Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Absolute value. The absolute value of X by definition says it's going to be X if X is greater than or equal to 0, and it's going to be the opposite of X if X is less than 0. Basically, the absolute value of X, whatever X is on the inside, comes out as zero or positive. So if the absolute value of X = a positive number then X is just P or negative P. If P is 0 then it's just zero and if P is negative then it's not possible because we can't have an absolute value which by definition is always positive equaling a negative number. So here absolute value of X = 8, the inside X equal 8 or the opposite of X equal 8. If we solve that we can see X is 8 and X equal -8. Because it's a listing of numbers. We usually use set notation and it's just -8 8. If we wanted to check it. Think about does the absolute value of eight equaling 8 and that answers true. The absolute value of -8 also equals 8 and that's true the absolute value of X equaling 0. So X = 0 and the opposite of X = 0. When we solve this, we can see that it's zero in either case. So the absolute value of 0 is 0, absolute value of X equal -7. So this one's going to be not possible because I can never have a positive number on the inside equaling negative number on the outside. So the absolute value of something which always is positive or zero can't ever equal negative. Now this next one, I have five X + 7 absolute value equaling 4X plus three. We're going to have five X + 7 equaling 4X plus three. And then we're going to take the opposite of one of the two sides. It does not matter which side. So let's take the opposite of five X + 7 equaling the four X + 3. And then we're going to solve each of these. So if I subtract 4 XI get X. If I subtract 7, I get -4 over here. I'm going to go ahead and distribute the negative 1st. I'm going to take all the X's to one side, and I'm going to take that three to the other. So when I divide, I get -10 nights. So this is going to have two solutions. We're going to have -4 and nine goes into ten one time with one leftover so -4 and -1 and one ninth. If we look at this next one, we're going to do the same concept. So 7 -, 4 X equaling four X + 5. Then I'm going to take the opposite of one side just to show it doesn't matter. I'm going to do the right side this time. So one case I sometimes think of as a positive case equaling a positive, and the other one is one of them being positive equaling the opposite of the other. So when we solve here, we get 8X equaling 2, and X is going to be 1/4. Over here, 7 -, 4 X equaling -4 X -5. If I add 4X to each side, I get 7 equaling -5. This part's not true, so I'm not going to worry about this piece here. The only solution is going to be 1/4 because one piece had a true solution, the other piece did not. So we go with whatever made one or the other work. This next example X -, 4 equal 4 -, X and take the opposite of one of the two. It does not matter which side oops X -, 4 equal 4 -, X opposite X - 4 equaling 4 -, X. So here when I add XI get 2X and I add four, I get 8 and I get X equal 4 over here distribute that negative. If I add X to each side, I get 4 equal 4. Now this is a true statement. So this is true always. So this one is true from negative Infinity to Infinity. So when I look at the two answers, we know that the final solution is going to be negative Infinity to Infinity because four was already included here. Or all real numbers sometimes written as X such that X is an element of the real numbers if we do inequalities. So absolute value of inequality, the absolute value of X is less than P. This is an and statement. So we're going to split it up into two parts, and then we're going to look where they're both true. So I'm going to have two X -, 3 less than 9, and the opposite of two X -, 3 less than nine. Now sometimes you see this written immediately as two X -, 3 greater than -9. And we can see that if we multiplied or divided each side by a -1, that would indeed be the equation we came up with. So solving both these, we get 2X less than, 12X less than six. Remember it's an AND statement. 2X less than -6 X oh sorry, X greater than -3 so two X -, 3 we add three to each side. 2X greater than -6 X greater than -3 and statement. So if I put a -3 greater than and if I have a six and less than, if it's and I need them both to be true. So it's going to be all the XS such that -3 less than X less than six. For our set notation, I prefer interval notation, and that's going to be ( -3, 6 parenthesis. I'm going to let you try #8 absolute value of X greater than P Is an OR statement. The same exact steps. So X -, 9 greater than four or negative or the opposite of X -, 9 greater than four. So if I add nine, I get 13. Now I can divide it all by a negative if I wanted to, but I could have also distributed the negative. So if I had negative X + 9 greater than four, if I subtract the nine divide by a negative, I can see that I do indeed get the same solution whether I distribute the negative out first or whether I divide everything through by a -1. So here I need X greater than 13 or X less than five. So an interval notation reading left to right, it's negative Infinity to five, union 13 to Infinity. Set notation, it's X such that X is less than five or is done by the union sign X is greater than 13 and you try the last one. Thank you and have a wonderful day.