz it out. So again, we only care whether the inequality is true or false, and so we can evaluate whether minus three million is less than or equal to minus five million. And it turns out that this is false, so we do not want to include the interval that includes minus one million. Well, now we have the graphical solution of our inequality, and so from here it's easy to produce the solution in interval notation. And so our solution is x is in the interval from five negative five included off to infinity.