 So by now you should be familiar with a couple of different ways of expressing inequalities. Here's another one. It will be useful to express inequalities using interval notation. Now it's possible to go from a standard inequality to writing down the interval notation, but it's probably easiest to go from the graph of the inequality to interval notation. Each shaded region corresponds to an interval. The separate intervals are joined, unioned, together with a union symbol. We'll use a parentheses left or right if a boundary point is excluded, and we'll use a bracket left or right if a boundary point is included. Now sometimes our shaded regions include everything past a boundary point, so if we include everything past a boundary point, we indicate the upper boundary as our infinity symbol, and always use a right parenthesis. Likewise, if we include everything before a boundary point, we indicate the lower boundary as minus infinity and always use an open parenthesis. And finally, because what we get can look like the coordinates of a point to avoid confusion, we'll indicate the solution using set notation. So for example, suppose we want to express an interval notation x greater than or equal to 5. So first we'll graph the inequality. Since 5 is an included left boundary point, we'll use a square bracket. Since the interval includes everything past the boundary point, we'll use our infinity symbol with a right parenthesis. And to avoid confusion, we'll always indicate that the variable is in the interval. So for an interval like this, first we'll graph it. So the boundary points are negative 3, which is not included, so we'll use a parenthesis. And 2, which is included, so we'll use a square bracket. And we'll indicate that our variable is in the interval. Similarly, if we have an OR statement, we'll start by graphing our interval. Since we've shaded two intervals, our interval notation will consist of two parts joined together with a union symbol. This first interval starts at negative 1, but not including it, and goes all the way to the left. So we'll express that as negative infinity up to negative 1, both using parenthesis. The second interval starts at 3 included and goes forever to the right. And so we'll express this as the interval from 3 using square brackets off to infinity, always using an open parenthesis. It's also helpful to go backwards. So suppose I have the inequality x in the interval negative 1 to 4. Let's graph it and then express this as an inequality. So the first thing to notice is our boundary points are at negative 1 and at 4. Since we use a parenthesis at each, neither boundary is included, so we graph using an open circle. And finally, since there's just one interval, we graph the region between our boundaries. And finally, we can write this as an inequality. We go from negative 1 less than x less than 4. It's worth remembering that any compound inequality like this can be expressed using and, so an alternative expression of this is negative 1 is less than x, and x is less than 4.