So in this section we're gonna be talking about how we solve absolute value inequalities. Now, if you remember from last chapter, when we solve the absolute value equations remember, we would end up having to set up two different equations because most of the time there were two answers. Well, you're gonna find something very similar is gonna happen when we solve absolute value inequalities. We're gonna essentially end up having a compound in the quality. So let's try and figure out how this is going to work. So let's start with this first example. The absolute value of X is less than three. Or remember, the absolute value means whose distance from zero is less than three in this particular case. So let's think about who's distance from zero would be less than three. Well, it could be one. It could be, too. It could be 2.5 everything up until three. But in this particular case, not including three. So all of these values, starting at zero up until three, all have an absolute value. Who's less than three because their distance from zero is less than three? Well, it's kind of the same idea going in the left direction, meaning our values of extra getting smaller because negative one would be included because the distance from negative 120 is only one. Well, that's less than three. Same thing with negative to same thing with negative 2.5 all the way up to just not including negative three. So all of these values between zero and negative three would also be included in our solution set. Okay, so now let's try and write out and inequality to represent our number line. Well, we have two critical points here at negative three and positive three, and we're shaving all the values in between. Remember, this is when we usually use an and statement. So in this particular case, we would say that negative three is less than X, which is less than three. Now. What I want to do is break it down into the and statement. So our first inequality says negative three is less than X, so we have negative. Three is less than X, and our second in part is X is less than three. So what I want to do for the first inequality is flip it around so that way the exes first. Well, in that case, we would have X is greater than negative three and X is less than three. So let's take a look at our original problem. Compared to our solution we have, the absolute value of X is less than three, so notice the inside is less than three. That's one of our inequalities, or what's on the inside is the opposite and the reversal of the sign of the, um, original value. So notice we flipped inequality sign and which this our value to be a negative. And you're gonna find that that's going to be very useful when we start setting up equations and involved less than so, The key here is always going to be in an statement, because we're all always gonna be talking about the values in between these two numbers for this particular case in between three and negative three. All right, let's take a look at our second example. Thea Absolute value of X is greater than three. That means those numbers whose distance from zero is greater than three units away. So, for example, four would work because its distance from zero was bigger than three same thing with five, basically anything that's greater than three. Because all of these values in our shaded area has a distance. Who's from zero? That's greater than three. Well, we can think about on the opposite direction, starting at negative three again Open circle because negative three is exactly three units way. But all of these values to the left and negative three would also work because their distance from zero is greater than three. Okay, so now let's come up with an inequality to represent our number line. Well, for this first piece we have, the X is less than negative three. And for the second piece, this is one X is greater than three. And because we have these two separate shaded areas, remember, this will be an or statement, so kind of like the first example to get our first or the positive answer. We take what's inside our absolute value, and we keep the sign and the number on the opposite side, as opposed to our first one. The negative direction. We flipped it inequality sign and switch our value to be negative. So that's going to be the general rule. And whenever it's greater than it will always be in or a statement. So let's try and summarize this for us. Let's put a little separator in here. So whenever we have the absolute value off and the expression, I'm gonna call it X, and it's going to be less than any particular number I'm going to call it. See, our solutions will be that X could be lesson see, and X could be greater than negative. See? So notice we flip the sign and change the seed to be a negative value, as opposed to when we have the absolute value of X is greater than some constant. So our first inequality will be X is greater than see. Yeah, but this time it's gonna be an or statement. So we're still gonna have what's inside the absolute value. We're gonna flip part in the quality sign to be less than, and we're gonna take the opposite of that constant. So we'll have negative C. So these are two general forms, so whenever it's less, then it'll be in an statement greater than is or and it doesn't matter if it's a less than or equal to or greater than or equal to less than or less than or equal to will always be in and statement and greater or greater than or equal to will always be in your statement. So in the next couple of videos will go well. We have to solve absolute value inequalities. It's very much just like solving absolute value equations offer. Then we need to know if they're going to be an and or nor statement, and in the very last example we do. Example Number four, I'll show you the special cases you might potentially have.

