 Hello and welcome to the session. Let's discuss the following question. It says, solve the following system of inequalities graphically. Let us now move on to the solution. The first inequality given to us is x plus 2y less than equal to 10 and its corresponding equation of line is x plus 2y is equal to 10. Now to draw this line we need to have two points. So if x is 0 then y is equal to 5 and if y is equal to 0 then x is equal to 10. So we need to plot the ordered pairs 0 5 10 0 to draw the line x plus 2y is equal to 10. Let us now draw the line x plus 2y is equal to 10. For that we need to plot the ordered pairs 0 5 and 10 0. Now if y is 0 x is 10 and if x is 0 y is 5. Now we join these two points to get the line x plus 2y is equal to 10. Now we have to identify the region for the inequality x plus 2y less than equal to 10. For that we take any point not lying on the line x plus 2y is equal to 10 and we check whether that point satisfies this inequality or not. If that point satisfies this inequality we will shape the region which contains that point and if that point doesn't satisfy this inequality we will shape the region which doesn't contain that point and we know that 0 0 does not lie on the line x plus 2y is equal to 10. So we take that point to be 0 0. So if x is 0 y is 0 the inequality becomes x plus 2y that is 0 plus 2 into 0 is less than equal to 10 that is 0 is less than equal to 10 which is true. That means the point 0 0 satisfies the inequality x plus 2y less than equal to 10. So we will shape the region which contains the point 0 0 for the inequality x plus 2y less than equal to 10. Now this is the region which contains the point 0 0 for the inequality x plus 2y less than equal to 10 so we will shape this region. Also the inequality contains the less than equal to sign which shows that the line x plus 2y is also included in the solution region. So we need to darken this line to show that this line is also included in the solution region of the inequality x plus 2y less than equal to 10. Now the second inequality given to us is x plus y greater than equal to 1 and the corresponding equation of line is x plus y is equal to 1. Now if x is 0 then y is 1 and if y is 0 then x is equal to 1. So we need to plot the ordered pairs 0 1 and 1 0 to draw the line x plus y is equal to 1. Let us now draw the line x plus y is equal to 1. So if x is 0 y is 1 and if y is 0 x is 1. So we join these two points to get the line x plus y is equal to 1. Now to identify the region for the inequality x plus y greater than equal to 1 we take the point not lying on the line x plus y is equal to 1. 0 0 does not lie on the line x plus y is equal to 1. So if x is 0 y is 0 then inequality becomes 0 plus 0 greater than equal to 1 that is 0 greater than equal to 1 which is not true. That implies the point 0 0 does not satisfy the inequality x plus y greater than equal to 1. So we will shape the region which does not contain the point 0 0 for the inequality x plus y greater than equal to 1. Now this is the region which does not contain the point 0 0 for the inequality x plus y greater than equal to 1. That means the region above the line x plus y is equal to 1. So we shape this region. So the region above the line x plus y is equal to 1 is the solution region for the inequality x plus y greater than equal to 1 which also includes the line x plus y is equal to 1. So we darken this line. Now the third inequality given to us is x minus y less than equal to 0 and its corresponding equation of line is x minus y is equal to 0. That implies x is equal to y. So we need to draw the line x is equal to y and we know that this is the line passing through the region that is if x is 0 then y is also 0. If x is 1 then y is 1. Let us now draw the line x is equal to y. It passes through the region. So we draw the line x is equal to y. Now we have to identify the region for the inequality x minus y less than equal to 0. For that we take any point not line on the line x minus y is equal to 0. Now we know that the point 0 0 lies on the line x minus y is equal to 0 that is x is equal to y. So we take any other point let's say 0.12 and if x is 1 y is 2. So the inequality becomes 1 minus 2 less than equal to 0 that implies minus 1 is less than equal to 0. Which is true that means the point 1 2 satisfies inequality x minus y less than equal to 0. So we will shape the region which contains the point 1 2 so the inequality x minus y less than equal to 0. Now the point is 1 2 that is x is 1 y is 2 it is this point. So we will shape the region which contains this point. So let us now shape this region. Now we need to darken the line x is equal to y because the line is also included in the solution region. Now we are also given that x is greater than equal to 0 and y is greater than equal to 0. That is each point in the solution region lies in first quadrant and the line x is equal to 0 and y is equal to 0 also included in the solution region. Now we see that the region in dark green is common to all the solution regions and this is the required solution region. And this completes the question. Hope you enjoyed this session with Diane Take Care.