Worked example: absolute value equations with one solution

we're told solve the absolute value of 3x minus 9 is equal to 0 and graph the solution on a number line so let's just rewrite the absolute value equation there told us that the absolute value of 3x minus 9 is equal to the to 0 so we're told it the absolute value of something in this case the something is 3x minus 9 is equal to 0 if I tell you that the absolute value of something is equal to 0 I'm telling that something has to be exactly 0 away from 0 or 0 away from the origin on the number line so the only thing that something could be is 0 if I tell you that the absolute value of x is equal to 0 you know that X has to be equal to 0 that's the only value whose absolute value is 0 so if I tell you that the absolute value of 3x minus 9 is 0 then we know that 3x minus 9 has to be equal to 0 and that's kind of unique about 0 is that's the only number that has a unique that that's that's only the absolute value of 0 if you had say a 1 here you could say oh well then this thing could be a 1 or a negative 1 but here if you have a 0 this thing can only be 0 so solving this equation is fairly straightforward if we want to isolate the 3x get rid of the negative 9 on the left hand side we add 9 to both sides of the equation add 9 to both sides of the equation these 9s cancel out that's the whole point on the left hand side you're just left with 3x and on the right hand side you are just left with 9 now we want to solve for X so we have 3 times X let's divide it by 3 because 3 times X divided by 3 is just going to be X well if we divide the left side by 3 we have to divide the right side by 3 so we are left with these guys cancel out X is equal to 9 over 3 which is 3 and that's our solution now let's try out let's make sure that this actually works so let's substitute it back into our original equation so we have the absolute value of 3 time X instead of X I'll just put in our actual answer that we got three times three minus nine minus nine has got to be equal to zero so what's this going to be equal to three times three is nine so it's the absolute value of 9 minus 9 which is the absolute value of 0 which is indeed 0 so it does indeed equal 0 and we are done