Solving absolute value inequalities 2

solve for P and we have the absolute value of p minus 12 plus 4 is less than 14 so let's just do this one step at a time the first thing we want to do is we really just want to isolate the part that has the absolute value so let's do that we can just get rid of this positive 4 here we can do that by subtracting 4 from both sides of the inequality and so the left-hand side the positive 4 and the negative 4 cancel out we're just left with the absolute value of P minus 12 and on the right hand side we get 14 minus 4 is 10 and we still have the less than sign so we have the absolute value of P minus 12 is less than 10 so let's just think about it a little bit if I were to tell you if I were to tell you that the absolute value of x is less than 10 what does that mean that means that the distance from X to 0 has to be less than 10 so if I were to draw a number line if I were to draw a number line and put 0 here we can only go up to 10 away and even that's too far it has to be less than 10 so this is positive 10 it would have to be less than positive 10 because 10 is exactly 10 away but we have to be less than 10 away from 0 and then we could go all the way to the left until negative 10 and even that we wouldn't be able to include because its absolute value is 10 it's not less than 10 but negative 9 negative nine point nine nine 9 we can include all of those things all the absolute value of any of those things is going to be less than 10 so another way to write this this absolute value inequality is that X could be X could be greater than negative 10 X could be greater than negative 10 and X needs to be less than 10 or we could write this as X is between so negative 10 is kind of the bottom boundary and we're not going to include it X is going to be greater than that and it is less than 10 so this is another way to write the absolute value of x is less than 10 is essentially saying that X has to be between negative 10 and 10 and it can't be either negative 10 or 10 we're not there's no equal sign here so the same exact logic instead of X we have a P minus 12 here so we can write the absolute value of P minus 12 is less than 10 is saying that negative 10 is less than P minus 12 which is less than 10 and we can just solve this compound inequality all at once by isolating the P in the middle and the best way to isolate the P in the middle we want to get rid of this negative 12 so let's add 12 to all three all three sections of this compound inequality and so we get negative 10 plus 12 is positive 2 and then is less than P minus 12 plus 12 is just P and then is less than 10 plus 12 is 22 so P is greater than 2 and less than 22 so if we were to plot it on a number line our solution set looks like this so this might be 2 over here this might be 22 over here maybe 0 is sitting right over here P is greater than 2 P is greater than 2 it's not greater than or equal to so we have to not fill this in has to be an open circle since it's only greater than and it's less than 22 it's not less than or equal to so we're not going to fill that circle in and it's everything it's everything in between and we can verify it for ourselves let's try a value that might a work out well well 12 is in between these two numbers it's in our magenta region right here so let's try P is equal to 12 so if you have 12 minus 12 so it's the absolute value of 12 minus 12 plus 4 which should be less than 14 so this is 0 plus 4 which needs to be less than 14 and 4 is definitely less than 14 so 12 worked 0 shouldn't work let's try 0 0 minus 12 so it's the absolute value of 0 I'll introduce in a different color it is the absolute value of 0 minus 12 plus 4 this should not be less than 14 this should not work so we get the absolute value of negative 12 plus 4 should not be less than 14 then we should end up with a contradiction here so we have 12 plus 4 less than 14 we end up with 16 is less than 4 which is not true so zero does not work so at least we were feeling pretty good we took something outside of our solution set didn't work something inside of our solution set it did work