In our previous lessons, we've looked at writing inequalities for word problems, graphing inequalities. Now we're going to start working on solving inequalities. And in this video we'll work on inequalities that have addition and subtraction with them. Solving an inequality follows the same steps as solving an equation, except we want to pay attention to the inequality symbol. So as we look at the next couple of examples, it's going to look very much like solving an equation, except at the end we just have to pay attention to that inequality symbol. Our first example is m plus two is greater than seven. And our goal for solving inequalities is the same as solving equations. We want to isolate the variable, which means we want to get our variable, in this case m, by itself, and undo any numbers that are with it. And we'll use the opposite operation to undo that number, which in this case is the number two. Whatever we do to that side of the inequality, we have to do the same thing on both sides. And then once our variable is by itself, we'll check the inequality symbol to make sure that it does work for our final solution. So first I'm going to draw a line down through the inequality symbol so that I can clearly see the left side from the right side. And now I want to undo the two so that I can get the m by itself. On that left side, the opposite of adding two is to subtract two, and I'll need to do the same thing on both sides. And now I'll simplify this problem to see what we're left with. On the left, we have the plus two and minus two. When we combine those, they'll equal zero. So it's like they're just canceling each other out. So on the left side, I have just m left by itself, which is what we wanted. On the right side, I have seven minus two that simplifies to five. And now my last step is to bring down the inequality symbol in the middle. So the solution that we have here is m is greater than five. And this solution is telling us that m can be any number that's more than five. It could be five and a half, it could be six, it could be ten, it could be a million, any number that's greater than five. But before we finish this problem, let's do a check just to make sure that we do have the correct symbol here. So I'm going to copy down our original problem. And now we just need to come up with a number to substitute for m in this problem. And since m is greater than five, we can use any number that's greater than five to check our answer. I'm going to use the number six, but you can pick your own number if you want. So now mine becomes six, plus two is greater than seven. On the left. The six plus two is just eight. So this becomes eight is greater than seven and that's true, eight is greater than seven. So since that's correct, I know that I do have the correct inequality symbol in my answer. Our next example is x minus nine is less than or equal to 16. And our steps for solving this are going to be the same as before. We want to isolate the variable and once we've done that, we want to check to make sure that we have the correct symbol in our answer. So on the left side we have our variable x, but we also have minus nine. Now we want to undo that to get x by itself. So the opposite of subtracting nine is to add nine and we'll do that on both sides. Now we'll simplify this to see what we're left with. On the left minus nine and plus nine we'll cancel each other out and we'll have x by itself. And on the right side, 16 plus nine simplifies to 25 and then we'll bring down our inequality symbol in the middle. So our solution is x is less than or equal to 25. And this is saying that x can be any number that's less than 25. It could be ten, it could be zero, it could be a negative number, but it can also be equal to 25 as well. So let's do a check to make sure that that symbol does work for this problem. Now, when we come up with a number to plug in for x, we just need to make sure that it works based on the solution that we have. So we can use any number that is less than 25 or equal to 25. So you could plug 25 in for x as well. Choose whatever number you want that fits those requirements. I'm going to pick the number 20. So now I have 20 minus nine is less than or equal to 16, and 20 minus nine simplifies to eleven. So this becomes eleven is less than or equal to 16. And that's true because eleven is less than 16. So this works. That lets us know that we do have the correct symbol in our answer. And it's very important to do that last step by checking to make sure the symbol is correct. Because there are some situations where there's symbol will change. We might have a different symbol in our answer than we do in our original problem. We'll see those situations in another lesson.