Hello, amazing math people. We're going to look at absolute value. The absolute value by definition, says that whatever's on the inside is going to come out as a positive. So if X started out being greater than or equal to 0, then the absolute value of X is just going to come out as X. For example, if I had the absolute value of five, it's just going to come out as five. The absolute value of 0 is going to come out as 0. However, if X was less than 0, we're going to have the opposite of X. So if I said the absolute value of -4, I want the opposite of -4, which is 4. If I want the absolute value of -10 negative 10 is less than 0, so I want the opposite of -10 and the opposite of a negative is a positive. If the absolute value of X = P if P is a positive number then X equal P or X equal negative P If P is 0 then X = 0 and if P is negative then there are no solutions. So if we look at this and we say the absolute value of X equal 8, well what makes that true is if the inside was a +8 or the inside was a -8. So the absolute value of +8 is 8 and the absolute value of -8 is 8. So our solutions for this one would be -8 and eight. If we looked at the absolute value of X equaling 0, well, the absolute value of 0 is zero. There is no -0 so our only possibility there is 0. If we looked at the absolute value of X equaling -7, well, the absolute value by definition back here said that the inside was always going to come out as a positive number 50410. So the absolute value can't ever be a negative. So this one has a no solution. Now be very careful. That doesn't mean the inside value can't be negative. It means that the whole absolute value can't equal a negative. Up here we had the inside values being negatives and it worked just fine. We also want to look at absolute value inequalities and the absolute value inequalities. The absolute value of X less than P, so less than is an AND statement. So the inside could be positive or negative. So the inside X was less than P It's a less than and negative X less than P Now frequently you will see instead of this case, people will go ahead and multiply it through by a -1. If we multiplied that whole thing through by a -1, we could think of this as X greater than negative P, because when we multiply or divide by a negative. So sometimes people say, oh, leave this portion alone. For one case, write it exactly the way it is. For the other case, flip the signs. Well, that's true, but I think there's actually a little more logic behind knowing that the inside could be positive, the inside could be negative. So now you have the inside being the only thing that changes. And that's true by its definition. So X less than P and negative X less than P If we wanted to look at an example, say two X -, 3 less than nine. So we'd have two X -, 3 less than 9 and -2 X -3 less than nine. If we solve here 2X greater than 12, or sorry, less than X less than six here, go ahead and distribute the -3. Subtract that three divide by a -2. So we're going to get X less than six and X greater than -3. If we look at this, X greater than -3 an open dot at -3 shading off to the right, X less than six and opened out on 6, shading off to the left. This and statement is really important. It tells me that they both have to be true. Well, the only place that they're both true is in between. So our solution's going to be -3 to six. We need to look at the absolute value of X greater than P next. So the absolute value of X greater than P, Guess what? It's a great or. So it's going to be a or statement great or less than was an and great or was an OR. So if we had the absolute value of X -, 9 greater than four, we'd have the positive case where we leave it exactly the same. By definition, that inside could also be negative. So we'd have the opposite of X -, 9 greater than 4X greater than 13 or negative X + 9 greater than four, negative X greater than -5 X less than five. In this statement, X less than five, open dot shaded off to the left, X greater than 13, open dot shaded off to the right. The fact that it's an OR tells me my solution. So it's going to be negative Infinity to five, union 13 to Infinity. That or was important. If it had been an and, there wouldn't have been anywhere where they were both shaded, so it would have been no solution. Thank you and have a great day.