Solving absolute value inequalities 1

we're told to graph all possible values for H on the number line and this is a especially interesting inequality because we also have an absolute value here so the way we're going to do it we're going to solve for we're going to solve this inequality in terms of the absolute value of H and from there we can solve it for H so let's just get the absolute value of H on one side of the equation so the easiest way to do this is to add 19 and 1/2 to both sides of this equation I often like putting that as an improper fraction but 1/2 is pretty easy to deal with so let's add 19 and 1/2 to both sides of this inequality can I just say equation it's an inequality not an equation it's an inequality sign not an equal sign so plus 19 and 1/2 on the left-hand side these guys obviously cancel out that was the whole point and we are left with the absolute value of H on the left hand side is less than and then if we have what 19 and 1/2 essentially minus 12 19 minus 12 is 7 so it's going to be 7 and 1/2 7 and 1/2 so now we have that the absolute value of H is less than 7 and our 7 and 1/2 so what does this tell us this means that the distance another way to interpret this remember absolute value is the same thing as distance from 0 so another way to interpret this statement is that the distance the distance from from H to 0 it has to be less than less than 7 and 1/2 so what values of H are going to have less than a distance from 7 and 1/2 well it could be less than 7 and 1/2 and greater than 0 or equal to 0 so let me put it this way so H could be H could be less than 7 and 1/2 but if it gets too far negative if it goes to negative 3 we're cool negative 4 negative 5 negative 6 negative 7 we're still cool but then at negative 8 all of a sudden the absolute value isn't going to be less than this so it also has to be greater than negative seven and a half if you give me any number in this interval its absolute value is going to be less than seven and a half because all of these numbers are less than seven and a half away from zero let me draw it on the number line which they want us to do anyway so if this is the number line right there that is zero and let me draw some points let's say that this is seven that is eight that is negative seven that is negative eight what numbers are less than seven and a half away from zero well you have everything all the way up to seven and a half is exactly seven and a half away so that you can't count that so seven and a half you'll put a circle around it same thing true for negative seven and a half the absolute value it's exactly seven and a half away but we have to be less than seven and a half away so neither of those points are going to be included positive seven and 1/2 or negative seven and a half now everything in between is less than seven and a half away from zero so everything else counts everything else counts everything outside of it is clearly more than seven and a half away from zero and we're done