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I now wanted to solve some inequalities that also have absolute values in them and if there's any topic in algebra that probably confuses people the most it's this but if we kind of keep our head on straight about what absolute value really means I think you will find it it's not that bad so let's start with a nice fairly simple warm-up problem let's start with the absolute value of x is less than 12 so remember what I told you about the meaning of absolute value it means how far away you are from zero so one way to say this is what are all of the X's that are less than twelve away from zero let's draw a number line let's draw a number line so if we have zero here and we want all the numbers that are less than twelve away from zero well you can go all the way to positive twelve and you could have all the way to negative twelve anything that's in between these two numbers anything that's in between these two numbers is going to have an absolute value of less than twelve it's going to be less than twelve away from zero so this you could say this could be all of the numbers where X is greater than negative twelve those are definitely going to be these are definitely going to be have an absolute value less than twelve as long as they're also and X has to be less than twelve so if some if an X meets both of these constraints its absolute value is definitely going to be less than twelve you know you take the absolute value of negative six that's only six away from zero the absolute value of negative eleven only eleven away from zero so something that meets both of these constraints will satisfy the equation actually we've solved it because this is only a one-step equation there but I think it lays a good foundation for the next few problems and I could actually write it like this in interval notation it would be everything between negative twelve and positive 12 and not including those numbers or we could write it like this x is less than 12 and is greater than negative twelve that a solution set right there now let's do let's do one that's a little bit more complicated it allows us to think a little bit harder so let's say we have the absolute value of 7 X the absolute value of 7 X is greater than or equal to 21 so let's not even think about what's inside of the absolute value sign right now the absolute for the in order for the absolute value of anything to be greater than or equal to 21 what does it mean it means that whatever is inside of this absolute value sign whatever that is inside of our absolute value sign it must be more it must be 21 or more away from 0 let's draw our number line and you really should visualize a number line when you do this and you'll never get confused then you shouldn't be memorizing any rules so let's draw 0 here let's do positive 21 let's do a negative 21 a negative 21 here so we want all of the numbers so whatever this thing is that are greater than that are greater than 20 greater than or equal to 21 they're more than 21 away from 0 their absolute value is more than 21 well all of these negative numbers that are less than negative 21 when you take their absolute value when you get rid of the negative sign or when you when you find their distance from 0 they're all going to be greater than 21 if you take the absolute value of negative 30 is going to be greater than 21 likewise up here anything greater than positive 21 anything greater than positive 21 will also have an absolute value greater than 21 so what we could say is 7x needs to be equal to one of these numbers or 7x needs to be equal to one of these numbers out here so we could write we could write 7 X needs to be equal needs to be one of these numbers well what are these numbers these are all of the numbers that are less than or equal to negative 21 or or 7x let me do a different color here or 7x has to be one of these numbers and that means that 7 X has to be greater than or equal to positive 21 I really want you to kind of internalize what's going on here if our absolute value is greater than or equal to 21 that means that's what's inside the absolute value has to be either just straight-up greater than the positive 21 or less than negative 21 because if it's less than negative 21 when you take its absolute value it's going to be more than 21 away from zero hopefully that makes sense we'll do several of these practice problems so it really gets ingrained in your brain but once you have this set up and this just becomes a compound inequality divide both sides of this equation by seven you get x is less than or equal to negative three or you divide both sides of this by seven you get X is greater than or equal to three so I want to be very clear this what I drew here was not the solution set this is all of this is what 7x had to be equal to I just wanted you to visualize what it means to be have the absolute value be greater than 21 to be more than 21 away from zero this is the solution set X has to be greater than or equal to 3 or less than or equal to negative 3 so the actual solution set to this equation the actual solution set to this equation let me draw a number line let's say that's 0 that's 3 that is negative 3 X has to be either greater than or equal to 3 greater than or equal to 3 that's the equal sign or less than or equal to negative 3 or less than or equal to negative 3 and we're done let's do a couple more of these because they are I think