So in this example, we're being asked to solve the given absolute value inequality, and I'm gonna hafta solution set on the number one. So remember, solving absolute value inequalities is very similar to solving absolute value equations. The first thing we need to dio is isolate our absolute value simple. Well, right now, the absolute value is getting subtracted. Bye bye, subtracted by two. So we're gonna have to move that to to the other side. To do this, we're gonna add to to both sides of our inequality, because negative two plus two is zero, so these will cancel. So what we're left with is the absolute value of X ministry is greater than or equal to. Well, nine plus two is 11. All right. Perfect would now isolated their absolute value. Now we want to set up our two inequalities. So our first inequality will be what's inside the absolute value, which is X ministry. We're going to keep the inequality sign greater than or equal to, and we keep our constant 11. Okay, so now the question is, is this an or statement or an and statement? Well, remember is greater than or equal to which means it will be in your statement. Okay, let's set up on next. Inequality. So for the second inequality again, we keep what's inside the absolute value, which is X minus three. We're gonna flip are inequality sign. So we're gonna have less than or equal to, and we're gonna take the opposite of 11, which is negative. 11. Perfect. Now we have our compound inequality. We just need to go ahead and solve. So let's start with the first one. X minus three is greater than or equal to 11. So the solve for X I'm gonna add three to both sides of my inequality. Well, 11 plus three is 14, so we have X is greater than or equal to 14. Now, let's solve the second inequality. Well, to get X by itself again, we're gonna add three to both sides. However, in this case, we have a negative 11 plus three, which is negative. Eight. So we have X is less than or equal to negative vein. Perfect. We have now solved our compound inequality Well, our absolute value inequality. So we've found our solution is that X could be greater than or equal to 14 or X could be less than or equal than negative. Now we need to graph the solution set on a number line, so we'll set up our number line here. We'll put our Q critical values negative and positive. 14 And remember to graph or statements. You just graph both inequality separately, just on the same number line. Well, Halloween graph X is greater than or equal to 14. Well, we would have a close circle at 14 because it's included in solution set and because it's greater than or equal to 14, will shade to the right of 14. So all of these values in this red shaded area would be potential solutions. Now we have to Graft X is less than their equal the negativity again. We're gonna have a close circle because negative eight is included in the solution set. And because X is less than or equal to negative eight, we're going to shade everything to the left so everything in this red shaded area will also be part of our solution set. So now we have graft, the solution set to our original absolute value inequality

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So in this example, we're being asked to solve the given absolute value inequality, and I'm gonna hafta solution set on the number one. So remember, solving absolute value inequalities is very similar to solving absolute value equations. The first thing we need to dio is isolate our absolute value simple. Well, right now, the absolute value is getting subtracted. Bye bye, subtracted by two. So we're gonna have to move that to to the other side. To do this, we're gonna add to to both sides of our inequality, because negative two plus two is zero, so these will cancel. So what we're left with is the absolute value of X ministry is greater than or equal to. Well, nine plus two is 11. All right. Perfect would now isolated their absolute value. Now we want to set up our two inequalities. So our first inequality will be what's inside the absolute value, which is X ministry. We're going to keep the inequality sign greater than or equal to, and we keep our constant 11. Okay, so now the question is, is this an or statement or an and statement? Well, remember is greater than or equal to which means it will be in your statement. Okay, let's set up on next. Inequality. So for the second inequality again, we keep what's inside the absolute value, which is X minus three. We're gonna flip are inequality sign. So we're gonna have less than or equal to, and we're gonna take the opposite of 11, which is negative. 11. Perfect. Now we have our compound inequality. We just need to go ahead and solve. So let's start with the first one. X minus three is greater than or equal to 11. So the solve for X I'm gonna add three to both sides of my inequality. Well, 11 plus three is 14, so we have X is greater than or equal to 14. Now, let's solve the second inequality. Well, to get X by itself again, we're gonna add three to both sides. However, in this case, we have a negative 11 plus three, which is negative. Eight. So we have X is less than or equal to negative vein. Perfect. We have now solved our compound inequality Well, our absolute value inequality. So we've found our solution is that X could be greater than or equal to 14 or X could be less than or equal than negative. Now we need to graph the solution set on a number line, so we'll set up our number line here. We'll put our Q critical values negative and positive. 14 And remember to graph or statements. You just graph both inequality separately, just on the same number line. Well, Halloween graph X is greater than or equal to 14. Well, we would have a close circle at 14 because it's included in solution set and because it's greater than or equal to 14, will shade to the right of 14. So all of these values in this red shaded area would be potential solutions. Now we have to Graft X is less than their equal the negativity again. We're gonna have a close circle because negative eight is included in the solution set. And because X is less than or equal to negative eight, we're going to shade everything to the left so everything in this red shaded area will also be part of our solution set. So now we have graft, the solution set to our original absolute value inequality

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