 Hello and welcome to the session. In this session we will discuss how to solve equations and in equations involving absolute value and write the solution set as indicated. Now in one of our sessions we have already discussed how to solve linear equations involving absolute values. So here let us discuss how to solve linear in equations involving absolute values and write the solution set. Now we know that absolute value of a variable x denoted by this symbol is the distance of x from 0 on a number line. It is never negative because it measures the distance not the direction. Now let us discuss steps for solving an absolute value inequality. First is, unsolute the absolute value expression. Second is, by using the definition of absolute value we settle the inequality without absolute values. And third is solve this linear inequality that is set up in the previous step. Now let us see a basic example of absolute value inequality. Since absolute value measures distance from 0 on the number line. So, absolute value of x is greater than 4 indicates that x is more than 4 units from 0 that is any number to the right of 4 and to the left of minus 4 is more than 4 units from 0. So, absolute value of x is greater than 4 is equivalent to x is greater than 4 or x is less than minus 4 and the solution set of this inequality that is this inequality is the union of the solution sets of the two simple inequalities that is these two simple inequalities. Now we know that solution set of the inequality x is less than minus 4 is the open interval minus infinity to minus 4 and the solution set of the inequality x is greater than 4 is the open interval 4 to infinity. So, the solution set of the inequality absolute value of x is greater than 4 is given by the union of these two solution sets that is open interval minus infinity to minus 4 union open interval 4 to infinity and here on the number line we have represented this solution set of the inequality absolute value of x is greater than 4. Here we have shaded all the position of the number line which is greater than 4 and which is less than minus 4 and at minus 4 and 4 we have drawn hollow circles which indicate that these points are not included in the solution set. Now let us see this table for basic absolute inequalities when k is greater than 0 for the inequality absolute value of x is greater than k equivalent inequality is x is greater than k or x is less than minus k and its solution set is given by open interval minus infinity to minus k union open interval k to infinity and this is the representation of the solution set of the given absolute value inequality on the number line for the inequality absolute value of x is greater than equal to k equivalent inequality is given by x is greater than equal to k or x is less than equal to minus k its solution set is given as open interval minus infinity to k union sunny closed interval k to infinity and this is the representation of the solution set of the given inequality absolute value of x is greater than equal to k. Here we have drawn dark circles at the points minus k at k which means that these points are included in the solution set then for the inequality absolute value of x is less than k the equivalent inequality is given as minus k is less than x is less than k and its solution set is given as open interval minus k to k and this is the representation of the solution set of the inequality absolute value of x is less than k on the number line then for the inequality absolute value of x is less than equal to k the equivalent inequality is given as minus k is less than equal to x is less than equal to k and its solution set is given by the closed interval minus k to k and this is the representation of its solution set on the number line. Now let us discuss an example where we have to solve the inequality absolute value of x minus 6 is less than 2 and draw the solution set on the number line. Now this inequality is of the time absolute value of x is less than k and we know that for this type of inequality the equivalent inequality is given as absolute value of k is less than x is less than k so this inequality can be written as minus 2 is less than x minus 6 is less than 2 now adding 6 to each part of this inequality we get minus 2 plus 6 is less than x minus 6 plus 6 is less than 2 plus 6 which implies now minus 2 plus 6 is 4 is less than x is less than 2 plus 6 that is 8 so the solution set of the given inequality is given as the other interval 4 to 8 now let us represent this solution on the number line now this solution includes all real numbers between 4 and 8 so we will shape the number line from 4 to 8 we will put hollow circles at 4 and 8 that is the points 4 and 8 which indicate that these points are not included in the solution set so here the shaded portion of the number line represents the solution of the given inequality so in this session we have discussed how to solve inequalities involving absolute value and this completes our session hope you all have enjoyed this session