Solving absolute value equations
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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. They told us that the absolute value of 3x minus 9 is equal to 0. So we're told that the absolute value of the something-- in this case the something is 3x minus 9-- is equal to 0. If I told you that the absolute value of something is equal to 0, I'm telling you 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 that something could be is 0. If I told 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 told you that the absolute value of 3x minus 9 is 0, than we know that 3x minus 9 has to be equal to 0, and that's kind of unique about the 0 is that, it's the only number the has a unique, 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 9's 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. But 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. Let's substitute it back into our original equation. So we have the absolute value of 3 times x. Instead of x, I'll just put in our actual answer that we got, 3 times 3 minus 9 has got to be equal to 0. So what's this going to be equal to? 3 times 3 is 9. 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.