 And then welcome to this session. In this session we will discuss combining equations. We already know what an equation is. An equation is a sentence, an expression that compares two expressions of different values. The inequalities not equal to less than, less than equal to greater than, greater than equal to are used to compare the expressions that is two expressions of different values and so they form an equation. So consider the examples of equations 2x plus 3 greater than equal to 5 this is an equation or 51 not equal to 81 is an equation and so on. Let us see how we solve two equations. Suppose we need to solve 5x plus 10 less than 40 plus 3 less than equal to 11 that we are given two equations that is 5x plus 10 less than 40 and 2x plus 3 less than equal to 11 we need to solve these two equations. So solving the combination of these two equations first step would be to solve each of the equations individually. So for this we will suppose equal to the set containing the assertion that is less than 30 and the set of real numbers have considered the first inequality 5x plus 10 less than 40. We will consider the second inequality for this we suppose a set B equal to the set containing element x such that that 3 is less than equal to 11 and x belongs to the set of real numbers are represented by the set A and second inequality is represented by the set B. Two equations we have add two sets so we can solve the equal to equation set B. Now is less than 40. Let us now solve this equation. So for this is less than 40 minus 10. So for the we get we divide both sides by 5. So 5x upon 5 is less than 30 upon 5. Now the 5 cancels with 5 and 5 6 times is 30. So we now have x is less than. Now we solve the second inequality 2x plus 3 less than equal to 11. So consider plus 3 less than equal to 11. Now we subtract both sides by 3. So 2x plus 3 minus 3 is less than equal to 11 minus 3. This gives us is less than equal to. So 2x upon 2 is less than equal to 8 upon. This 2 cancels with 2 and 2 4 times is 8. So this gives us x is less than equal to 4. We require solution set that is the solution set obtained on solving the combination of the two equations which is given by A intersection B. Now the solution set for the set A the set containing the element x such that x is less than 6 and this x belongs to the set of real numbers section. The solution set for the set B which is x less than equal to x is less than equal to 4 and this x belongs to the set of real numbers r. Now the intersection of these two sets would be equal to the set contained into 4 where this x belongs to the set of real numbers r. So this is our solution set. We can graph the solution set. So this we will go on number 9. This is our number 9. Now as the required solution set has elements x where this x is less than equal to 4. To present this solution set we will shade the portion to the left of 3 is included in the graph. The shaded portion including in between the two equations we can solve the two equations in this way and we can also graph them. Less than less than 30 plus 3 less than equal to 11. Now solving the combination of these two equations would be same. The only difference is that we have all in between the two equations. Now our implies union would be equal to union that is all without the solution set for the sets A and B individually. So this would be equal to the set containing the element x such that x is less than 6 where this x belongs to the set of real numbers r. This is the solution set for the set A union. The solution set for the set B which is x such that x belongs to the set of real numbers such that x is less than x belongs to the set of real numbers equation to the left is not included in the graph. The combination of two equations.