 So in the previous video, we learned about graphing absolute value functions. Now, and we did that via transformations. Every absolute value function could be graphed by transformations. But the question I wanna talk about in this video is how does one solve equations involving absolute value? Because we might ask ourselves questions like the following. What are the x intercepts of the function, the absolute value of two times x minus three minus two? Now we solve them on the graph and we graphed it, right? We found the vertex as three comma two and then using our slope, we kind of accidentally found these x intercepts. There's this x intercept at four comma zero and another one at two comma zero. So we can see that the x intercepts from the graph are two and four, but is there a purely algebraic way of finding these x intercepts? Well, remember x intercepts mean we're setting the y coordinate equal to zero. The x intercepts are those points of the graph that are on the x axis and the x axis is the line y equals zero. So if we wanna solve for the x intercepts, we're essentially just trying to solve the equation, my function equals zero. This is true for any function, but what particularly happens when you try to solve this with absolute value? Well, trying to solve this right here, my first thing to do is I would add two to both sides. That's fairly routine from what we've done in the past. Then we get two times x minus three all inside the absolute value is equal to two. And so what we're tasked to do here is we have to then solve this equation when it has absolute value. How do you get rid of the absolute value expression here? What do we do? Well, absolute value, if we remember, it's not a one-to-one function, right? This graph does fail the horizontal line test. So we can't just necessarily do the inverse function, but it turns out we can reason with ourselves on how to proceed here. If the absolute value of my quantity equals two, that means that the number was originally two or it was negative two. The only way the absolute value is two is if the number in question is two or negative two. And so we really just kind of consider both situations and we can do that simultaneously by saying plus or minus two. Really, when you see like plus or minus two right here, this represents two numbers, positive two and negative two, but we're just doing shorthand so we don't have to write the two numbers all the time. So it basically to get rid of absolute value, you have to take plus or minus the number that's on the right-hand side. There is an important caveat to that, which we'll talk about in just a minute, but let's proceed with the question right here. So now we basically have two equations, two times x minus three equals two and two times x minus three equals negative two, these two linear equations, but we can solve them simultaneously because we're gonna divide both sides by two. On the left-hand side, the twos will cancel leaving just an x minus three. On the right-hand side, plus two divided by two is gonna be plus one and minus two divided by two is gonna be a minus one. So we still get plus or minus one. The next thing to do is then add three to both sides in which case minus three plus three cancels out and then the right-hand side, we're gonna get x is equal to three plus or minus one and this is about as far as we can get without considering the two cases. So really we have the two cases, three plus one and three minus one for which we then get four and two like we solve visually on the graph. So algebraically, we can solve equations involving absolute value. It just means that when you wanna get rid of the absolute value symbol on the left-hand side, you're gonna have to insert a plus or minus into the equation. But like I said, there is a slight caveat which I wanna explore in this other example we did. So let's take the function f of x equals two times the absolute value of x minus three plus four and if we try to solve that one, we'll see what happens. So we're gonna get minus four on both sides. The four is on the left with cancel. This then gives us two times the absolute value of x minus three is equal to negative four. And next divide both sides by two. So the two cancels right here and then we get the absolute value of x minus three is equal to negative two. I'm now gonna switch my ink to red here because the things I'm gonna be doing next are actually wrong and I want you to see red here to think of blood. So death will come if we proceed in the following manner. Because now we're tempted to say the following. Hey, look, the absolute value of x minus three equals negative two. What we learned on the previous slide is to get rid of absolute value, we just take plus or minus. And so you're gonna get x minus three equals plus or minus negative two. I guess that's really just like minus plus two which is really just plus minus two again. So I'm gonna add three to both sides. I get x equals three plus or minus two which means five, three plus two and one, three minus two. But conveniently over here you can see the graph of our function and you can see that one and one, two, three, four, five are not x intercepts of the graph. What just happened? Well, the issue came back to, I'm gonna switch to red, right? The issue came to here because this is completely bonkers. The issue came to here. The absolute, let's pause and think about this for a second. The absolute value of x minus three is equal to negative two. Is that even possible? Can absolute value equal negative two? Well by construction, absolute value always produces something positive. The only exception of that is the absolute value of zero is itself zero. But absolute value cannot produce a negative. Basically the issue here is that negative two is not in the range of your absolute value function. I will say absolute value like that. You can't get two from absolute value or I really should say it in just simpler terms. The absolute value of x minus three cannot equal negative two. If your absolute value equals a negative it actually turns out that there's gonna be no solution to that equation, right? So we draw, I shouldn't write this in red. This is now actually the correct stuff, right? So we should be saying no solution in this situation. No solution. The empty set is the solution set to this equation. And now since this equation had no solution this would mean for our function that this function has no x intercepts which we can then see on the graph that, oh yeah, because the V is pointing upward and my vertex is above the x-axis there is no x intercepts on this graph. Now the geometry and the algebra come in harmony with each other. If you have an absolute value equal to a negative you have no solution. If you have an absolute value equal to a positive or equal to zero then in that situation you take plus or minus and you proceed from there. So make sure you check for negative as before you get rid of the absolute value. So looking at some more examples of this if you have the absolute value of x plus four equal to 13 we wanna solve this equation. We first have to check is my left hand side negative? Nuh-uh, this is greater than or equal to zero so I can proceed. I'm gonna get x plus four equals plus or minus 13. Subtract four from both sides you get negative four plus or minus 13. You can never leave your answer with a plus or minus here. We have to write this as two individual numbers. This is negative four plus 13 and negative four minus 13. In which case we look at the two situations 13 take away four is a nine and negative four take away 13 is a negative 17. And so those are your two values for x. And you know if we wanted to then we ought to we should plug those back into the original equation and check what happens. Nine plus four is 13, it's absolute value is 13. Negative 17 plus four is negative 13 whose absolute value is 13, it checks out. And then as another example right here when it comes to absolute value you wanna get the absolute value all by itself. So whatever might be attached to the absolute value whether it's like additions, attraction, multiplication, division perform inverse operations until you get absolute value all by itself. So subtract two from both sides we see that the absolute value of two x minus three is equal to seven take away two which is five. That is a positive five. So there is a solution we'll proceed forward by getting rid of the absolute value we'll take plus or minus five two x minus three equals plus or minus five. Then solving this linear equation we'll add three to both sides. That'll give us two x equals three plus or minus five for which if you want to we actually could then look at the two options, right? Three plus five is eight and three minus five is a negative two. You don't have to keep this plus or minus if you don't want to. But then whichever option you choose the next thing to do is divide by two, divide by two. And so we get that x is equal to three plus five over two which I mean we can't type that in as an answer. We're gonna have to simplify that which that gives us eight over two and negative two over two. And so we see that the two solutions here are gonna be four and negative one. And we could check in the original solution again. And that's all it takes to solve an absolute value function that you isolate the absolute value and then you make a check. If the absolute value is equal to a negative, nope, problems done, no solution. If the absolute value is equal to zero or positive you get rid of the absolute value by doing plus or minus on the right hand side and then you can solve the two cases either together or separately whichever works best for you. And this is how we can solve an absolute value function or absolute value equation.