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 equal, greater than equal. I think we've seen these symbols before, but let's just make it official. What does it mean? So 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. Positive real number. On the number line, of course you can think of this is just meaning that a is to the left of b. If I draw this, 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 gap, 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 rulebook 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, it's call the law tracheotomy. Either you're less than a number, greater than a 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 3. 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, that will 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've got to watch out for, we're 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'll agree that 3 is less than 4, 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? Of course the sign switches. So 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've 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. So let's move it over to the left by subtracting 4 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 3 on the right side. Let's add 1 to both sides. Again additions 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 2 up. I just said it divide by a negative. I'm going to put this in red so we are super aware that we're doing this. So watch out when you do this. 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 2. If I were to try to graph this on the number line, I'm heading back infinitely many choices here, I plot negative 2 and I look at it as an open circle and I go off to the right. Any number greater than negative 2 will satisfy the inequality. You can always test, this plug in, pick your favorite number greater than negative 2 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 1 is less than 2x plus 3, which is less than or equal to 5. When you see something like this, you just realized it's a lazy person's way of writing 2 inequality's at once. You can split this off 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 have both of them the same way, adding, subtracting, isolating xs 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 3 from both sides. So addition, subtraction is not a problem, so we have minus 4 is greater than 2x, and 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, so this 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, that'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 equal to 1. If I were to draw this on the number line, I'd put negative 2 on the map, I put 1 on the map. I don't want negative 2, I've got to be strictly greater. So draw an open circle and then I'll head over to 1, and since I'm allowed to have 1 in here, I'll close the circle up and I'll write this way the number line this is the set of numbers that I'm looking for. While drawing the number line, and these open and close 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 and this means include the number, include the end point. If you don't want to include the end point, if you have an open circle, we tend to use parentheses and this means we do not include the end point. For this particular example, since I don't want negative 2, I would write this as parentheses and negative 2, and since I want 1, I would write a bracket around 1. You can use a bracket or parentheses 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, (negative 2, 1] or you can do an inequality. This is saying negative 2 less than x and x is less equal negative to 1. Maybe I'll right that just even a little more fancy and put it back together in a compound inequality. So write it in inequality notation versus interval notation, and 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, our 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, if you think about all the ways you can include or not include endpoints using parentheses or not, you can have parentheses on the left, brackets on the right, you can have brackets on the right, parentheses 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 the second, we always 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 be an included it so we can have include a to infinity. I think these are all of them now, but there's a couple of things I want to point out. If you use infinity, if you have infinity or negative infinity anywhere, we always use parentheses. We can't get to infinity, I can't include infinity, so this 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 or equal to b, and of course you can start mixing and matching I want x greater than a and less than equal to b and on the side I want to include a but I don't want to x 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, on the same side, I can be less than 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 to be able to become comfortable with intervals and inequalities. Either way you want to write it there fine, it's just a matter of preference. Sometimes the directions will be clear, but otherwise you'll 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 this a little differently. For a quick minute and this is perhaps a little counterintuitive so maybe I'll see you come off on the side secret, you pretend that it's actually equal to 0. I know this is an inequality, but we set it equal to 0 for just a minute and then you solve, and this is why you don't foil it, because we like it and factored 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 the sign graph. We graph the numbers that we just found, 1 and 3, and we put them on the real number. So this is called a sign 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. I mean, don't pick like square of 2 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 in 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 tend to try to pick the easiest ones. When you make a sign graph, the first thing you do is you find the places where x is actually equal to 0 and you graph them. Then you pick your test points, which we've just done, and then here's the nice thing about this, we're going to plug in. You 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 mistake or some other mistake here, so let's just plug in. We're testing to see if it works. So let's see the first one, our 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 asked myself, I say, "Self, is this thing greater than 0?" Let's find out. We get minus 1 times minus 3. Is that greater than 0? Hey, that's just 3. Is 3 great than 0? Absolutely, and so 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. We have 2 minus 1 times 2 minus 3, some plugging in to the original equation. I am not foiling, I'm not doing algebra, I'm just plugging in. I'm 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. So that means I don't want the interval containing the test point. I don't want that middle interval, and 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. 