A solution set is a collection of numbers that satisfy some criterion. So, it's very general and we talk about the solution set of an equation or inequality. So, for example, the solution set of the equation x plus 1 times x minus 3 equals 0 is the set consisting of minus one and three. Now, there's a few things to notice here. These curly brackets denote set or collection, and there's a comma that separates the elements. Let's write down the solution set corresponding to an inequality. So, let's write down the set of all numbers x that satisfy the inequality x plus 1 times x minus 3 is greater than zero. So, once again, we have curly brackets to denote set or collection. Let me write x in R such that x is less than minus one or x is greater than three, and this needs a bit of explanation. The curly brackets as I said, denote set. This funny symbol that looks like a Greek Epsilon is an abbreviation for element of. This strange looking R symbol is just an abbreviation for the set of all real numbers or the real number line, and this vertical line is an abbreviation for such that. A lot of Math teachers use a colon, but if I try and do a colon on a black board or a white board often it is not very clear, and a vertical line turns out to be very clear. So, you read this as the set of real numbers such that x is less than minus one or x is greater than three. Now, where did this come from? Well, that came from the sign diagram. If you go back to the sign diagram, Where you see a plus sign for the expression that corresponds to x being less than minus one, to the left of minus one or x being bigger than three to the right of three. By contrast, the solution set of the inequality where the expression we had before is less than zero, if we go back to the sign diagram, this expression is less than zero if x is between minus one and three. So, when we describe the solution set, we get curly brackets, the set of x in the real number system such that minus one is less than x which is less than three. Okay. So, this condition here, this captures precisely the fact that x is to the right of minus one and to the left of three. This set notation that I've introduced you to takes a little bit of getting used to and probably looks like a foreign language. There is a very succinct way of describing lots of real numbers using interval notation. Now, if I take, for example, the set of real numbers such that x is greater than three. If I picture the real number line and I find the number three, all the numbers that are greater than three are the numbers that appear to the right of three on the number line. I'm not including three so I'll put a little circle around three to exclude three, and then I'll just imagine covering the whole real line all the way off towards indefinitely to the right, towards infinity. Okay. This is called an infinite interval to the right, and it's denoted very succinctly with a round bracket three comma and then an infinity symbol. The round bracket means that you exclude three. We, of course, exclude infinity because that's not a number. So, I'll just introduce, in general, interval notation. There are several possibilities. Suppose I've located points a and b on the real number line. If I want to capture succinctly all the numbers between a and b, including a and b, this is denoted using square brackets. Square bracket a comma square bracket b is the set of all real numbers x such that a is less than or equal to x which is less than or equal to b. Square brackets indicate that both a and b are included as possible values of x. Okay. Now, if we want to exclude a and b, we imagine a little tiny little circles that exclude a and b, then we include everything in between. Then, we use what's called round bracket notation. Round bracket a comma b is the set of all real numbers x such that a is less than x which is less than b. Now, you can have mixtures of round and square brackets. For example, I could include a, which I indicate with a dot, and I could exclude b which I'll include with a little circle, and then include all the numbers in between. Using interval notation, this is denoted by square bracket a comma b round bracket, and that's all the real numbers such that a is less than or equal to x which is less than b. Another possibility is that, I put a little circle around a and dot at b and then include everything in between, and that's captured with round bracket a, b square brackets, and that's all x in R, such that a is less than x which is less than or equal to b. Another possibility which is, in fact, an example we had before where you take some point, say b, on the real line and you exclude it, and then you go on forever to the right. That's captured with round bracket b comma infinity round bracket. So, that's all x in R such that x is greater than b. If you wanted to include b, then we would use square bracket notation. Square brackets b infinity, that's all x in R, such that x is greater than or equal to b. There's a similar notation to the left but we use minus infinity so we could take some point a, we could exclude it or include it, and move infinitely to the left. Okay. In the first case where we exclude a, we use the notation round bracket minus infinity, indicating moving indefinitely to the left, comma a, round bracket to exclude a, that's all x in R such that x is less than a. Or if we wanted to include a, we have the square bracket notation or round bracket negative infinity comma a, square brackets to include a, that's x in R such that x is less than or equal to a. Okay. There's one other case. What happens if we want to include all real numbers? Well, of course, we get the whole real line denoted by R, using interval notation that's round bracket negative infinity comma infinity round brackets. Now, let's finish by applying interval notation to one of the sign diagrams. Remember that we had this sign diagram for the expression x plus 1 times x minus 3, and we can read off information and express it using interval notation. For example, we could look at this collection of real numbers x such that this expression is negative. Okay. So, on the sign diagram, that's all the x between but not including minus one and three. So, in interval notation, it's denoted by minus one comma three. If we look at all the real numbers x for which the expression is positive, then that comes in two pieces. All the numbers bigger than three or less than minus one, and we capture that using two intervals. All the numbers to the left of minus one is captured by round bracket minus infinity comma minus one. All the numbers to the right of three is captured by the interval notation three comma infinity with round brackets. Then, we glue them together using what's called a union symbol. When you use the union symbol, then you pull together all the elements in the two sets that are involved. There's a lot of material to digest in just a few minutes especially if you've never seen the notation or never heard any of this terminology before. Please carefully read and digest the notes accompanying this video, and then, when you're ready, please attempt the exercises. We look forward to see you again soon.
