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# Worked example: absolute value equation with two solutions

## Video transcript

we're asked to solve for X so let me just rewrite to this equation so that the absolute value is really pop out so this is 8 times the absolute value of x plus 7 plus 4 and that same color plus 4 is equal to negative 6 times the absolute value of x plus 7 plus 6 plus 6 now the key here at first it looks all it looks kind of daunting is this complex equation you have these absolute values in it but the way to think about this is if you could solve for the absolute value expression you could then it then turns into a much simpler problem than you can take it from there so you can almost treat this expression the absolute value of x plus 7 you can just really treat it as a variable and then once you solve for that you it becomes a simpler absolute value problem so let's try to do that let's try to solve for not X see first we're going to solve for the absolute value of x plus 7 you'll see what I mean so I want to get all of the absolute values of x plus 7 on the left hand side so I want to get rid of this 1 on the right hand side easiest way to get rid of it is to add is to add 6 times the absolute value of x plus 7 to the right hand side we can't of course only do that to the right hand side if these two things are equal and we're being told that they are then if you add something on this side the only way that the Equality will hold it if you still do it on the left hand side so let's do that so plus 6 times the absolute value of x plus 7 and I want to get all of these constant terms on to the right-hand side so I want to get rid of this positive 4 easiest ways to subtract 4 right over there but if we do it on the left hand side we have to do it on the right hand side as well and so what does this what does this get us what does this get us so our left hand side if I have 8 of something in this case the something is absolute values of X plus 7s but if I have 8 of something and I add 6 of that same something I now have 14 of that something so that's going to be 14 absolute values of X plus 7 14 times the absolute value of x plus 7 before in the negative for canceled out and that was intentional the negative 6 and the 6x plus 7s cancel out or absolute values of X plus 7s cancel out and that was intentional and then we're left with 6 minus 4 which is just 2 so that's going to be equal to equal to 2 now just as promised we want to solve for the absolute value of x plus 7 so let's divide both sides by 14 to get rid of that coefficient there or that that factor whatever you want to call it with the thing that's multiplying the absolute value of x plus 7 so we'll just divide both sides by 14 and we are left with the absolute value of x plus 7 is equal to is equal to 2 over 14 they're both divisible by 2 so this is the same thing as 1/7 so just as promised we've now solved for the absolute value of x plus 7 but we really need to solve for X so how can we reason through this so if I take the absolute value of something and I got you 1/7 there's two possible things that I took the absolute value of I could have taken the absolute value of positive 1/7 or I could have taken the absolute value of negative 1/7 so this thing that we're taking the absolute value of so X plus 7 could be equal to positive 1/7 could be equal to positive 1 7 or or X plus 7 could be equal to negative 1/7 could be equal to negative 1/7 and just think about that for a second if this thing if this thing right over here we're equal to 1/7 you take its absolute value would be 1/7 if this thing it was negative 1/7 you take its absolute value it would be positive 1/7 so that's how we got this so now let's just solve for x so if we subtract 7 from both sides for this left-hand equation we get X is equal to 1/7 minus and 7 we can rewrite as 49 over 7 which is equal to negative 48 over 7 so that's one possibility for X and then the other possibility the other possibility we would get x is equal to so we have negative 1/7 negative 1 7 - 49 / 7 we're just subtracting 7 from both sides that's what 49 over 7 is and then this gets us to negative 50 50 over 7 so the two solutions to this this what we thought was a complicated equation are negative 48 over 7 and negative 50 over 7