All right, Hi everyone and welcome to our lecture on linear inequalities in one variable. Let's just talk about the title for a second. Inequalities just means not equal. When you work with inequalities, you're going to be working with equations of the form x is less than something, who knows what. You could, of course, have that x is greater than something, who knows what, or you can always add the less than or equal or greater than or equal. I think we've seen these symbols before, but let's just make it official. What does it mean? Let's define one of them and the rest will follow immediately. We're going to let a and b be real numbers and we'll say that a is less than b, written as a is less than b. If b minus a, if I take the difference, then I get a positive number, so positive real number. On the number line, of course, you can think of this as just meaning that a is to the left of b if I draw this, so when I subtract, there is some gap, there is some space. From this, you can immediately write the opposite and say that b is greater than or less than, equal to, so you can mix these all up. This is our basic definition and it's interesting to think of this as positioning on the real line and when you subtract, you can capture that difference. Now that we have this notion of inequality, what can we actually do with it? Let's talk about some properties of inequality. This will be our rule book that we have to follow when working and solving inequality. We're going to list out four of them, here we go. The first one says either a is less than b, b is less than a, or a is equal to b. One of these three things has to happen. This is called the law trichotomy. Either you're less than a number, greater than number or equal to the number, nothing else can happen. Think about that for a minute, convince yourself why that is true. If I start with an inequality, if a is less than b and c is some real number, then if I add c to both sides, the inequality does not change. The sign stays the same. Number three, if a is less than b, and now c is a positive real number, then if I multiply both sides, ac will still be less than bc. If you multiply by a positive number, there'll be exactly the same and last one, if a is less than b and c is negative, then ac is greater than bc. This last one is the one we got to watch out for. I'm going to put a little star next to it. The key here is notice the sign changed when you multiply by a negative. Remember division is the same as multiplication by a reciprocal. If we divide or multiply by a negative, we must switch the sign. This is the only algebraic thing you got to watch out for. However, it's extremely important just to show you a quick example of that, hopefully you agree that three is less than four. But if I take c to be minus 1, a nice negative number and multiply it by both sides, minus 3 and minus 4. Stare at this for a second. Which way does the sign go? Which is bigger and of course, the sign switches. If you multiply both sides or divide both sides by a negative, the sign must switch. Let's do some examples to see that in action. Let's solve for the inequality 2x minus 1 is less than 4x plus 3.We treat these almost the same as equalities. However, we got to remember the one exception, watch out when you multiply or divide by a negative. As usual, we have x on both sides. We want to isolate x, we want to put it on one side. Let's move it over to the left by subtracting four from both sides of the equation. When I do that, subtraction is fine proceed as usual 2x minus 4x is good old, negative 2x minus 1, keep the sign the same and now we're left with the three on the right side. Let's add one to both sides. Again, addition of numbers does not make me nervous, doesn't change anything. I get negative 2x is less than 4 and finally, divide both sides by negative two up, I just said it divide by a negative and then put this in red. We are super aware that we're doing this, so watch out. When you do this, you get x is, change the sign, greater than negative. Final answer here is all x that is greater than negative two. If I just to try to graph this on the number line, remember we're heading back infinitely many choices here. I plot negative two and I look at it as an open circle and I go off to the right. Any number greater than negative two will satisfy the inequality. You can always test this plugin. Pick your favorite number greater than negative two and plug it into the original one and I promise you it will work. As another example, let's look at a little more challenging problem called the compound inequalities. I'm going to write negative one is less than 2x plus 3, which is less than or equal to five. When you see something like this, just realized it's a lazy person's way of writing two inequalities at once. You can split this office saying minus 1 is less than 2x plus 3 and, you got to get both conditions to hold 2x plus 3 is less than or equal to 5. When you see this compound inequality, it probably makes sense to treat it into two separate equations. We're going to handle both of them the same way. Adding, subtracting, isolating