in this example. We're being asked to solve the given absolute value inequality. And then we're gonna graft the solution set on a number line. Well, remember, solving absolute value inequalities is just like solving the absolute value equations. The first thing we need to dio is isolate our absolute value. So we're gonna have to improve the five and negative three to the right hand side. So we're going to start by moving the five. Well, to move the five, we're going to subtract five from both sides of our inequality because five minus 5 to 0. So those were canceled. So now we have negative three. What's right? That a little neither. Negative. Three times the absolute value of eight minus X is less than or equal to. Well, negative seven minus five is negative 12. Okay, well, our absolute value is an isolated, yet it's getting multiplied by negative three. So what we need to do to undo that is divide both sides of our inequality by negative three, because then the negative threes will cancel. So we're gonna bring down the absolute value of eight minus X. Now, don't forget, we just divided both sides of our inequality by a negative value, which means we need to switch our inequality sign so it's really gonna end up being greater than or equal to. And then we have negative 12 divided by negative three, which is positive for perfect. We have now set up our absolute value inequality. We got our absolute value by itself. So what we can do now is set up our two inequalities. So for the first one, remember, we're gonna take what's inside the absolute value, which is eight minus sex. And then we're gonna keep you in the quality side, greater than or equal to, and we're gonna keep our constant. For now, we have to determine. Is this an and or nor statement? Well, because our inequality sign is greater than or equal to is going to be in your statement notice we don't go back to the original problem. We're gonna go. We're only going to decide if it's an Andhra or statement from this step when our absolute value is isolated. So as I've mentioned, this is an overstatement. Now we have to set up our second inequality. Well, to do that would keep the inside, or we keep the expression inside the absolute value, which was eight minus x. Then we're gonna flip. Are inequality sign so less than or equal to and we'll take the opposite of our constant of four. So that will be negative for perfect. Now we've set up our compound inequality. We just need to go ahead and solve. So let's start with the first one. So to get X by itself, I'm going to start by subtracting eight from both sides. Don't forget to bring down the negative. So we have Negative X is greater than or equal to Well, four minus a is negative for then to get X by itself, we're going to divide both sides by negative one. Don't forget, we're divided by a negative value. So we have to flip art in the quality sign so it's gonna be less than or equal to. And then we have negative four divided by negative one, which is positive for all. Right now we need to solve our second inequality. So the steps of the same, we're gonna start by subtracting eight from both sides of the inequality again, bring down the negative sign, and then we have negative four minus eight, which is negative. 12. Then we'll divide both sides of our inequality by negative one and again because we're divided by a negative. We're gonna flip, are in the quality sign, so it's going to be greater than or equal to. And negative 12. Divided by negative one is positive. 12. Perfect. Now we found our solution. X is less than equal before or X could be greater than or equal to 12. So now we want to graph our solution set on a number line so we'll draw our number line. Here. We'll put our critical values. Will have four in 12. And remember, because it's an or statement, we're just gonna graph both inequality separately. So let's start with X is less than or equal to four. Well, that means that four will have a close circle because four is included in the solution set and we're talking about values that are less than four meaning on our number line to the left. So initiate all these values to the left. Okay, Now we want to graft. Inequality X is greater than or equal to 12. We're also gonna have a sorry close circle a 12 because 12 is concluded in the solution set. And because X is greater than or equal to 12, we're going to shade everything to the right. So now we have graft, a solution set to our original absolute value inequality.