Intro to absolute value equations and graphs

let's do some equations that deal with absolute values and just as a bit of a review when you take the absolute value of a number let's say I take the absolute value of negative 1 what you're really doing is you're saying how far is that number from zero and in the case of negative 1 if we draw a number line right there that's a very badly drawn number line so if we draw a number line right there that's zero you have a negative 1 right there well it's one away from zero so the absolute value of negative one is 1 and the absolute value of 1 is also it's also one away from zero it's also equal to 1 so on some level absolute values you know it's the distance from zero but another I guess simpler way to think of it it always results in the positive version of the number the absolute value the absolute value of negative seven thousand three hundred and forty six is equal to seven thousand three hundred and forty six so with that in mind let's try to solve some equations with an app with absolute values in them so let's say add the equation the absolute value of X minus five is equal to ten and one way you can interpret this and I want you to think about this this is actually saying that the distance between X and five is equal to ten so how many numbers that are exactly 10 away from five and you can already think of the solution to this equation but I'll show you how to solve it systematically now this is going to be true in two of two situations either X minus 5 is equal to positive 10 right if this evaluates out to positive 10 that when you take the absolute value of it you're going to get positive 10 or or X minus 5 might evaluate to negative 10 if X minus 5 evaluated to negative 10 when you take the absolute value of it you would get 10 again so X minus 5 could also be equal to negative 10 both of these would satisfy this equation now to solve this one add five to both sides of this equation you get X is 2:15 to solve this one add five to both sides of this equation X is equal to negative five so our solution there's two X's that satisfy this equation X could be 15 15 minus 5 is 10 take the absolute value you're going to get 10 or X could be negative 5 negative 5 minus 5 is negative 10 take the absolute value you get 10 and notice both of these numbers are exactly 10 away from the number 5 let's do another one of these let's do another one let's say we have the absolute value of X plus 2 is equal to 6 so what does that tell us that tells us that either X plus 2 that the thing inside the absolute value sign is equal to 6 or the thing inside of the absolute value sign the X plus 2 could also be negative 6 right if this whole thing evaluated negative 6 you take the absolute value you would get 6 so or X plus 2 could equal negative 6 and then if you subtract 2 from both sides of this equation you get X could be equal to 4 if you subtract 2 from both sides of this equation you get X could be equal to negative 8 so these are the two solutions to the equation and just to kind of have it gel in your mind that absolute value you can kind of view it as a distance you could rewrite this problem as the absolute value of x minus negative 2 is equal to 6 and so this is telling you this is asking me what are the X's that are exactly 6 away from negative 2 remember up here we said what are the X's that are exactly 10 away from positive 5 right whatever number you're subtracting from positive 5 these are both 10 away from positive 5 this is asking what is exactly 6 away from negative 2 and it's going to be 4 or negative 8 you could try those numbers out for yourself let's do another one of these let's do another one and we'll do it in purple let's say we have the absolute value of 4x I'm going to change this problem up a little bit 4x minus 1 the absolute value of 4x minus 1 is equal to actually this cube is equal to 19 so just like the last few problems 4x minus 1 could be equal to 19 for X minus 1 could be equal to 19 or 4x minus 1 might evaluate to negative 19 because then we take the absolute value you're going to get 19 again or 4x minus 1 could be equal to negative 19 then you just solve these two equations add 1 to both sides of this equation we can do them simultaneously even add 1 to both sides of this you get 4x is equal to 20 add 1 to both sides of this equation you get 4x is equal to negative 18 divide both sides of this by 4 you get X is equal to 5 divide both sides of this by 4 you get X is equal to negative 18 over 4 which is equal to negative negative 9 halves so these both of these X values satisfy the equation try it out negative 9 halves times 4 this will become a negative 18 negative 18 minus 1 is negative 19 take the absolute value you get 19 you put a 5 here 4 times 5 is 20 minus 1 is positive 19 so you take the absolute value once again you'll get a 19 let's try to graph one of these just for fun so let's say I have let's say I have y is equal to the absolute value of x plus 3 so this is a function or a graph with an absolute value in it so let's think about two scenarios there's one scenario where the thing inside of the absolute value is positive so you have the scenario where X plus 3 I'll write it over here X plus 3 is greater than 0 and then you have the scenario where X plus 3 is less than 0 right what X plus 3 is greater than 0 this graph or this line or I guess we can't call it a line this function is the same thing as Y is equal to X plus three right if this thing over here is greater than zero then the absolute value sign is irrelevant so then this thing is the same thing as Y is equal to X plus three but when is X plus 3 greater than zero well if you subtract 3 from both sides you get X is greater than negative 3 so when X is greater than negative 3 this graph is going to look just like Y is equal to X plus 3 now when X plus 3 is less than 0 in the situation where this the inside of our absolute value sign is negative in that situation this equation is going to be y is equal to the negative the negative of X plus 3 how can I say that well look if this is going to be a negative number if X plus 3 is going to be a negative number right that's what we're assuming here if it's going to be a negative number then when you take the absolute value of a negative number you're going to make it positive that's just like multiplying it by negative 1 well if you're if you know you're taking the absolute value of a negative number it's just like multiplying it by negative 1 because you're going to make it positive and this is going to be the situation this is the X plus 3 is less than 0 if we subtract 3 from both sides when X is less than negative 3 so when X is less than negative 3 the graph will look like this when X is greater than negative 3 the graph will look like that so let's see what that will make the entire graph look like let me draw let me draw my axes that's my x-axis that's my y-axis so just to let me multiply this out just so that we have it in MX plus B form so this is equal to negative X minus 3 negative X minus 3 so let's just figure out what this graph would look like in general negative X minus 3 negative X minus 3 the y-intercept is negative 3 so 1 2 3 and negative x means it goes it slopes downward has a downward slope of 1 so look like this it would look like this the x-intercept the x-intercept would be at X is equal to so if you say y is equal to zero that would happen that would happen when X is equal to negative three so it's going to go through that line that point right there and the graph if we didn't have this constraint right here would look something like this it would look something like that that's if we didn't constrain it to a certain interval on the x-axis now this graph what does it look like let's see it has this y-intercept at positive 3 positive 3 just like that and where's its x-intercept when Y is equal to 0 X is negative 3 so it also goes to that point right there and has a slope of 1 so it would look something like this that's what this graph looks like now what we figured out is that this absolute value function it looks like this purple graph when X is less than negative 3 so when X is less than negative 3 that's X is equal to negative 3 right there when X is less than negative 3 it looks like this purple graph right there so that's when X is less than negative 3 but when X is greater than negative 3 it looks like the green graph it looks like that so this graph looks like the strange V when X is greater than negative 3 this is positive so we have the graph of we have a positive slope but then when X is less than negative 3 we're essentially taking the negative of the function if you want to view it that way and so we have this negative slope so you kind of have this v-shaped function this v-shaped graph which is indicative of an absolute value function