x as we need, just remember that if you multiply or divide by a negative, we must switch the sign. Let's go through that now, so let's subtract three from both sides. Addition subtractions is not a problem, so we have minus 4 is greater than 2x. Now, I'm going to divide by 2, but I'm dividing, so I just think about it, 2 is positive. I don't have to worry about the sign and I get negative 2 is greater than x. The left side is now done. Moving over to the right side. Let's subtract 3 from both sides. When I do that, I get 2x is less than or equal to 2. Divide both sides by 2. It's a positive number, that's perfectly fine and I get x is less than or equal to 1. Remember there's an and statement in here. I'm looking for all x's that is greater than negative 2 and less than or equal to 1. If I were to draw this on the number line, I put negative 2 on the map, I put 1 on the map. I don't want negative 2. I got to be strictly greater, so I'll draw an open circle and then I'll head over to one. Since I'm allowed to have one in here, I'll close the circle up and I'll write this this way on the number line. This is the set of numbers that I'm looking for. While drawing the number line and these open and closed circles is nice you get to see what numbers are included and not, I tend to write this using intervals. If you want the number you use a bracket. This is a bracket. This means include the number, include the endpoint. If you don't want to include the endpoint if you have an open circle we tend to use parenthesis and this means we do not include the endpoint. For this particular example, since I don't want negative 2, I would write this as parenthesis negative 2. Since I want one, I would write a bracket around one. You can use a bracket or parenthesis on either the left side or the right side. We've seen two answers here depending on what flavor you want to do it. You can do an interval notation, parenthesis negative 2,1 bracket or you can do an inequality. This is saying negative 2 less than x and x is less than or equal to negative 1. Maybe I'll write that just even a little more fancy and put it back together in a compound inequality. Write it in inequality notation versus interval notation. Of course, we even have a little picture on the number line to represent either of these situations. Intervals are going to be our preferred notation going further. The pictures are nice but we tend not to draw pictures. Interval is just more concise and it's really important that you get these down. In general, there's a few different combinations of ways to write this. You can think about all the ways you can include or not include endpoints using parentheses or not. You can have parenthesis on the left, brackets on the right. You can have brackets on the right, parenthesis on the left. Pick your favorite combination. In addition, sometimes we also want infinity as an interval. When we do that, you'll see in a second, since we can't actually include infinity, we always use parentheses. Here's just a bunch of them. I'm sure I'm missing a couple. Maybe we could even have negative infinity to b and include it or we can have include a to infinity. I think these are all of them now, but there's a couple things I want to point out. If you use infinity, if you have infinity or negative infinity anywhere, we always use parenthesis. You can't get to infinity, I can't include infinity, so this is just notation. We're always going to include parentheses with that and just a reminder if you have brackets, that means include the endpoint. If you have parentheses, that means don't. Any interval can be written, of course, in inequality notation. Let's think about the other way you can see this. This says I want a less than x, less than b. If I go the other way here, this says I want a less than or equal to x include the endpoint less than or equal to b. Of course, you can start mixing and matching. 1a x greater than a and less than or equal to b. On this side I want to include a, but I don't want x to equal b. Now over here, a to infinity, what does that mean? That just means that x is greater than a. Any number you want. There's no upper bound. On the other side, if I have b as my upper bound here, this just says that x is less than. When I have parentheses I use the less than sign. If I have the brackets, then I can be less than or equal to a and on the same side I can be less than or equal to b. This very last case when you have all possible numbers negative infinity to infinity. This might be worth writing out again. If you have this, just realize this is the same as the set of real numbers. This is saying all reals. Our goal is going to be able to become comfortable with intervals and inequalities. Either way you want to write it, they're fine. It's just a matter of preference. Sometimes the directions will be clear, but otherwise you write it the way that you know how. Let's do another example. Let's look at x minus 1 times x minus 3, and we'll set this greater than infinity. Now, we have a nice inequality here. We have a product. One of the things I want you to