 Hello and welcome to the session. Let's discuss the following question. It says solve the following system of inequalities graphically. So let's move on to the solution. The first inequality given to us is 4x plus 3y less than equal to 60 and its corresponding equation of line is 4x plus 3y is equal to 60. Now to draw this line we need to have two points. So if y is 0 then x is equal to 15 and if x is 0 then y is equal to 20. So in order to draw the line 4x plus 3y is equal to 60 we need to plot the ordered pairs 15 0 and 0 20. Let's now draw the line 4x plus 3y is equal to 60 for that we need to plot the points 15 0 and 0 20. Now when x is 15 y is 0. Here is x is equal to 15 and y is 0 and when x is 0 y is 20 this point. Now we join these two points to get the required line. Now we have to identify the region for the inequality 4x plus 3y less than equal to 60. For that we take any point not lying on the line 4x plus 3y is equal to 60 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. Now we take that point to be 0 0 and we see that the point 0 0 satisfies this inequality because 4 into 0 plus 3 into 0 is less than equal to 60 that is 0 is less than equal to 60 is true. That means 0 0 satisfies inequality 4x plus 3y less than equal to 60. So we will shape the region which contains the point 0 0 for the inequality 4x plus 3y less than equal to 60. Now this is the region which contains the point 0 0 so we shape this region this is the solution region for the inequality 4x plus 3y less than equal to 60 and this solution region also includes the line 4x plus 3y is equal to 60 because the inequality contains less than equal to sign. So we make a dark line to show that this line is included in the solution region. Now the second inequality given to us is y greater than equal to 2x and its corresponding equation of line is y is equal to 2x. Now if x is 0 then y is also 0 and if x is 2 then y is 4 and if y is 6. Now we draw the line y is equal to 2x and we see that it passes through the origin and we need to plot the ordered pair 0 0 2 4 3 6 and so on. Now when x is 0 y is 0 when x is 2 y is 4 is this point when x is 3 y is 6 and x is 4 y will be 8. Now we join these points to get the line y is equal to 2x. Now to identify the region for the inequality y greater than equal to 2x we take any point not lying on the line y is equal to 2x. Now we have 0 0 lying on the line y is equal to 2x so we take any point which does not lie on the line y is equal to 2x. So let's take that point to be minus 1 1 that is x is minus 1 y is 1. So the inequality becomes 1 is greater than equal to 2 into minus 1 that is 1 is greater than equal to minus 2 which is true. That means the point minus 1 1 satisfies inequality y greater than equal to 2x. So we will shade the region which contains the point minus 1 1 for the inequality y greater than equal to 2x. Now here we have the point minus 1 1 and we have to shade the region which contains the point minus 1 1 for the inequality y greater than equal to 2x. So this is the region which we have to shade this region. The solution region for the inequality y greater than equal to 2x which also includes the line y is equal to 2x because the inequality contains the sign greater than equal to. That means the line is included in the solution region. Now the third inequality given to us is x greater than equal to 3 and its corresponding equation of line is x is equal to 3. Now we draw the line x is equal to 3. Here we have x is equal to 3 so we draw the line x is equal to 3. Now we have to shade the region for the inequality x greater than equal to 3. Now this is the region where x are greater than equal to 3 so we need to shade this region. So let's now shade this region. The solution region for the inequality x greater than equal to 3 and this solution region also includes the line x is equal to 3. We make a dark line. We have also given that x and y greater than equal to 0. That means every point in the shaded region is in the first quadrant. This is the first quadrant. So first quadrant is the solution region for the inequality xy greater than equal to 0. Now we see that the region in brown is the region which is common to all the four regions. That means this is the required solution region. And this completes the question. Bye for now. Take care. Have a good day.