We've mentioned in previous lessons that sometimes the inequality sign will change when we're solving inequality problems. In this lesson, we're going to look at when that happens, and it does involve negative numbers. So depending on what we have to do to solve an inequality, if those steps involve doing something with a negative number, we will have to pay attention to whether that inequality sign will stay the same or whether it will change. So we're going to play around with some numbers here just so that we can get a sense of when the inequality sign will change and when it stays the same. So we start off with two numbers, eight and five. And let's see what inequality symbol do we need to put in the middle? Well, eight is greater than five, so we'll use the greater than symbol. Now let's see what happens when we add a negative number to both sides of this inequality. Let's add negative two and see what we get. On the left side, we have eight plus negative two, which gives us six. On the right side, we have five plus negative two, that gives us three. And we know that six is greater than three, so we'll put the greater than sign. Now look at the symbol that we started with and the symbol that we ended up with. Both the greater than symbol. So our inequality symbol stayed the same. So we could say that adding negative two to both sides did not change the sign of the inequality. This time we'll start with the same numbers, but we'll do something a little different. Instead of adding negative two, we're going to multiply by negative two on both sides. So let's see what we end up with. On the left side, we have eight times negative two, that gives us negative 16. And on the right, five times negative two gives us negative ten. And now let's see what symbol should go in the middle. Well, negative 16 is less than negative ten, it's more negative, so that means it's a smaller number. So we'll need to use the less than symbol. And now let's look at the symbol that we started with and that we ended up with. We started with the greater than symbol, but then after we multiplied both sides by negative two, we needed to use the less than symbol. So the symbol changed. It flipped directions. So we could say that multiplying by negative two on both sides does change the sign of the inequality. So when we added negative two on both sides, the symbol did not change, but when we multiplied by negative two, it did change. So let's put all this information together so that you can understand when you'll need to keep the sign the same and when you'll need to change it. So when we add or subtract a positive number to both sides, that will make the inequality sign stay the same. Also, if we add or subtract a negative number, that will also cause the sign to stay the same. So no matter what we add or subtract, the symbol is going to stay the same, whether the number is positive or negative. When we multiply or divide by a positive number, that will also cause the symbol to stay the same. However, just like we saw with our last example, multiplying by negative two did cause the symbol to change. And that also happens with dividing by a negative. So when we multiply, multiply, or divide by a negative number on both sides, the inequality sign will change. Or we could say that it flips to the other direction. So just keep these rules in mind as you're solving inequalities and you'll know exactly what symbol you'll need to use.