 So remember that we can collapse AND inequalities, provided the middle parts are the same. So if I have the inequality A less than or equal to B, and also B strictly less than C, since the middle part is the same, I can merge the two inequalities into a single three-part inequality. Now remember in order to do so, the inequalities have to run in the same direction, so I might have to switch them around. But as long as the middle is the same, I can merge the inequalities. It's important to remember we can never collapse an OR inequality this way. And here's a nice way to remember that OR is also a type of rock, at least if you add an E. So these two inequalities have a rock in between them that will keep you from joining them. Now the other side of this is we can expand a three-part inequality as an AND inequality. So this A less than or equal to B strictly less than C becomes A less than or equal to B, and also B strictly less than C. And similarly for the other joint inequality. What this means is that if we have a three-part inequality, we can solve it as a compound inequality. So let's take the inequality 25 less than or equal to 8x plus 1 strictly less than 65. So first we'll rewrite this as an AND inequality, 25 less than or equal to 8x plus 1, and also 8x plus 1 strictly less than 65. Now we have two inequalities to solve, so let's solve this first inequality. Our first step will be subtracting 1, then we'll divide by 8. Now if we want to solve the second inequality, we subtract 1, and then divide by 8. And here's the thing to notice. We did the same things to the equations. Remember we were able to write the original three-part inequality because the middle parts were the same and we merged them. Well here the middle parts are the same, so we can merge them. And this leads to a very useful simplification. Suppose you have a three-part inequality where the variable expression is in the middle. If you apply the same operations to all three parts and respect the rules about dividing or multiplying by a negative, you can solve both inequalities simultaneously. So let's solve the inequality, 35 less than or equal to 12 minus 4x less than or equal to 55. Now remember if you multiply or divide an inequality by a negative number, the direction of the inequality changes. So if we look ahead we see that the coefficient of x is going to be negative 4, and so we will in this current form have to divide by negative 4 unless we can make that into a positive. So let's add 4x to both, wait no, all three sides. And let's simplify where we don't change our inequalities. And now if I want to isolate the x, I need to subtract 35 from all three sides. And I get, oh wait, now I have to subtract 20 from all three sides. Oh wait, I haven't isolated the x yet so now I have to add 20. And you can kind of see how this is going to go. And this leads to an important idea, to solve the three-part inequality this way, you must keep the variables in the middle part, otherwise you won't be able to isolate them. Alright, so let's keep the variables in the middle part. The first thing we have to do is subtract 12 from all three sides. Now we do have that coefficient of negative 4, so now we need to divide all three sides by negative 4 And remember if you multiply or divide an inequality by a negative number, the direction of the inequality changes. So what we have here is wrong, we need to fix it. So let's clean this up a little bit. Now while this is a perfectly good way to leave our answer, one of the things we like to do is to read our inequalities so that they increase as we go from left to right. So let's rearrange this where we've reversed our inequality so it reads negative 43 fourths less than or equal to x, which must also be less than or equal to negative 23 fourths.