 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that find the range of the values of x which satisfy the inequality minus 1 upon 3 is less than 2x upon 9 plus 1 is less than equal to 2 for x belonging to R the set of real numbers graph the solution set on the number line. Consider the inequality minus 1 upon 3 is less than 2x upon 9 plus 1 is less than equal to 2. Here the LCM of 3 and 9 is 9 therefore we multiply the whole inequality by 9. As we know if we multiply an inequality by the same positive real number it does not change the inequality so we get minus 1 by 3 into 9 is less than 2x upon 9 into 9 plus 1 into 9 is less than equal to 2 into 9 which implies minus 3 is less than 2x plus 9 is less than equal to 18. Now on separating the two inequalities we have minus 3 is less than 2x plus 9 and 2x plus 9 is less than equal to 18. First let us consider minus 3 is less than 2x plus 9 we know that on adding or subtracting any positive real number to an inequality does not change the inequality so we subtract 9 from both sides of the inequality we obtain minus 3 minus 9 is less than 2x plus 9 minus 9 which implies minus 12 is less than 2x which implies minus 12 upon 2 is less than 2x upon 2 which implies minus 6 is less than x mark this in equation as 1. Now consider 2x plus 9 is less than equal to 18 here again we subtract 9 from both sides of the inequality we get x plus 9 minus 9 is less than equal to 18 minus 9 which implies that 2x is less than equal to 9 which implies x is less than equal to 9 by 2 which is equal to 4.5 so we have x is less than equal to 4.5 mark this in equation as 2 now on combining in equations 1 and 2 we obtain minus 6 is less than x is less than equal to 4.5 now as x belongs to R the set of real numbers range of values of x is all real numbers lying between minus 6 is less than x is less than equal to 4.5 where 4.5 is included in the range and minus 6 is not included in the range as minus 6 is strictly less than x and x is less than equal to 4.5 now let us graph the solution set on the number line first of all draw a number line and mark values as 0, 1, 2, 3 and so on and minus 1, minus 2, minus 3 and so on on the number line at equal distances. As minus 6 is strictly less than x therefore mark minus 6 with an open dot which shows that minus 6 is not included in the range also x is less than equal to 4.5 therefore mark 4.5 with a closed dot which shows that 4.5 is included in the range now shape the number line between minus 6 and 4.5 the shaded portion shows the solution set of the given in equation which includes 4.5 but does not include minus 6. This completes our session hope you enjoyed this session.