In this lesson, we'll look at solving inequalities that have multiplication and division in them. Here's our first example. Four y is less than or equal to 36. As we're solving this inequality, our steps are going to be the same as solving an equation, which means that we want to focus on isolating the variable. We wants to get that variable y by itself. And to do that, we'll have to undo the number that's with it using the opposite operation. And whatever we do to that side, we have to do the same thing on both sides of the inequality. And once we do have our variable y by itself, we'll need to check the inequality symbol to make sure that we do have the correct symbol in our answer. So first I'm going to draw a line down through our inequality symbol just so that I can clearly see the left from the right side of this problem. And now I'm going to focus on the side that has the variable y. We can see that we have the number four right next to the y. There's no symbol, no plus sign or minus sign or anything in the middle, so that lets us know that they are being multiplied together. So, to undo the four and get y by itself, we need to do the opposite of multiplication. So we'll need to divide by four. And I'm going to use the fraction line to represent division here. It just helps us to keep our work nice and organized. And since I'm going to divide the left side by four, I also have to divide the right side by four. And now I can simplify what I have and see what we end up with. So on the left, we have a four in the numerator and a four in the denominator of this fraction. That means they're essentially going to cancel each other out. Four divided by four is equal to just one, so we would have one times y, which is the same as just y. So all of that simplifies to just y. So we have our variable by itself, which is what we wanted. Now, on the right side, we have 36 divided by four that simplifies to nine. And now we just need to bring our inequality symbol down in the middle. So our solution is y is less than or equal to nine. And before we finish this problem, we should check to make sure that we do have the correct inequality symbol in our answer. Because sometimes the symbol will change for certain problems. So it's always a good idea to check and make sure you have the correct one. And to do that, we'll copy down our original problem. And now we need to substitute a number in for y so that we can test and make sure that this does work. Since our solution says that y is less than or equal to nine, we can use any number that is less than or equal to nine. I'm going to use the number eight. So here I substituted eight for Y, and now I'll simplify this. Four times eight gives us 32. So we have 32 is less than or equal to 36. And then we ask ourselves if that's true. And it is true because 32 is less than 36. So that tells us that our answer is correct. We did choose the correct inequality symbol for our solution. For our next example, we have k over five is greater than twelve, and our steps for solving this inequality will be the same as before. We want to focus on isolating the variable k. So let's draw a line down through our inequality symbol and focus on the side that has the variable which is the left side, where we have the variable k. Remember, a fraction represents division, so k over five really represents k divided by five. So to undo that five, we'll have to do the opposite of division, which is multiplication. So I'm going to multiply that side by five. And notice that I place the five that we're multiplying by up with the k in the numerator. It's very important that you put it in the numerator to show that we're multiplying by the whole number five. And whatever we do to that side, we have to do to the right side as well. So we'll also multiply that side by five. And now we'll simplify this and see what we're left with. So on the left side, we have five in the numerator and five in the denominator. They will cancel each other out, just like in the last problem we saw. We have the same number in the numerator and denominator. It simplifies to just one, and we would end up with one times k for this problem, which is the same as k. And on the right side, twelve times five gives us 60 and we'll bring down our greater than symbol in the middle. So our solution is k is greater than 60. Now let's check just to be sure that our symbol is correct. Start with our original problem. Now we need to pick a number to plug in for k. You can pick any number you want as long as it is greater than 60. I'm going to use 100. So now we have 100 divided by five on the left side, which simplifies to 20. So now I have 20 is greater than twelve, and that is true. So that tells me that we do have the correct any quality symbol. And our solution is k is greater than 60.