 Hello friends, let's work out the following problem. It says solve the following system of inequalities graphically. So let's move on to the solution. The first inequality given to us is x plus y is less than equal to 6 and its corresponding equation of line is x plus y is equal to 6. Now to draw this line we need to have two points. So if y is 0 then x is equal to 6 and if x is 0 then y is equal to 6. So if we want to draw this line we need to plot the points 6, 0 and 0, 6. Let's now draw the line x plus y is equal to 6. For that we need to plot the points 6, 0 and 0, 6. Now when x is 6 y is 0 which is this point and if x is 0 y is 6. So it is this point. Now we join these two points to get the line x plus y is equal to 6. This is the line x plus y is equal to 6. Now we have to shade the region for the inequality x plus y less than equal to 6. For that we take any point not lying on the line x plus y is equal to 6 and we'll check whether that point satisfies this inequality or not. If that point satisfies this inequality we'll shade the region which contains that point. If that point doesn't satisfy this inequality we'll shade the region which doesn't contain that point. Now we take that point to be 0, 0. So when x is 0 y is 0 then inequality becomes 0 plus 0 less than equal to 6 that is 0 is less than equal to 6 which is true. Thimpize the point 0, 0 satisfies the inequality or the region x plus y less than equal to 6. So we'll shade the region which contains the point 0, 0. Now we shade the region for the inequality x plus y less than equal to 6 which contains the point 0, 0. So this is the region which contains the point 0, 0. So let's shade this region. This is the solution region for the first inequality and the solution region also includes this line x plus y is equal to 6 because the inequality contains the sign less than equal to. So we darken this line to show that the line is included in the solution region. Now the second inequality given to us is x plus y greater than equal to 4 and its corresponding equation of line is x plus y is equal to 4. So when y is 0, x is 4 and when x is 0, y is 4. So in order to draw the line x plus y is equal to 4 we need to plot the points 4, 0 and 0, 4. So now we draw the line x plus y is equal to 4 and for that we need to plot the points 4, 0 and 0, 4. So when x is 4, y is 0, so it is this point and when x is 0, y is 4. So it is this point. Now we join these two points to get the line x plus y is equal to 4. Now we have to identify the region for the inequality x plus y greater than equal to 4. So we take the point 0, 0 not lying on the line x plus y is equal to 4. So that the inequality becomes 0 plus 0 greater than equal to 4, that is 0 is greater than equal to 4 which is not true. That implies the point 0, 0 does not satisfy plus y greater than 4. So we shape the region for the inequality x plus y greater than equal to 4 which does not contain 0, 0. So we have to shape the region for the inequality x plus y greater than equal to 4 which does not contain 0, 0. So it is this region. So let us now shape this region. Now again for the inequality x plus y greater than equal to 4, we need to darken the line x plus y is equal to 4 because it shows that this line is included in the solution region. So we darken this line. The region in red is common to both the shaded regions and it is the solution region for the system of inequalities 1 and 2 and this is the required solution. So this completes the question. Bye for now. Take care. Have a great day.