confusing but if you really start to get the gist of what absolute values saying they become I think intuitive so let's say that we have the absolute value let me get a good one let's say the absolute value of 5x plus 3 is less than 7 so that's telling us that whatever is inside of our absolute value sign has to be less than 7 away from 0 so the ways that we can be less than seven away from zero let me draw a number line so the ways that you can be less than seven away from zero you could be less than seven and greater than negative seven right you have to be in this range so in order to satisfy this this thing in this absolute value sign it has to be so the thing in the absolute value sign which is 5x plus 3 it has to be greater than negative 7 greater than negative 7 and and it has to be less than 7 in order for its absolute value to be less than 7 if this thing this 5x plus 3 evaluates anywhere over here its absolute value its distance from 0 will be less than 7 and then we can just solve these you subtract 3 from both sides 5x is greater than negative 10 divide both sides by 5 X is greater than negative 2 now over here subtract 3 from both sides 5 X is less than 4 divide both sides by 5 you get X is less than 4/5 and then we can draw the solution set we have to be greater than negative 2 so we have to be greater than negative 2 not greater than or equal to and less than 4/5 so this might look like a coordinate but this is also interval notation for saying all of the X is between negative 2 and 4/5 or you could write it all of the X's that are greater than negative 2 and less than 4/5 that is these are the sexes that satisfy this equation and I really want you to internalize this visualization here now you could you might already be seeing a bit of a rule here you might already be seeing a bit of a rule here I don't want you to just memorize it but I'll give it to you just in case you want it if you have something like f of X the absolute value of f of X is less than let's say some number a right so this was a situation we have some f of X less than a that means that the absolute value of f of X or f of X has to be less than a away from 0 so that means that f of X f of X has to be less than positive a or greater than negative a that translates to that which translates to f of X greater than negative a and f of X less than a but it comes from the same logic this has to evaluate to something that is less than a away from zero now if we go the other side if you have something of the form f of X is greater than a that means that the that this thing has to evaluate to something that is further than a away from zero so that means that f of X is either just straight up a greater than positive a or f of X is less than negative a right if it's less than negative a maybe it's negative a minus another one or negative you know maybe maybe it's you know negative five plus negative a then when you take this absolute value it'll become a plus five so its absolute value is going to be greater than a so I just want to you know you could memorize this if you want but I really want you to think about this is just saying okay this has to evaluate be less than a way from zero this has to be more than a away from zero let's do one more because I know this is can be a little bit confusing and I encourage you to watch this video over and over and over again if it helps let's say we have the absolute value let's say we have the absolute value of 2x let me do another one over here let's do a harder one say the absolute value of 2x over 7 plus 9 is greater than 5 7 so this thing has to evaluate to something greater it has to be evaluated something that's more than 5/7 away from 0 so this thing 2x over 7 plus 9 it could just be straight up greater than 5/7 or it could be less than negative 5 7 because if it's less than negative 5/7 its absolute value is going to be greater than 5/7 or 2x over 7 plus 9 will be less than negative 5 7s we're doing this case right here and then we just solve both of these equations see if we subtract let's just multiply everything by seven just to get these denominators out of the way so this will if we multiply both sides by seven you get 2x plus 9 times 7 is 63 is greater than five let's do it over here - you'll get 2x plus 63 is less than negative five let's subtract 63 from both sides of this equation and you get 2x let's see if you five minus 63 is 58 2x is greater than 58 if you subtract 63 from both sides of this equation you get 2x is less than is less than or sorry if you subtract 63 from both sides you get negative 68 oh I just realized I made a mistake here you subtract 63 from both sides and this five minus 63 is negative 58 don't want to make a careless mistake there and then divide both sides by two you get in this case X is greater than we don't have to swap the inequality because we're dividing by a positive number negative 58 over 2 is negative 29 or here if you divide both sides by 2 or X is less than X is less than negative 34 less than 68 divided by 2 is 34 and so on the number line the solution set to that equation will look like this that's my number line I have negative 29 negative 29 I have negative 34 so the solution is I can either be greater than 29 not greater than or equal to so greater than 29 that is that right there or I could be less than negative 34 or I could be less than negative 34 so any of those are going to satisfy this absolute value inequality