 Hello and welcome to this session. In this session we will discuss how to apply linear equations and inequality equations in very very easy in real life. First of all let us discuss solving and creating equations. Now suppose we have given a trapezoid having właśnie has a trapezoid number which is equal to 54 cm square. Now here sides b1 and b2 are parallel to each other and here we have given b1 is equal to 10 cm also we are given We are here height h is equal to 6 centimeters and here we have to find length of side b2 that is we have to find b2 here. Now we know that area of trapezoid is equal to 1 by 2 into height of the trapezoid that is h into b1 plus b2 the whole. Here b1 and b2 are the two permanent sides of the trapezoid. Now we can write this as area A is equal to 1 by 2 into h into b1 plus b2 the whole. Now we have to set up an equation to find the length of base b2. First of all we will multiply this equation on both sides by 2. So we have 2 into A is equal to 2 into 1 by 2 into h into b1 plus b2 the whole. 2A is equal to h into b1 plus b2 the whole. Now we will divide both sides of this equation by h. So we have 2A upon h is equal to h into b1 plus b2 the whole. The whole upon h and this further implies 2A upon h is equal to b1 plus b2. Now we will subtract b1 from both sides of this equation. So we have 2A upon h minus b1 is equal to b1 plus b2 minus b1. And this further implies 2A upon h minus b1 is equal to b2. So this is the required equation like this b equation 1. Now in this equation we will put A is equal to 24 cm square h is equal to 6 cm and b1 is equal to 10 cm. So putting these values we have 2 into 54 whole upon 6 minus 10 is equal to b2. Now we know that 2 into 3 is 6 and 3 into 18 is 54. So this implies 18 minus 10 is equal to b2. This further implies b2 is equal to 8. So b2 is equal to 8 cm so b1 is equal to 10 cm, h is equal to 6 cm and b2 is equal to 8 cm. Now while writing a linear equation we have to model the given situation with an algebraic model of the problem by identifying the actions that suggest operations. Then we define one or more variables to represent the unknown and then we write the algebraic model that represents the action using the variables and the operations but in the algebraic model we solve each variable if possible. Now let us see an illustration which uses an algebraic model. Now you start with 20 values and save 6 values each week. How many weeks will it take to solve this problem? Now let us start with its solution. Now let us use an algebraic model. First we write off the problem. Now here total amount saved equals started amount plus amount saved each week times number of weeks. So here we have related all the parts of the given problem. Now let us define what is given to us. Now here unknown is number of weeks so let it be w. Now here total amount saved is 200 dollars equals started amount that is 20 dollars plus amount saved each week that is 6 dollars times number of weeks that is w. And now we write the algebraic model that represents the action using the variables and operations. So we have 200 dollars is equal to 20 dollars plus 6 dollars into w. Now let us solve this equation from which it is equal to 20 plus 6 into w. Now we subtract 20 from both sides of this equation. And here we have 200 minus 20 is equal to 20 plus 6 w minus 20 which implies 180 is equal to 6 w. Now dividing both sides of this equation by 6. We have 180 upon 6 is equal to 6 w upon 6. Further on solving this now we know that 6 into 1 is 6 and 6 into 30 is 180. So we have 30 is equal to w thus it will take 30 weeks to save 200 dollars. Now we use the similar method of modeling a linear inequality and solving it. Now let us see the following illustration. Samantha wants to earn 29 dollars to buy an outfit. Her mother agrees to pay her 6 dollars an hour for gardening in addition to her 5 dollars weekly allowance along the house. Watching the minimum number of hours, Samantha must work at gardening to receive at least 29 dollars this week. Let us first understand the problem. We are given that Samantha's weekly allowance is 5 dollars. She will get 6 dollars for gardening and we have to find she will have to work to receive at least 29 dollars. Now see at least means the minimum amount it means amount will be greater than or equal to 29 dollars. So here sign will be of greater than or equal to. Now first of all let us relate all parts of the given problem. Now here Samantha's weekly allowance plus amount earned per hour by gardening times number of hours is at least 29 dollars. So we have related all parts of the given problem. Now next we define what is given to us here. We have to find number of hours that is the unknown here is number of hours. So let it be denoted by H is Samantha's weekly allowance that is 5 dollars plus by gardening that is 6 dollars times number of hours that is H at least 29 dollars. Now let us write the variables and substitute values. So it is 5 dollars plus into is at least now for at least we will use the sign of greater than or equal to 29 dollars. Now let us solve this inequality now 5 plus 6 H is greater than equal to 29. Now subtract 5 from both sides. So we have 5 plus 6 H minus 5 is greater than or equal to 29 minus this implies 6 H is greater than or equal to 24. Now let us divide both sides of this inequality by 6. So this implies 6 H upon 6 is greater than or equal to 24 upon 6. Now here since we are dividing by a person number so the inequality remains same. Now this implies is greater than or equal to now 6 into 4 is 24. So we get H is greater than or equal to 4. So minimum number of hours Samantha must work to receive at least 29 dollars is 4. Now let us check our result for this what H is equal to 4 in this inequality. So we have 5 plus 6 into 4 is greater than or equal to 29 which implies 5 plus 24 is greater than or equal to 29. And this implies 29 is greater than or equal to 29 which is true. Thus our answer is correct. So in this session we have discussed how to apply linear equations and in equations in real life. And this completes our session. Hope you all have enjoyed the session.