 Hello and welcome to the session. In this session, we shall study about linear iniquities and their properties. We know that equal to is a sign of an equality and equation is a statement involving the sign of equality. Now we learn about iniquities. First let us see some signs of inequalities. Needs greater than, needs less than, needs equal to, needs greater than or equal to, needs less than or equal to a statement involving sign of inequality either greater than or greater than equal to or less than or less than equal to is called an iniquition. It may contain one or more than one variable or number will satisfy an inequality. If it makes the inequality true, the values of the variables which satisfy an inequality are called the solutions of the inequality. Now suppose we want natural numbers that satisfy the inequalities x less than equal to 15 and x greater than 10, then natural numbers greater than 10 are 11, 12, 13, 14, 15, 16, 17 and so on and numbers satisfying both x less than equal to 15 and x greater than 10 are 11, 12, 13, 14 and 15. So the solution of the inequality x less than equal to 15 and x greater than 10 is 11, 12, 13, 14, 15. Let us now discuss some properties of inequality. For this consider an inequality 10 greater than 5 and we shall examine if the inequality remains the same if we add same number to both the sides. First add 3 to both the sides that is 10 plus 3 is greater than 5 plus 3 that is 30 is greater than 8 which is true so property 1 is an inequality remains true on adding the same number on both the sides. Second subtract 3 from both the sides therefore 10 minus 3 is greater than 5 minus 3 that is 7 is greater than 2 which is true so property 2 is an inequality remains true if the same number subtracted from each side. Now multiply both sides by 5 we get 10 into 5 is greater than 5 into 5 that is 50 is greater than 25 which is true so property 3 is an inequality remains true on multiplying both sides by the same positive number now multiply both sides by minus 5 we get 10 into minus 5 is greater than 5 into minus 5 which is not true that is minus 50 is greater than minus 25 which is not true but minus 25 is greater than minus 50 is true the statement is true if the inequality is reversed hence property 4 is an inequality is reversed on multiplying both sides by the same negative number divide both sides by 5 we get 10 divided by 5 is greater than 5 divided by 5 that is 2 is greater than 1 which is true so property 5 inequality remains true on dividing both sides by the same positive number now divide both sides by minus 5 we get 10 divided by minus 5 is greater than 5 divided by minus 5 which is not true that is minus 2 is greater than minus 1 which is not true but minus 1 greater than minus 2 is true so here also the statement is true if the inequality is reversed so property 6 is an inequality is reversed on dividing both sides by the same negative number inequalities can be solved as equations are solved using the properties of equations for example 2x plus 2 is greater than equal to 8 subtract 2 from both the sides we get 2x plus 2 minus 2 is greater than equal to 8 minus 2 that is is greater than equal to 6 divide both sides by 2 we get 2x divided by 2 is greater than equal to 6 divided by 2 so x is greater than equal to 3 this completes our session hope you enjoyed this session