 In this video, I want to talk about how we can solve absolute value inequalities. Now, like when we solved linear inequalities in a previous video, to solve an absolute value inequality, we're going to begin by finding the markers along the real line. And we do that by solving the corresponding absolute value equation. So with the example right here, if you're trying to solve the inequality, the absolute value of x is greater than 3. First, switch it over to the equation, the absolute value of x equals 3. And like we saw in the previous video, since 3 is positive, we get that this is solvable and we'll get x is plus or minus 3. This is going to give us our markers. And so what we're going to do is then mark these on the real line. So we're going to get x is negative 3 right here, and we're going to get x is positive 3 right here. So we make these markers. And we've learned previously how we could do this using like a sign chart of some kind. Take the x-axis and divide it into three places, negative 3 and 3 by our two markers. I'm actually going to do this because the sign chart requires test points. And I'm going to proceed to do this with a graph. But just as a reminder, if we wanted to solve this using the sign chart, we have our markers 3 and negative 3. We didn't have to pick a number that's on these intervals. Like between negative 3 and 3, we get a 0. If you want something bigger than 3, we could do like a 4. That would work. And then less than negative 3, we could get a negative 4. And then you plug these back into the original inequality. That's what we want to do. So if you take the absolute value of negative 4, which is itself 4, is that greater than 3? The answer is yes. So we're going to put a check mark right there on our sign chart. Great. Let's plug in 0. The absolute value of 0 is equal to 0. Is that greater than 3? And that's a nuh-uh. That doesn't work. So we're going to put an x right here in our sign chart. And then lastly, if you want to take something bigger than 3, take the absolute value of 4, which again is 4, which is greater than 3, that works off. That works here. And so it passes off. And so we're going to put a check right here. And so what this tells us is we want numbers less than negative 3 and greater than 3. So we build the interval negative, negative infinity up to negative 3. Union 3 to infinity. Now do we put brackets or parentheses on the 3? Well, if you're exactly equal to 3, it doesn't work because we don't have a greater than or equal to. We have strictly greater than. So since it's a strict inequality, the markers are not included. Therefore, we're going to put parentheses, thus giving you the answer you can always see on the screen. So we can solve this using the sign charts we talked about before. But I actually am a big proponent of solving these inequalities graphically because the test points require doing arithmetic. This one wasn't so bad because it was just the absolute value of 3. The absolute value of x. But as these things get more and more complicated, the arithmetic for your test points get more and more complicated. And the less arithmetic we can do, the less likely are to make mistakes because in mathematics, that's where the primary mistakes happen when we do arithmetic. That's why man invented calculator to try to avoid those arithmetic mistakes. So what we're going to do here is think about the following. We have our markers x is negative 3 and x is 3. And if I were to rewrite this inequality in a very different way, I could write this as the absolute value of x minus 3 is greater than 0. The reason I'm doing that is let's introduce a function. Let's introduce the function f of x is equal to the absolute value of x minus 3. The reason that's significant is that we're really trying to solve the inequality f of x is greater than 0, which really means we're looking for where it is above the x-axis. Above the x-axis, we should mention that. Where is it above the x-axis? Now, if you graph this function right here, this is just the standard absolute value function where we shifted down the vertex by 3. So the vertex is now at 0, negative 3. It's still going to be going up. It's going to have x intercepts at 3 and negative 3 because we found the markers already. And so you see this graph right here. If you have a V that's pointing up, where is it going to be above the x-axis? It'll happen when you are past 3 right here. And it'll also be above the x-axis when you're to the left of negative 3. So by looking at the graph, we can see very quickly where on the graph are we above the x-axis? Where is the y-coordinate bigger than 0? That's what we mean by above the x-axis. So we can very quickly see that we want to be less than negative 3 and we want to be greater than positive 3, thus giving us the interval negative infinity to negative 3, union 3 to infinity. So other than solving for the markers, which is basically unavoidable, you don't have to do any arithmetic if you solve it graphically. And now that seemed like a little bit much, but let me show you some more examples where I'm actually going to draw these ones by hand. So considering this inequality here, let's first solve the equation. The absolute value of x minus 4 minus 4 is equal to 0. So we want to solve the equation, the absolute value of x equals 4. So x equals plus or minus 4. I'm going to play markers on my x-axis based upon that. We get a 4 and we then get a negative 4 like so. Now the next thing to look at is what's the coefficient of the x right here? It's a 1, which means my absolute value is going to look something like this. Oh no, my drawing is