 So let's take a look at solving absolute value equations the hard way. And if you don't want to or don't like to draw pictures and do the geometry, you can also solve an absolute value equation with only algebra. And this relies on the following. Suppose the absolute value of x is equal to a, where x is some expression. If a is negative, this equation has no solution. And that's because absolute value, I mean absolute difference, is a distance, and the distance can't be negative. If a is greater than or equal to zero, then the observation to make is the absolute value of a is a, and the absolute value of negative a is a. And if we compare these two statements to our original equation, that tells us that whatever our expression was, it's either equal to a, or it's equal to negative a. And what that means is that in general, an absolute value equation corresponds to two equations without absolute value, unless it doesn't. And we'll see that sometimes we may end up with more equations. So we might solve absolute value x minus 5 equals 3. There are two equations, x minus 5 equals 3, and x minus 5 equals negative 3. And we'll solve both equations, and that gives us two solutions. Now as a general rule, we always want to check our solutions, and this is especially important for absolute value equations. So we'll check, we'll let x equals 8 in our original equation, and see if we end up with a true statement. And we do, so x equals 8 is a solution, we'll let x equal 2 in the original equation, and see if our statement is true, which will also give us a true statement, so x equals 2 is also a solution. So both solutions check, and we can circle, highlight, or otherwise indicate that we're happy with these solutions. Or for this equation, we have 5 minus x equals 3, or 5 minus x equals negative 3. And so we can solve both equations. And as before, we should check both solutions, and they both check, so we are happy with both as solutions. So again, we can solve this absolute value equation, because that's going to give us two equations, 3x plus 2 equals 8, 3x plus 2 equals negative 8, and we can solve both. And again, we find that both are solutions. So this is a good example of why the worst way to learn math is to memorize procedures, because our procedure for solving absolute value equation is to set it equal to plus or minus whatever's on the right hand side. But if we do that, and solve, we should always check our solutions. If x equals 3, then we obtain the statement, which is false, so x equals 3 is not a solution. Likewise, we need to check x equals 7. If we let x equals 7, we obtain the statement, which is also false, so x equals 7 is not a solution either. So neither x equals 3 nor x equals 7 solves the equation. So this equation has no solution. Now you'll notice we've gone through a lot of work, and at the very end of it we have nothing to show for it other than our knowledge that this equation has no solution. And this is exactly what you can expect to happen when you don't understand what you're doing. So rather than just following the procedure for solving an absolute value equation, it's worth keeping in mind that because we're dealing with an absolute value, the absolute value is always non-negative. And so as soon as we see that our equation has a setting an absolute value equal to a negative number, we can conclude immediately that this equation will have no solution. And if we do that, we won't have to go through all this work. How about an equation like this? Well, we know what to do if we have an equation in the form absolute value equals number, but this is absolute value in a bunch of other things equals number. So what can we do? We can do a little bit of algebra so that we have an equation in the form absolute value equals number. So again the first thing to notice is that what we have over on the left-hand side is a sum. So we can begin by getting rid of the sum by subtraction. Now we have a product three times something so we can get rid of the product by dividing. And now we have an equation in the form absolute value equals number. So we know that x minus four is either equal to five or x minus four is equal to negative five. So solving gives us our two solutions.