 So what happens when we include variables in our mathematical expression? It turns out it will be very useful to identify the type of expression that you have. So let's introduce a few terms. Something like a plus b is a sum, a minus b is a difference, a times b is a product, and a divided by b is a quotient. And here's the important idea. If an expression involves several operations, the last operation performed determines the type of expression. For example, let's identify the type of expression, 5 plus 3 times 8 minus 4. Now, if we were to evaluate this expression, we'd have to first multiply 3 times 8 to get 24, add 5 plus 24 to get 29, and then subtract 29 minus 4 to get 25. And since the last operation performed is a subtraction, this is a difference. Now, when the expression involves a variable, we perform the same analysis without actually performing the calculations. So let's consider the expression 5 plus 3x minus 4. So again, if we were to evaluate this expression, we would have to do the multiplication first 3 times x. Now, here's an important idea. You might get stuck on this idea, well, how can I multiply 3x when I don't know what x is? And the good news is, you're an mathematics. And what that means is we don't actually need to know what 3 times x is to be able to write it down, 3x, and work with it. So what is the next thing we have to do? Well, we do have to take care of this addition. We have to add 5 plus 3x. And again, we don't actually care what the value of 5 plus 3x is. The important thing is that we have done that addition by writing it down. And finally, if we want to get this expression 5 plus 3x minus 4, the last thing we have to do is we have to subtract 4. And since the last operation performed is a subtraction, this expression is a difference. How about something more horrifying? Well, if we were to evaluate this expression, parentheses have to be taken care of first. So inside this first set of parentheses, we have to multiply 3x, then add to get 3x plus 5. But wait, there's a second set of parentheses, and we have to take care of what's inside this set of parentheses next. So we multiply 2x and then subtract. At this point, we've taken care of all the stuff inside the parentheses, so our last step is to multiply the two things together. And since the last operation performed is a multiplication, this is a product. Or let's take a look at an even more horrifying expression. So remember that when we have a fraction, we have to assume that numerator and denominator are enclosed in a set of parentheses. So there's a parenthesis around the top and around the bottom. And that means we have to do these things first. In the numerator, we have to take care of the multiply first, then add. In the denominator, there's a square root. So we have to consider there's a set of parentheses around the radicand. And that means we have to deal with the 2x minus 1 first. So we have to multiply 2x and then subtract. Then we have to take care of the square root and then add 3. And that gets us our denominator. Finally, the last thing we have to do is divide the numerator by the denominator. And since the last operation performed is a division, this is a quotient.