In this video, we discuss Sign Diagrams which indicate where mathematical expressions are positive or negative. Solution sets a formal way of describing collections of real numbers that satisfy a specific criterion or criteria, often associated with sign diagrams and interval notation which is a succinct way of describing lots and lots of real numbers that fill up certain parts of the real line. We begin by discussing the notions of positive and negative when applied to real numbers. The real lines splits naturally into two halves divided by the number zero, which we think of as exactly in the middle. We often denote positive with a big plus sign and negative with a big minus sign. Now, zero is neither positive nor negative. It's just stuck there in the middle separating the positive from the negative real numbers. Now, in an earlier video, I discussed interpreting a product ab as an area of the rectangle, where one of the side lengths is a and the other side length is b. Now, notice when I labeled the edges of this rectangle, I was careful not to assign any sense of direction. If you include in your mathematics an idea of direction, then you get this notion of positive and negative. When you do that, then areas can also have a positive or a negative aspect. Positive and negative orientations of numbers occur quite naturally in everyday life. For example, if you drive along a straight road, you can move forwards, or you can move backwards, and if you think of moving forwards as positive, and backwards as negative, then when you add up all the effects you may end up with some net displacement. That is modeled in the mathematics of positive and negative numbers. You're going to see this to great effect towards the end of this course when we do integral calculus, and there are areas under curves like areas of rectangles, but areas more generally under curves has physical interpretations and it's important to be able to assign a positive, or a negative direction, or orientation. This area can be positive or negative. The rules are, that ab is positive, and that's expressed by saying ab is greater than zero. Precisely when a and b are both positive, or both negative. The product ab is positive precisely when both the factors have the same sign, they're either both positive or both negative. On the other hand, the product ab is negative, precisely if a is positive and b is negative, or a is negative and b is positive. In other words, if the two factors a and b have a different sign. For example, two times three is six. Two times minus three is minus six, and that's of course the same as three lots of minus two, and that's the same as minus two times three. You introduce another minus, say I take minus two lots of minus three, you don't get minus six, you get positive six. Product of two negative numbers is a positive number. If we revisit an example from the previous video, we saw in two steps that x is equal to one, or x is equal to two. Okay. So, the expression x minus one times x minus two is equal to zero precisely when x equals one, or x equals two. But, what happens if we vary x from, if we change x it's not one or two. Like I said if it's not zero, if the expression is not zero, then it will either be positive or negative. It's very important in calculus to know when certain mathematical expressions become positive or negative. That's related to slopes of curves. Once we introduce calculus, you'll see lots of applications of knowing when an expression is positive or negative. The situation where an expression is positive or negative can be expressed very concisely using something called a sign diagram. Now, I'll explain it using examples, and you'll catch on very quickly. I draw what looks like a table, x on top, and I have the expression that we're investigating below, and there's a line in the middle which is meant to represent the real number line, in as much as it captures the important points where the expression becomes zero. So, in this example, the expression is zero when x is one or two. So, I label those points on the x side as one and two. On the expression side, I put the expression evaluates to zero. If we consider x which are bigger than two, to the right of two on the number line, then both the factors x minus one and x minus two are positive, and a product to positive numbers is positive. Now, if we consider x between one and two, then x minus one is positive, but x minus two is negative. A product of a positive and a negative number is negative. So, on the sign diagram, I put a minus sign between one and two. Now, if x is to the left of one, then both x minus one and x minus two are negative. And remember, a product of two negative numbers is positive, and that completes the sign diagram. Gives you complete information about the sign of the expression as x varies over the real line. Let's do a slightly more elaborate example, just gradually increase the complexity. Now, let's find the sign diagram for the expression x plus one times x minus three, and the technique is you first of all work out where the expression is equal to zero. Then it's very similar to the previous example, where you've got a factorization of zero, that implies that at least one of the factors is equal to zero, and you quickly deduce that x is either minus one or three. So, now we build our sign diagram with the critical points with expression is zero is x equals minus one and x equals three. So, I draw points that correspond to minus one and three, and the expression evaluates to zero. Then I just think, about the sign of the expression as I move about the real line backwards and forwards pass three between minus one and three and beyond minus one. So let's do that. So, if x is bigger than three, then both x plus one and x minus three are positive. So, the product is positive. If x is between minus one and three, x plus one is positive, but x minus three is negative. So, the product is negative. Finally, if x is less than minus one, then both factors are negative. Product of two negative numbers is positive. That's the complete sign diagram. Notice that it's exactly the same as the previous sign diagram except that the numbers, the critical points where the expression is zero are minus one and three. If you look at the notes that accompany this video, you'll see some more variations and an increasing complexity in the types of problems. Once you got sign diagram you can ask yourself, which real numbers satisfy certain criterion, like being positive, or negative, or non-negative which means positive or zero, or non positive which means negative or zero? There's lots of other criteria, and we captured this idea of a collection of numbers satisfying a criterion using a notion of a solution set.
