 Hello and welcome to the session. In this session we will discuss how to write an interval in set notation, interval notation and as a graph. We will also learn how to apply and use the concept of open and closed intervals. First of all we will discuss inequality. We know that inequality is a mathematical statement that says one quantity is not equal to another quantity. And there are four types of inequalities and these are less than, greater than, less than or equal to and greater than or equal to. We must note that the solution of the inequality is represented in the form of intervals. Now we are going to discuss intervals. An interval is a set that consists of all real numbers between a given pair of numbers. It can also be thought of as a segment of the real number line. An end point of an interval is either of the two points that mark the end of the line segment. An interval can include either end point, both end points or no end point. For example, the set of all numbers x satisfying 0 is less than equal to x is less than equal to 1 is an interval which contains 0 and 1 as well as the numbers between them. There are three ways to write an interval and these are set notation, interval notation or by graph. First of all we shall discuss set notation. In this the intervals are written as a set of points. Now in set notation the interval x is less than minus 2 can be written as set of all x values such that x is a real number and x is less than minus 2 or it can be written as set of all x values such that x belongs to the set of real numbers and x is less than minus 2. Now we will discuss interval notation. In this notation the end points used are always written from left to right. That is the smaller number is written first followed by comma and then the larger number round brackets or parenthesis are used to indicate end point which is excluded from the interval. Here parenthesis corresponds to greater than or less than symbols. Now square brackets indicates that an end point is included in the interval. Also square brackets corresponds to greater than equal to and less than equal to symbols. Also open brackets are used with minus infinity and infinity. Now suppose we have the interval x is less than minus 2 then in interval notation it is written as open interval from minus infinity to minus 2. Here the round brackets are used to indicate that the end point minus 2 is not included and with minus infinity we always use round brackets. An open interval means an interval which does not include its end points which is denoted by round brackets and if we have the interval minus 2 is less than equal to x is less than equal to minus 1 then in interval notation it is written as which is pronounced as closed interval from minus 2 to minus 1. Here square brackets indicate that the end points minus 2 and minus 1 are included and closed interval means the interval that includes its end points and it is denoted by square brackets. Also we have half closed half open intervals that is the intervals which include one end point and not the other. For example we have the semi closed interval from 2 to 3 and semi open interval from 5 to 6. Now we shall learn how to represent intervals by graph. On the number line we draw a thick line to show the included values. Here a closed or darkened circle is used when the end point is included and a hollow circle is used when the end point is not included. Suppose we have an interval minus 2 is less than equal to x is less than equal to 5 and we have to represent it on the graph. For this first we draw the number line. Now the interval is minus 2 is less than equal to x is less than equal to 5 so we darken this number line from minus 2 to 5. Now since we have less than equal to sign so we will include the end points minus 2 and 5 for this we draw 2 dark circles at the points minus 2 and 5. So this number line shows the interval minus 2 is less than equal to x is less than equal to 5. Thus in this session we have discussed how to write an interval in set notation, interval notation and as a graph. Also we have learnt to apply and use the concept of open and closed intervals. This completes our session. Hope you enjoyed this session.