okay in this video I'd like to talk to you about linear inequalities a lot of this may be review for you but that's okay we're going to hit it all a very detailed formatting go pretty quickly so feel free to stop the video and replay as you need to I've got several examples I'm going to start out with something like 2x plus 5 less than 17 so that he here is these work almost like equations pretty much the same there's one little rule that you have to keep up with and so just make yourself some kind of note here if you multiply or divide both sides of this inequality by a negative by a negative number on a negative 1 and negative 5 negative something negative anything by a negative reverse or switch the inequality that's the key to the lesson if you multiply or divide both sides by negative not every example do I have that in here so we'll just work through and I'll just show you some of the some of the steps I want you to be able to do okay so back over here I would start out subtract 5 from both sides 17 minus 5 is 12 divided by 2 now I said / so you've got it this one strike your radar it says wait dividing by 1/2 I didn't take negative 2 so the rule doesn't apply / - and you're good this is called the inequality we're gonna put a graph with it so let me draw just a standard old number line that goes from minus infinity to infinity so we're looking for the number line graph this particular exercise says everything less than 6 well here set 6 now I'm from the the school of if you don't include it there's no equal to then I just put an open circle and then I say less than less than means I shade to the left 6 5 4 3 2 1 all those are less than 6 so you can see the emphasis on the number line the next thing is what's called the interval notation and this is important if you're going on to take more math you just speak math period interval notation is very important so that's why I do this number line and I look at the number line and it says hey all my solutions are from over here from minus infinity up to but not including the 6 so you have 4 minus infinity you read it left to right you you write it left to right so from minus infinity up to 6 the notation says a parenthesis here because you can't ever get to negative infinity and a parenthesis on the 6 because you never get to the 6 either you get very close five point nine nine nine nine nine but never six so that's what the parentheses means be watching if you did include the six it would be a bracket so we'll get plenty of examples we'll won't practice that so what's the requirement this linear inequality named solve it like a regular equation keep in mind this rule here we multiply or divide those sides by negative make sure you switch the inequality I'll practice that with you but when you finally get it solved then you draw the number line and write the interval notation so there's really three parts to the answer that's what you need to practice second example it says negative 5x is less than or equal to 30 well the easiest way to solve this for X is to divide both sides by a negative five and so you get X now because I divided by a negative we switch the direction of the inequality and positive divided by negative is negative 30 divided by 5 there we go so I've solved the inequality now I draw the number line I draw the same number line every time from minus infinity to infinity and I read this and it says you need to put a negative 6 on there the equal to means go ahead and include it so that's why I color it in says I need that point and greater than means we need to shade to the right that's where the bigger numbers are so looking at this read it from left to right all the shaded part would be from negative 6 to infinity put a bracket on the negative 6 because you can include it in a parenthesis because where you never put brackets on infinities all right well there's another with 3 parts to the answer let me clear off some space and do some more examples with you so for this example well what I would do is load the X is 2 one side look what happens when I try to subtract 4x from both sides to get it on the other side see like I'm trying to move that one away well that one went away too so what I'm left with is this statement that says 1 that symbol which means less than 7 1 is less than 7 yeah 1 is less than 7 that's true that's a true statement therefore all real numbers all real numbers will work for X so you say all reals we practice doing it this way when I mentioned that somewhere in a lesson back you said what is the number line look like well my number of my generic number line goes from minus infinity to infinity you say yeah but what do you call her in the whole number line okay but what does the interval notation look like well you just right now what you color in so you read it from left to right I've gone from minus infinity to infinity and that's how you're working let's take a look at another example here this one two times X plus three is greater than or equal to 2x plus 11 well if I'm solving this like an equation the first thing I would do is just distribute my two nothing to do on the right side kind of falls back into that example so I subtract 2x from both sides and that gets rid of my X's and what I'm left with is 6 greater than or equal to 11 and this is a false statement so therefore there is no value of x at work there is no value of x at work no solution so what does the graph look like it looks like a number line that never got touched what is your interval look like there is no interval because there is no solution so there's a couple of you know a little crazier cases I've got a few more just to show you how some of the algebra works let's clean off the board here and let me pick up with another example take a minute and write that one down and now let's go solve it there's an understood one sitting right here before the parenthesis so you wanted to stream you that - through there want to get rid of those parentheses if you can there's nothing to do on the right side yet so let's distribute that's a minus X and a minus seventh now I go ahead and combine your 1 minus 7 so that would be a negative X minus 6 or negative 6 months X okay my next step I like to get all my X's on one side here so and it doesn't matter which side put them on there 2 minus 1 that's 1 X move your 6 over and we're looking at X is greater than or equal to 10 we've solved the inequality my number line I choose 10 question is do you fill it in yes because of the equal to symbol and it says greater than which means you need to shade to the larger you if it's greater you shade to the right and then my interval notation reflects what I've shaded well on the left is a 10 on the right is an infinity I include the 10 and of course not the infinity there your three ways to write the answer okay I've got one more example that would kind of fit in this category it says X minus 4 over 6 so this is going back to almost like the first lesson we've had less than X minus 2 over 9 plus 5 18 well that here from a few lessons back I think is less than 1 in this course what we do is we put some parentheses when there's multiple terms sitting on the same fraction multiple terms here and what we try to do is build up and get a common denominator the key is common denominator now you could just multiply all these together you're gonna get a huge number and that's your common denominator way too complicated just stand back and look at it and say 18 is my larger value so what if what if I just multiply this 1 by 2 and that 1 by 3 that would do it so why not just say multiply this 1 by 2 and this 1 by 3 and that will put you an 18 here in 18 here an 18 here you notice I'm multiplying by a positive to a positive 3 so this rule still doesn't apply so since I have all the same denominators we're going to work this as look at the numerators 3 times X minus 4 [Music] less than 2 times X minus 2 plus 5 and now just work through like we've been doing I'll distribute I can combine my like terms right here so that wouldn't take me just a second move my X's to one side doesn't matter which so that gives me X minus 12 add the 12 to that other side and that gives me X on the side there on the left and then this ends up being a 13 so we draw a number line put your 13 somewhere on your number line can you include it no because there's no equal to and then it says no less than less than is to the left less than is always to the left there so now you look at your shaded part to write your interval you read it left to right so we're going all the way from minus infinity up to 13 you cannot include infinities and in this example there's no equal to so there's my interval notation so good good review and bringing the concepts together from a previous lesson okay I have one last example here it's just a little bit different it's a short one though and now I put it over there let me clean that space and show you this last example take a second and write this down and I want to ask you how do you read that in plain English the most common response I've heard is 2 is less than X minus 7 is less than 5 that's true but boy that just doesn't register in my head what I want you to think about is speaking from the standpoint of the variable X which is attached to this minus 7 so let's call it X minus 7 X minus 7 is between 2 & 5 that's what I want you to catch X minus 7 is between 2 & 5 it's greater than the 2 and it's less than the 5 so that means it's got to be between 2 & 5 it's called a compound inequality compound because there's two other well these solve out pretty fast all you need to do is get the X in the middle by itself so this case is really fast you're going to add seven that'll put the X by itself then make sure you add seven to both ends but from the left and the right and so we end up with the X is between okay practice that with me X is between nine and twelve it's bigger than the nine less than the twelve so that means it's between nine and twelve here's what the number line would look like you know I draw this standard old number line here I don't need the infinities X is between nine and twelve it does not include the nine nor does it include the twelve it's bigger than the nine but less than the twelve so to see how that makes sense and then you read that shaded part that goes from nine to twelve not including the nine more than twelve and that's your interval notation good that's your quick review of linear inequalities