hideous, right? I didn't even hit my x-intercept. Shame on me. It turns out you don't actually have to draw a really good graph right here. We just need to know the basic shape of the graph. Because this absolute value is going to be pointing upwards, we see that the slope of the absolute value, there's no reflection on it. We know that this thing is going to be pointing upwards and it's going to cross the x-axis at 4 and negative 4. Well, what are we looking for? When you look at this right here, less than 0, this means we're looking for those points below the x-axis. And if we want things below the x-axis, that's going to be this sector right here. So despite the crudeness of my graph, I can see that the solution is going to be negative 4 to 4. We want all numbers between negative 4 and 4 because that gives me below the x-axis. And because, again, it's a strict inequality, no equality is allowed there, it's going to be parentheses negative 4 to 4. So we could do this by test points, but I can actually get it very quickly without any test points whatsoever. Let's look at another example. So we want to solve the inequality, the absolute value of 2x plus 4 is less than or equal to 3. So solving the equation, the absolute value of 2x plus 4 equals 3. 3 is positive, so I can proceed. We get 2x plus 4 equals plus or minus 3. This will give us 2x equals minus 4 plus or minus 3. And I'm actually going to break this up in the two cases. You get negative 4 plus 3, which is going to be a negative 1. And you're going to get negative 4 minus 3, which is a negative 7. Divide both sides by 2. My markers are going to be negative 1 half and negative 7 halves. In which case I labeled these on the screen right here. Negative 1 half is right there. And then negative 7 halves. Let's see, that's almost negative 4, 1, 2, 3, 4, but it's just shy of it. So we get a point that looks like this. And again, looking at the original thing, your original slope is going to be positive. This thing is going to be pointing upward, like that. And so you get this picture. Your absolute value graph is going to look something like this. Honestly, the vertex is going to be lower. I drew this to scale. It would actually be about right here, wouldn't it? Because we get a shift down by 3. But I don't want to say you need to worry about that. This doesn't have to be a very good graph. Honestly, when I draw these things, I don't even bother labeling the x-axis. I just draw a line like this. And then I'm like, oh, the absolute value v points upward. So it's like, but boom, like this. Oh, this is the number negative 1 half. Oh, this is the number negative 7 halves. Your graph doesn't have to be very good. We really just need to know the direction that the graph is going. Now, if we want to be less than, less than tells us we want to be below, below, and always will be below the x-axis. That's always the reference we're looking for. So if we're below the x-axis, we're talking about this sector right here, or this sector right here, depending on which graph you're looking at. And therefore, we then see the solutions will be negative 7 halves to negative 1 half, because we want the stuff between. But since we have less than or equal to, this tells us that we're going to have brackets, brackets. And so this then gives us our solution. Negative, we take negative 7 halves to negative 1 half inclusive. We include both end points. And so whenever you have your absolute value is less than or equal to some number, right? This is going to tell you below the x-axis. If we have less than, it's going to be below the x-axis right here. The difference between less than or equal to, right? Just a means do we include the markers? If in contrast you have a greater than something, that would actually be above the x-axis or greater than or equal to will be again above the x-axis. So assuming the absolute value is on the left, greater than means above and less than means below, which kind of makes sense, because that's what the numbers, that's what the symbols mean. Greater than means above, less than means below. And so let's look at a few more examples right here. Let's take the absolute value of 1 minus 4x is less than 5. Because we have a less than right here, I can already tell you that we're going to be looking for things below the x-axis. And we're going to have no mark, and we're not going to include the markers because we are strictly less than 5 here, okay? And so then solving the inequality, the absolute value of 1 minus 4x is equal to 5. 5 is positive so we can proceed. 1 minus 4x is equal to plus or minus 5. Minus 1 from both sides, we get negative 4 equals minus 1 plus or minus 5. Again, I'm going to simplify this thing right here. Negative 1 plus 5 is a 4 and negative 1 minus 5 is a negative 6. Divide both sides by negative 4. We get x is equal to negative 1. And then we're going to get 6 over 4, which is the same thing as 3 halves. It'll be positive like so. And so we can either draw these to scale or we don't even have to do that. Negative 1 and 3 halves like this. Again, looking at the original expression, do I see any negative numbers in front of my absolute value symbol? The answer is no. So this thing will point upwards like so. Again, is this a perfectly drawn graph? No, it's hideous. It's so not good, but it doesn't need to because only the direction matters. If you want to see something drawn perfectly to scale, you can actually look at the graphics in the lecture notes attached to this video, but I'm intentionally not doing that because we don't need them right here. Because we're looking for the things below the x-axis, we're