fight the urge to do is to foil this. When you have an inequality we approach these a little differently. For a quick minute, and this is perhaps a little counter-intuitive, so maybe I'll say we come off on the side. Secret, you pretend that it's actually equal to 0. I know this is inequality, but we set it equal to 0 for just a minute and then you solve. This is why you don't foil it because we'd like it in factor form. When you solve it, you get x equals 1 and x equals 3. That you go off on the side to get those numbers, see where it's equal to 0 even though you're working with an inequality, then you come back to your equality. We're going to draw what's called a sine graph. We graph the numbers that we just found, 1 and 3, we put them on the real number. This is called a sine graph. These numbers will partition, they'll split the real number line into different segments. When you have your different segments, you then grab any number you want as your test points. Always pick easy numbers, don't pick like square root of two or something like that. Pick nice easy numbers. What's a nice easy number to pick left of 1, how about x is 0? What's a nice, easy number to pick between 1 and 3, how about x is 2? What's a nice, easy number to pick greater than 3, how about x equals 4? There's no wrong choice here. I just send to try to pick the easiest ones. When you make a sine graph, the first thing you do is you find the places where it's actually equal to 0 and graph them, then you pick your test points, which we've just done. Then here's the nice thing about this. We're going to plug in to test if they work. Plug in. Let's see. Does the numbers actually work? The beautiful thing about this is that there's no algebra. Very hard to make a minus sign mistake or some other mistake here, so let's just plug in. We're testing to see if it works. Let's do the first one. X equals 0. Remember you plug in your test points. When I plug in my test points, I get 0 minus 1 and I get 0 minus 3 and I ask myself, I say self is this thing greater than 0? Let's find out so we get minus 1 times minus 3, is that greater than 0? Hey, that's just 3. Is 3 greater than 0? Absolutely, and so I keep, I put a little check next to the spot that works. That means I want this interval, I want the interval at x equals 0. Let's do the other test point. Remember the other one we picked was x equals 2. Here we go and we have 2 minus 1 times 2 minus 3 so I'm plugging in to the original equation, I am not foiling, I'm not doing algebra, I'm just plugging in and doing arithmetic. 2 minus 1 is 1, 2 minus 3 is minus 1, and I ask myself, I say self is this greater than 0? Hey, is minus 1 greater than 0? No. That means I don't want the interval containing the test point. I don't want that middle interval. Then last but not least, let's test x equals 4. When I do that, I get 4 minus 1, 4 minus 3. Is that greater than 0? That becomes 3 times 1, which of course is 3. Is that greater than 0? Yes, absolutely, so I want them back, so our final answer will then be the combination of the two intervals containing our test points where this inequality was satisfied. That means we want everything to the left of 1. We don't want 1. Why don't we want 1? If you plug in 1, well, remember we said we get 0, and we want this to be strictly greater than 1. I want all numbers to the left. That's like negative infinity up to one parenthesis, or I want everything to the right of 3. I don't want 3 if I plug in 3, I'm going to get equal 0 and had to be strictly greater than 0, so I want 3 to infinity. I have these two integrals that I want to hand back. I can write that with either an or just write out OR that's fine. If you want to get fancy, you can certainly write the union of two sets, 3 to infinity. This is the union and that's perfectly fine as well. You can write the union of the two sets to say, or, this interval, or this interval. Again, pick your favorite number. I promise it'll work there. Let's do another example. Let's look at x minus 1, x minus 3, and 5 minus x is less than or equal to 0. We're going to make it a little more difficult. Now we're going to put less than or equal and I have three pieces multiplied together. Remember do not put these together. Fight the urge we like factor form. Why? We go off on the side and we set it equal to 0 for a second to try to find the roots or zeros of this equality so we set it equal to 0. Don't tell anyone. I know it's inequality, but when we do that, we get x is 1, x is 3, and of course, x is 5. Those numbers then go on our sine graph. We love these questions. There's no algebra to do. We've mapped these, put them into the right order so we have 1, we have 3, and we have 5. Then once you have your sine graph, you pick some nice test points. Pick your favorite points, something nice and easy, 0 seems good, 2 seems good, 4 seems good and again, 6 seems good. With three solutions, we're going to have four test points to do. Once you have your test points, you test the test points. Pick your favorite value. Let's pick x equals 0 and we'll plug in and you ask yourself, is it true, does