Worked example: absolute value equations with no solution

so we're asked to solve for X and we have this equation with absolute values in it so it's 4 times the absolute value of x plus 10 plus 4 is equal to 6 times the absolute value 6 times the absolute value of x plus 10 plus 10 and at first this looks really daunting but the key is to just solve for this absolute value expression and then go from there let me just rewrite it so the absolute value expression really jumps out so this is 4 times the absolute value of x plus 10 plus 4 plus 4 is equal to 6 times the absolute value of x plus 10 plus 10 plus 10 so let's get all of the let's get all of the absolute values of x plus 10 on the left hand side so I want to get rid of the 6 times the absolute value of x plus 10 on the right well how would I do that well I could subtract 6 times the absolute value of x plus 10 from the right but we've already seen this multiple times if these two things are equal and if I want to keep them equal if I subtract 6 from the right hand side I've got to subtract or if I subtract 6 times the absolute value of x plus 10 from the right hand side I have to subtract the same thing from the left hand side so we're going to have minus 6 times the absolute value of x plus 10 and likewise I want to get all my constant terms I want to get this 4 out of the left hand side so let me subtract 4 from the left and then I have to also do it on the right otherwise my equality wouldn't hold and now let's see what we end up with what we end up with so on the left hand side the 4 minus 4 that's 0 you have 4 of something minus 6 of something that means you're going to end up with negative 2 of that something negative 2 of the absolute value of x plus 10 remember this might seem a little confusing but remember if you had 4 times if you had 4 apples and you subtract 6 apples you now have negative 2 apples I guess you owe someone the apples same way you have 4 of this expression you take away 6 of this expression you now have negative 2 of this expression on the right a little bit neater so it's negative 2 times the absolute value of x plus 10 is equal to well the whole point of this of the of the 6 times X the absolute value of x plus 10 and minus 6 times the absolute value of x plus 10 is to make those cancel out and then you have 10 minus 4 which is equal to 6 now we want to solve for the absolute value of x plus 10 so let's get rid of this negative 2 and we can do that by dividing both sides by negative 2 you might realize everything we've done so far is just treating this red expression as almost just like a variable and we're going to solve for that red expression and then take it from there so negative 2 divided by negative 2 is 1 6 divided by negative 2 is negative 3 so we get the absolute value of x plus 10 is equal to is equal to negative 3 now this gets us to a very interesting situation you might jump to hey maybe this could be the positive version of the negative but remember absolute value absolute value is always is always non-negative it could when you take took the absolute value of 0 you would get 0 but the absolute value of anything else is going to be positive so this thing right over here is definitely going to be greater than or equal to 0 it doesn't matter what X you put in there when you take its absolute value you're going to get a value that's greater than or equal to 0 so there's no X that you could find that somehow going to that let you put it there you add 10 you take the absolute value of it you're actually getting a negative value so this right over here has absolutely no no solution and I'll put some exclamation marks there for emphasis