INSTRUCTOR: What is the absolute value of 5? And what is the absolute value of negative 8? The absolute value of 5 is 5. The absolute value of negative 8 is positive 8. The absolute value cannot produce a negative result. It will always produce a positive result. Or if you have 0, it will give you 0. 

Now let's say if you have an equation, for example, the absolute value of x is equal to 7. What are the possible values of x? x can equal two things, seven or negative 7, because if you plug these numbers in, the equation will be true. The absolute value of 7 is 7, so this works. The absolute value of negative 7 is also 7. So x could be 7 or negative 7. Both will make the equation true. 

So now how can we use this to solve a typical problem that involves an absolute value symbol? So let's say if we have the function the absolute value of x plus 2 is equal to 9. How can we solve this equation? Well, the first thing you should do is write two equations, the original one and another one, where the inside function is equal to negative 9 instead of positive 9, and solve for both. So in the first one, we can see that x can be equal to 7. And for the second one, x is equal to negative 11. So these are the solutions. 

Find the value of x given this equation. So just like the last example, let's set 2x minus 5 equal to 1 and negative 1. So first, let's add 5. 1 plus 5 is 6. And then let's divide both sides by 2. So x is equal to 3. And in the other example, let's add 5 to both sides. Negative 1 plus 5 is 4 for. And then divide by 2. x is 2. 

So if you check the work, you can see that it's going to work out. So if we plug in 2, 2 times 2 is 4. 4 minus 5 is negative 1. The absolute value of negative 1 is 1. So the equation is true. Now, if we plug in 3, it should also work. 2 times 3 is equal to 6. 6 minus 5 is 1. The absolute value of 1 is 1. So that works. 

Now, what about this one? The absolute value of 5x minus 1 is equal to negative 2. What's the solution? The absolute value function can never equal a negative number. So there's no solution for this particular example. It can only equal 0 or a positive number. 

Now let's try one more example for this particular lesson. Find the value of x. Now, what you don't want to do is write this equation on the left is equal to 7 or negative 7. First, you want to modify the equation. You want to get the absolute value function by itself. So let's add 2 first. So we're going to get 9 on the right side. 

Next, let's divide by 3. Now, you want the absolute value symbol by itself on one side. Once you have it, at this point, now you want to write two equations. So the first one is going to be x minus 1 is equal to 3. And the second-- x minus 1 is equal to negative 3. So if we add 1, x is equal to 4. And if we add 1 to the other one, negative 3 plus 1 is negative 2. So this is the solution.