## Comments

## Video Transcript

So in this section we're gonna be talking about how we solve absolute value inequalities. Now, if you remember from last chapter, when we solve the absolute value equations remember, we would end up having to set up two different equations because most of the time there were two answers. Well, you're gonna find something very similar is gonna happen when we solve absolute value inequalities. We're gonna essentially end up having a compound in the quality. So let's try and figure out how this is going to work. So let's start with this first example. The absolute value of X is less than three. Or remember, the absolute value means whose distance from zero is less than three in this particular case. So let's think about who's distance from zero would be less than three. Well, it could be one. It could be, too. It could be 2.5 everything up until three. But in this particular case, not including three. So all of these values, starting at zero up until three, all have an absolute value. Who's less than three because their distance from zero is less than three? Well, it's kind of the same idea going in the left direction, meaning our values of extra getting smaller because negative one would be included because the distance from negative 120 is only one. Well, that's less than three. Same thing with negative to same thing with negative 2.5 all the way up to just not including negative three. So all of these values between zero and negative three would also be included in our solution set. Okay, so now let's try and write out and inequality to represent our number line. Well, we have two critical points here at negative three and positive three, and we're shaving all the values in between. Remember, this is when we usually use an and statement. So in this particular case, we would say that negative three is less than X, which is less than three. Now. What I want to do is break it down into the and statement. So our first inequality says negative three is less than X, so we have negative. Three is less than X, and our second in part is X is less than three. So what I want to do for the first inequality is flip it around so that way the exes first. Well, in that case, we would have X is greater than negative three and X is less than three. So let's take a look at our original problem. Compared to our solution we have, the absolute value of X is less than three, so notice the inside is less than three. That's one of our inequalities, or what's on the inside is the opposite and the reversal of the sign of the, um, original value. So notice we flipped inequality sign and which this our value to be a negative. And you're gonna find that that's going to be very useful when we start setting up equations and involved less than so, The key here is always going to be in an statement, because we're all always gonna be talking about the values in between these two numbers for this particular case in between three and negative three. All right, let's take a look at our second example. Thea Absolute value of X is greater than three. That means those numbers whose distance from zero is greater than three units away. So, for example, four would work because its distance from zero was bigger than three same thing with five, basically anything that's greater than three. Because all of these values in our shaded area has a distance. Who's from zero? That's greater than three. Well, we can think about on the opposite direction, starting at negative three again Open circle because negative three is exactly three units way. But all of these values to the left and negative three would also work because their distance from zero is greater than three. Okay, so now let's come up with an inequality to represent our number line. Well, for this first piece we have, the X is less than negative three. And for the second piece, this is one X is greater than three. And because we have these two separate shaded areas, remember, this will be an or statement, so kind of like the first example to get our first or the positive answer. We take what's inside our absolute value, and we keep the sign and the number on the opposite side, as opposed to our first one. The negative direction. We flipped it inequality sign and switch our value to be negative. So that's going to be the general rule. And whenever it's greater than it will always be in or a statement. So let's try and summarize this for us. Let's put a little separator in here. So whenever we have the absolute value off and the expression, I'm gonna call it X, and it's going to be less than any particular number I'm going to call it. See, our solutions will be that X could be lesson see, and X could be greater than negative. See? So notice we flip the sign and change the seed to be a negative value, as opposed to when we have the absolute value of X is greater than some constant. So our first inequality will be X is greater than see. Yeah, but this time it's gonna be an or statement. So we're still gonna have what's inside the absolute value. We're gonna flip part in the quality sign to be less than, and we're gonna take the opposite of that constant. So we'll have negative C. So these are two general forms, so whenever it's less, then it'll be in an statement greater than is or and it doesn't matter if it's a less than or equal to or greater than or equal to less than or less than or equal to will always be in and statement and greater or greater than or equal to will always be in your statement. So in the next couple of videos will go well. We have to solve absolute value inequalities. It's very much just like solving absolute value equations offer. Then we need to know if they're going to be an and or nor statement, and in the very last example we do. Example Number four, I'll show you the special cases you might potentially have.

## Next Lectures