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, but we don't want 1. Why don't we want 1? If you plug in 1, well, remember we said we get 0, so I want all numbers to the left. Is like negative infinity up to 1 parentheses 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 have to be strictly greater than 0, so I want 3 to infinity. So I have these two inner rules that I want to hand back and I could write that with either and or just write out, 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. So you can write the union of the two sets to say or this interval, and again, pick your favorite number. I promise it'll all work there. Let's do another example. This is a variation on the last one. But instead of having 2 in factor form, we have a little bit of work to do to put it in that form. I want to know where x squared minus 4x plus 5 is greater than 2. In general, you want to put a 0 on the right side. If you look at the last question, this was done for us already. Usually you've got to put 0 on one side, pick the right side just because, so let's subtract 2 to both sides. That's fine. X squared minus 4x plus 3 is greater than 0, and then you factor. Now we factor. I want to split this somehow some way as x minus 3 times x minus 1, that factorization may not be immediately obvious. Stare at that for a second. Just convince yourself if that factorization is correct and then realize we're in the exact same spot as the last question, we're in the exact same spot. Just be ready to do a little bit of manipulation to put a 0 on the right side or to factor and then you have the same answer as before, so then we're back to the same process as the last question. 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 factored form, why we go off on the side and we set it equal to zero for a second to try to find the roots or the zeros of this equality. We set it equal to 0. Don't tell anyone I know it's an 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 sign graph. We love these kind of questions there's no algebra to do. We mapped these put them in the right order so we have 1, we have 3, and we have 5. Then once you have your sign 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 our heck 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 values. Let's pick x equals 0 and we'll plug in, and then 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 1, negative 3, and then 5. Two negatives make a positive, and I just get 15, and you can ask yourself, is 15 less than equal to 0? No. I do not want this section over here. We'll do one more and I'll leave the rest to you, but let's test 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, and you ask yourself is that less than or equal to 0? When you simplify you get 1, minus 1, and then 3. Put that all together that's negative 3, and is negative 3 less than equal to 0? Yes. It is. 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 do want the number 1? Do I want the number 3? I want to write this out as 1, 3, but I want to know do I want these parentheses or brackets? We saw before that if you plug in x equals 1 you actually do get 0, and 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 three I'm going to get zero and that is allowed. We use brackets here from 1-3. 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. We said before that five turns it to zero. I want five so let's put them together with the big fancy union sign, and this will be our final answer. I want the interval containing one of the three union bracket five to infinity, and of course, 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 saying any one of the values in this interval will work. Let's do another example. Let's try to make it even more complicated here. We'll take x squared minus 4 over x, 4 minus x, and we'll set all that greater than or equal to 0. Once again fight the urge to do algebra or do division. Let's go off on the side don't tell anyone. I want to know where x squared minus 4 is equal to 0. That's my numerator. You set the numerator equal to 0. When you do that you get x squared equals 4, which gives that x is plus or minus 2. A friendly reminder when you take a square root you have to put the plus or minus. I have plus or minus 2 that I'm going to put on my sign graph, but now I have a denominator. Whenever I have some fractional expression like this same thing as before also set it equal to zero, and when you move things over now you have to do the same for the denominator. Don't tell anyone we're going to set it equal to zero so I have x and 4 minus x equals 0. Two things multiply together to zero so that means that x is either 0 or x is equal to 4. These are the marks that I'm going to put on my sign graph. When I do this I have negative 2, I have 0, I have positive 2, and I have 4. I put all these intervals on the sign graph, and of course, I'm going to test. I pick some test points in between. I care more about the setup here than anything else. A lot of people forget to test the denominator for the sign graphs so just watch out for that, and if it wasn't equal to 0 I would move some things over once I have that. But then you pick your favorite test points, and I'll leave this to you as a check. You don't want the interval to left the negative 2, you do want negative 2-0. You don't want this one. You do want the next one, and you don't want that one. I'll leave this to you to test. Pause the video and do this just to make sure you agree. When you have back these two intervals that you want, ask yourself am I allowed to have negative 2? Am I allowed to have zero? You got to be very careful here. What do you think the right answer is? Am I allowed to have negative 2, and 0, and 2, and 4? That depends on if they appear in the numerator or the denominator. Plugging in negative 2 is perfectly fine. Makes the numerator zero. The denominator is negative 2, and then 4 plus 2 gives you some number, but it's not 0. You do get actually zero, and since I'm allowed to have zero I want the negative 2. But I am not allowed to have zero. Why is that? I'm not allowed to have zero because if I plug in zero, and I'm going to divide by 0. That is not allowed. Zero is a parenthesis. For all the same reasons I'm going to union these two. I want the two. Two makes the numerator zero. I get 0 over some number that's just 0. 0 is greater or equal to 0 that's true, but I don't want the four. If you put the four in there, if you include it by accident you're saying go ahead divide by 0, and of course, that's pretty bad so we don't do that. The final answer here is bracket negative 2, 0 union bracket 2, 4 parentheses. This is as tough as they're going to get if you got this one. Great job. 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 3. 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 the inside thing were positive the absolute values do nothing. It's saying, "Solve for me where 2x minus 1 is less than 3?" That's if the inside is positive. But you got to remember and people forget that sometimes, the absolute value is really testing two things. Says 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. If the inside expression 2x minus 1 is negative I'm going to negate it 2x minus 1 is less than 3. This absolute 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. Pause for a second, but it's positive 2 no big deal and I get x is less than 2. On the other side I got a sum. When I divide my negative 1 I get 2x minus 1. The two negatives cancel, and I have to switch the sign 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, and I get 2x is greater than minus 3 plus 1 is minus 2, and then I divide both sides by positive 2. It's okay to divide by positive you don't have to 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 1 and 2 on the map. I want to be strictly greater than 2 so I want everything to the right of two but not including two. I want to be greater than negative 1. When we draw a number line I want to be less than 2 so I want to be everything to the left of two, but I also want to be greater than negative 1 and 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 you realize this is an inequality you would never write the two first as it comes to the right of the number line. The final answer here in the interval notation, negative 1-2 parentheses on both numbers. Go through some of these practice problems. Keep in mind the golden rule, and working with inequalities. If you multiply or divide by 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.