talking about this sector right here and therefore the solution would look like negative 1 to 3 halves because it's strict inequalities, the end points are not included, the markers are not included. And so I had a student once who sort of made the following observation when you look at these absolute values here and he's like, oh, when you look at the absolute value, it's kind of like you're taking a bird, right? You're always looking for either the beak of the bird or the wings of the bird. And as at SUU, our mascot is the Thunderbirds, right? T-birds. You can think of it exactly that. When you draw an absolute value graph and you're solving an inequality, you're either looking for the beak of the Thunderbird or you're looking for the wings of the Thunderbird. Those are always the options when you solve these inequalities right here. So looking at this right here, okay, we have greater than or equal to. So I'm gonna be looking for those that are above the x-axis and I do wanna include the markers this time. Solve the equation, the absolute value of 2x minus five equals three. Well, because we have equals three, we can continue to solve here. We're gonna get 2x minus five is equal to plus or minus three. Add five to both sides. You're gonna get five plus or minus three, which if we consider these two possibilities, you get eight and two. Divide both sides by two. You're gonna get four and one as your markers. So we'll make those on our graph. So we get a one, we get a four. And again, the coefficient in front of the absolute value is a one. So no reflections. That's all I care about. Did we reflect it? So the graph will look something like that. We want the things that are above the x-axis. So we're looking for the wings of the bird this time. There's above and there's above right there. So the answer would look like negative infinity up to one. Bracket, because we include the end points. Union four to infinity. Again, we use a bracket because we're including the end points in this discussion here. And so this is typically how one will solve an absolute value inequality. We can do this with, with, uh, uh, sign charts like we did the first example in this video. But if we think of it from a graphical perspective, we can get it much, much faster. And I think we're less error prone in that situation. There are some things that you should watch out for. Now, if we were to change this problem, if we change this to be negative, the absolute value of 2x minus five is greater than or equal to three. I would caution you if there's a negative sign in front, there's two ways to proceed. You basically, uh, you basically continue on with the way we did before. Um, you're going to find the same markers, but I, you can do that. But I think actually the best approach here is just start off the inequality times being both sides by negative one. In which case you're going to get the absolute value of 2x minus five is less than or equal to negative three. Like so. Uh, and so when you look at something like that, okay, you can then be like, hmm, wait a second. Absolute value less than a negative. That's not possible. There's no solution here, right? And so then we can be kind of done by looking at something like that. Um, so that's one possibility. If you're absolute value, if ever you get absolute value that's less than or less than or equal to a negative, then the answer is going to be no solution. That's not possible. Um, on the other hand, let's, let's work with this picture a little bit more. What if we had actually greater than equal to three in this situation? Again, times both sides by negative one, you're going to get absolute value of 2x minus five is greater than or equal to negative three. What happens in this situation? Absolute value greater than negative. That's not, that's not a no solution. That's actually all solutions, right? You have all real numbers work here. All reels are going to work in this situation because absolute value is always, it's always greater than a negative. And that's also true if you have greater than or equal to. So when you, when you compare an absolute value with a negative, my recommendation is before we start jumping to the equation, which the equation say, well, there's no solution. The direction does matter, right? Because if our absolute value is greater than a negative, we're probably in the situation where we have a line and our function is basically sitting above the line. And so it's like, oh, the function is always above the x-axis. So everything works. Yay, all real numbers. But the other situation could be like our absolute value is always below the x-axis. And if you're trying to look for things that are always above, you know, this thing isn't there. It's like, when is it, when is it above? Well, there's never anything above. So there are situations like that. And that can be handled in a graphical approach as well. So my recommendation is if ever you have a negative sign sitting in front of your absolute value, so you have a negative one, just multiply by that negative one to get rid of it and then switch the signs and act accordingly. It's because if you're getting an absolute value compared to a negative, then you basically have these options. If absolute value is less than a negative, no solution. If your absolute value is greater than negatives, then it's going to be all real numbers. And those are some special cases to look out for. They don't show up that often. But if they do, handle them accordingly. If you're comparing to a positive value though, like we saw in these examples, pick the beak or the wings of the Thunderbird.