it satisfy the inequality? When I do this, I get 0 minus 1, 0 minus 3, 5 minus 0, and I ask myself, is this less than or equal to 0? This turns out to be negative one, negative three, and then five, two negatives make a positive and I just get 15. You can ask yourself, is 15 less than or equal to 0? No. So I do not want this section over here. If we'll do one more and I'll leave the rest to you. But let's say x equals 2. What happens if I plug in x equals 2? Then you get 2 minus 1, 2 minus 3, 5 minus 2. You ask yourself, is that less than or equal to 0? When you simplify, you get one, minus one, and then three, put out altogether, that's negative three, and is negative three less than or equal to 0? Yes, it is. So we're going to want that middle interval for the same reasons and I'll let you go do this. You can check, you don't want the interval containing four and you do want the interval containing six. We want two intervals. We want the interval from 1-3. Now, be careful. You have to ask yourself, do I actually want the number 1? Do I want the number 3? I want to write this out as like 1, 3, but I want to know, do I want use parentheses or brackets? We saw before that if you plug in x equals 1, you actually do get 0. Since I want zero, I'm allowed to be zero because less than equals, yes, I want one. For all the same reasons, I want three. When I plug in 3, I'm going to get 0 and that is allowed. We use brackets here from one to three. I'm going to combine it with the other inequality. This is the one from five to infinity, so I want five to infinity. Whenever I have infinity, I always use parentheses. That's not what I'm worried about, five turns it to zero. We said before that five turns it to zero. I want five. Let's put them together with the big fancy union sign. This will be our final answer. I want the interval containing one or three union, bracket five to infinity, and parentheses on infinity of course. Realize what you're doing? You're handing back infinitely many answers for x. This is no longer the days of x equals 2, x equals 7. I have to hand you back a whole interval, seeing any one of the values in this interval will work. Let's do one more example with an absolute value just to see how these work. As an example, let's do 2x minus 1, in the absolute value is strictly less than three. Solve for x. When you have an absolute value, remember what this is saying? The inside could be positive or it could be negative. It's probably worth breaking this off into the two statements that the absolute value is trying to capture. For example, if 2x minus 1 we're positive, the inside thing, where positive of the absolute values do nothing. It's saying, "Hey, solve for me where 2x minus 1 is less than 3." That's if the inside is positive. But you've got to remember, and people forget that sometimes, the absolute values is really testing two things; says, well what if it's positive, then I'll step back and do nothing, but if it's negative, I multiply it by a negative, two negatives, then turn that value positive. So if the inside expression, 2x minus 1 is negative, I'm going to negate it, 2x minus 1, is less than 3. This absolute value expression is really two things written at once. We have to solve two things. That's fine. Let's do the left one first. We'll move the one over. I'm adding numbers to both sides. That's perfectly legal. We get 2x is less than 4. Divide both sides by 2. Just remember you're dividing, so pause for a second, but it's positive two, no big deal, and I get x is less than two. On the other side I get sum, when I divide by negative 1, I get 2x minus 1. The two negatives cancel. I have to switch the sign and I get negative 3. Be careful. Make sure you switch the sign when you divide or multiply by a negative. Once you're here, then it's business as usual. I get 2x is greater than, oops, minus 3 plus 1 is minus 2. Then I divide both sides by positive 2. It's okay to divide by positive. You don't switch the sign, and you get x is greater than negative 1. In terms of the number line, let's think about what we want here. I have negative one and two on the map. I want to be greater than two, strictly greater than two. I want everything to the right of two, but not including two. I want to be greater than negative one. When we draw a number line, I want to be less than two, so I want to be everything to the left of two. I also want to be greater than negative one. What ends up happening is you get the interval right in between them. The final answer, of course here is parentheses, negative 1-2. It's extremely important when you do your parentheses to realize this is an equality, you would never write the two first as it comes to the right of the number line. Final answer here in interval notation, negative 1-2, parentheses on both numbers. Go through some of these practice problems. Keep in mind the golden rule in working with inequalities. If you multiply or divide by a negative, you must switch the sign. Be careful writing and get comfortable using either interval or inequality notation. Great job on this video. We'll see you next time.
