Now we're going to look at solving inequalities with negative numbers and we're going to focus on problems that have multiplication and division in them. Our first example is negative three. X is greater than or equal to 18. So we'll draw a line through the middle of our inequality so we can see the left from the right side. And then we'll focus on the left side so that we can get x by itself, which means that we need to undo the negative three that's being multiplied by x. So we'll do the opposite, which is divide by negative three. And whatever we do to that side, we have to do to the right side as well. So on the left, negative three over negative three will cancel each other out because that will simplify to just one times x, which is the same as x. And on the right side, we have 18 divided by negative three, which is negative six. Now we have to decide what symbol goes in the middle. We learned from our last lesson that whenever we're multiplying or dividing by a negative number, we have to flip the inequality sign. And since we had to divide by negative three in this problem, that means we do have to flip our sign. So we start with the greater than or equal to symbol, and now it flips to the other direction and becomes a less than or equal to. Now we can see what our solution will look like on a number line. Since x is less than or equal to negative six, that means we'll need a closed circle at negative six to show that it's included as one of our answers. And then we'll shade everything to the left to represent all of the values that are less than negative six. So it would look like this. And our last step is to check our answer. So we'll start off with our original inequality and we'll need to figure out some number to substitute for x. And remember, we can choose any number that's less than or equal to negative six. I'm going to choose negative ten. So by plugging negative ten in for x, I have negative three times negative ten is greater than or equal to 18. By simplifying the left side, negative three times negative ten gives me positive 30. So now I have 30 is greater than or equal to 18, which is true. So now I know I do have the correct inequality symbol for this problem. We have m over negative four is less than two. We'll draw a line down through our inequality symbol and focus on isolating our variable m. Since that fraction represents division, we'll have to use multiplication to undo that negative four. So we'll multiply by negative four on both sides. On the left, negative four in the numerator cancels out the negative four in the denominator. So I have just M, and on the right side, two times negative four gives me negative eight. And once again, since we divided by a negative number, we'll have to flip our inequality sign. So the less than symbol flips around to become the greater than symbol. Now let's see what that looks like on the number line. So, since our solution says that m is greater than negative eight, I'm going to have an open circle at negative eight, because negative eight itself is not one of the solutions for m, but everything greater than negative eight is. So I'm going to shade all of the values that are to the right of negative eight, so it will look like this. Now let's check our answer. Start with our original problem and let's see what number do we want to plug in for m, it can be anything that's greater than negative eight. I'm going to choose the number zero. So now I have zero over negative four is less than twelve. Well, zero divided by negative four is just zero, so that becomes zero is less than twelve, which is true. So I know that I have the correct symbol. Now, it's very careful to pay attention to whether or not we are dividing or multiplying by a positive or negative number as we're solving our inequalities, because that determines if our symbol needs to change or not. So let's look at these two problems and see if we can figure out the difference. Here on the left, we have negative five. X is greater than negative ten. Negative five is being multiplied by x. So we need to divide both sides by negative five to get x by itself. So this problem will become x is less than negative two. So our symbols had to change because we divided by a negative five. Remember, dividing by negative five caused the sign to change. Now let's look at the right side. Now, on this problem, we have five. X is greater than negative ten. We still have a negative number in this problem, we have the negative ten on the right side. However, that's not what determines whether the sign will change or not. It's what we're actually doing to get our variable by itself. And since the coefficient of x this time is a positive five, we're dividing by a positive five on both sides. And dividing by a positive number does not cause the sign to change. So our solution will be x is greater than negative two. So remember, if we're dividing by a negative, the sign changes. But dividing by a positive five causes causes no sign change, it stays the same. So keep this in mind as you are solving your problems. Just ask yourself whether your step involves dividing or multiplying by a negative number, and that causes the sign to change. But if you divide or multiply by a positive, the sign will stay the same.