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what I want to do in this video is a handful of fairly simple inequality videos but the real value of it I think will be just to get you warmed up in the notation of inequality so let's just start with one so we have X minus 5 is less than 35 so let's see if we can find all of the X's that will satisfy this equation and that's one of the distinctions of an inequality in an equation you typically have one solution or at least the ones we've solved so far in the future we'll see equations where either have more than one solution but are the ones we've solved so far you solve for a particular X in the inequalities there's a whole set of X's that will satisfy this inequality so they're saying what are all the x's that when you subtract 5 from them it's going to be less than 35 and we can you already think about it I mean 0 minus 5 that's less than 35 minus 100 minus 5 that's less than 35 5 minus 5 that's less than 35 so there's clearly a lot of X's that will satisfy that and what we want to do is come up with a solution that essentially encompasses all of the X's so the way we do that is essentially the same way that we solved any equations we want to get just the X terms in this case on the left hand side so I want to get rid of this negative 5 and I can do that by adding 5 to both sides of this equation so I can add 5 to both sides of this equation that won't change the inequality it won't change the less than sign if something is less than something else something plus 5 is still going to be less than the other thing plus 5 so on the left hand side we just have an X this negative 5 and this positive 5 cancel out X is less than 35 plus 5 which is 40 and that's our solution and to just visualize the set of all numbers that represents let me draw a number line here let me draw a number line here and I'll do it around let's say that's 40 this 40 41 42 and then we could go below 40 39 38 you could just keep going below 40 it just keeps going on in both directions and NEX that is less than 40 will satisfy this so it can't be equal to 40 because if X is equal to 40 40 minus 5 is 35 it's not less than 35 so X has to be less than 40 and to show this on the number line we do a circle around 40 to show that we're not including 40 but then we can shade in everything below 40 so everything that's just exactly below 40 is included in our solution set so everything I've shaded in yellow is included in our solution set so thirty nine thirty nine point nine nine nine nine nine nine repeating which is about as close as you can get to 40 as possible that's in our solution set but 40 is not that's why I put that open circle around it let's do another one lets me do it in another color as well so let's say we have X let me do it over on this top right corner say we have X plus 15 is greater than or equal to negative 60 notice now we have greater than or equal so let's solve this the same way we solve that one over there we can subtract 15 from both sides and I like to switch up my notation here I added the 5 kind of on the same line you could also do your adding or subtracting below the line like this so if I subtract 15 from both sides so I do a minus 15 there and I do a minus 15 there then the left-hand side just becomes an X because obviously a 15 minus 15 that just cancels out and you get X is greater than or equal to negative 60 minus 15 is negative 75 if something is greater than or equal to something else if I take 15 away from this and from that the greater than or equal sign will still apply so our solution is X is greater than or equal to negative 75 let's graph it on the number line so let me draw a number line here I'll have let's say that's negative 75 that's negative 74 that's negative 70 3 that's negative 76 and so on and so forth I could keep plotting things now just to be greater than negative greater than or equal to negative 75 so the X can be equal to negative 75 so we can include the point because we have this greater than or equal sign notice we're not making it hollow like we did that we're making it filled in because it can equal negative 75 or it needs to be greater than so greater than or equal will shade in everything above negative 75 as well so an orange is the solution set and this obviously applet we could keep going to the right X could be a million it could be a billion it could be a Google it can be an arbitrarily large number as long as it's greater than or equal to negative 75 let's do a couple more let's do X X minus 2 X minus 2 is less than or equal to 1 once again we want to get just our X on the left hand side get rid of this negative 2 let's add 2 to both sides of this equation plus 2 plus 2 the left hand side just becomes an X you have a less than or equal sign that won't change by adding or subtracting the same thing to both sides of the inequality and then 1 plus 2 is 3 so X needs to be less than or equal to 3 any X that is less than or equal to 3 will satisfy this equation so let's plot it and I'll try out any X that's less than or equal to 3 and verify for yourself that it does indeed satisfy this inequality I shouldn't call it an equation this inequality so let me graph the solution set so let's say this is 0 1 2 3 4 that's negative 1 negative 2 so X has to be less than or equal to 3 can it can be equal to 3 so we fill in the dot or less than 3 so the solution set or over here is in pink anything less than or equal to 3 and verify it for yourself as f if X is equal to 3 you get 3 minus 2 which is equal to 1 and that that is valid because it could be less than or equal if you do 2 point 9 9 9 9 9 9 minus 2 you get point 9 9 9 9 9 9 which is less than 1 and you could keep trying for any of these numbers in this pink solution set here do one more let's do one more let's say we have X minus 32 is less than or equal to zero same drill as before let's add 32 to both sides of this equation let's add 32 to both sides of this equation the left-hand side just becomes X and then the right-hand side is less than or equal to 32 pretty straightforward same drill when we graph this equation let me draw the number line if this is 32 this is 33 this is 31 I could keep adding things above and below 32 but the solution set is everything less than or equal so we can it could be equal to 32 or less than so we fill in everything below that remember the reason why we're filling in the solid would the reason why 32 is an acceptable solution to this original inequality is because of this lesson or equal sign over here you didn't have less than or equal and that's why 40 wasn't part of the solution set