 Hello and welcome to the session. Let us discuss the following question. It says, sort the following system of inequality graphically. Let us now move on to the solution. The first inequality given to us is 2x plus y greater than equal to 4 and its corresponding equation of line is 2x plus y is equal to 4. Now to draw this line we need to have two points. So if y is 0 then x is equal to 2 and if x is 0 then y is equal to 4. So to draw this line we need to plot the ordered pairs 2, 0 and 0, 4. Let us now draw the line 2x plus y is equal to 4. For that we need to plot the ordered pairs 2, 0 and 0, 4. So if x is 2 then y is 0 and if x is 0 then y is 4. Let us now join these two points to get the line 2x plus y is equal to 4. Now we have to identify the region for the inequality 2x plus y greater than equal to 4. To identify the region we take any point not lying on the line 2x plus y is equal to 4 and we will 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 does not satisfy this inequality we will shape the region which does not contain that point. Now we take that point to be 0, 0 as it does not lie on the line 2x plus y is equal to 4. So if x is 0, y is 0 then inequality becomes 2 into 0 plus 0 greater than equal to 4 that is 0 greater than equal to 4 which is not true. That means point 0, 0 does not satisfy the inequality x, y greater than equal to 4. So we will shape the region which does not contain the point 0, 0 for the inequality 2x plus y greater than equal to 4. Now here we have the point 0, 0 and we have to shape the region which does not contain the point 0, 0 for the inequality 2x plus y greater than equal to 4. So this is the region which does not contain the point 0, 0 so we shape this region. So this is the solution region for the inequality 2x plus y greater than equal to 4 and this solution region also includes the line 2x plus y is equal to 4 because we have the sign greater than equal to which shows that the line is also included in the solution region. So we darken this line to show that it is included in the solution region. Now the second inequality given to us is x plus y less than equal to 3 and its corresponding equation of line is x plus y is equal to 3. So if x is 0 then y is equal to 3 and if y is 0 then x is equal to 3. So we plot the ordered pairs 0, 3 and 3, 0 to draw the line x plus y is equal to 3. Now when 0, y is 3 is this point and if y is 0, x is 3 so it is this point. Now we draw the line x plus y is equal to 3. Now to identify the region for the inequality x plus y less than equal to 3 we take any point not lying on the line x plus y is equal to 3. Let us take that point to be 0, 0 as it does not lie on the line x plus y is equal to 3. So if x is 0, y is 0 then inequality becomes 0 plus 0 less than equal to 3 that is 0 less than equal to 3 which is true that means the point 0, 0 satisfies the inequality x plus y less than equal to 3. So we will show that we have to shade the region which contains the point 0, 0 for the inequality x plus y less than equal to 3. Now we shade the region for the inequality x plus y less than equal to 3 and we have to shade the region which contains the point 0, 0. So this is the region which contains the point 0, 0 for this inequality so we need to shade this region below the the line so this is the solution region for the inequality x plus y less than equal to 3 and this solution region also includes the line x plus y is equal to 3 as the inequality contains the sign less than equal to. Now the third inequality given to us is 2x minus 3y less than equal to 6 and it's corresponding equation of minus 2x minus 3y is equal to 6. So if x is 0 then y is equal to minus 2 and if y is 0 then x is equal to 3. So we need to plot the ordered pairs 0 minus 2 and 3 0 to draw the line 2x minus 3y is equal to 6. Now we draw the line 2x minus 3y is equal to 6 for that we need to plot the ordered pairs 0 minus 2 and 3 0. So if x is 0 then y is minus 2 that is this point and if x is 3 then y is 0. Now we join these two points to get the line 2x minus 3y is equal to 6. Now we identify the region for the inequality 2x minus 3y less than equal to 6. So we take the point 0 0 as it does not lie on the line 2x minus 3y is equal to 6 and if x is 0 y is 0 then inequality becomes 2 into 0 minus 3 into 0 less than equal to 6 that is 0 is less than equal to 6 which is true. So the point 0 0 satisfies the inequality 2x minus 3y less than equal to 6. So we will shade the region which contains the point 0 0 for the inequality 2x minus 3y less than equal to 6. Now this is the region which contains the point 0 0 for the inequality 2x minus 3y less than equal to 6. So we shade this region solution region for the inequality 2x minus 3y less than equal to 6 and it also includes the line 2x minus 3y is equal to 6. So we darken this line to show that this line is included in the solution region. Now this triangular region is common to all the solution regions. So we shade this triangular region with some other color because this is the required solution region which is common to all the three regions. Now the triangular region in green is the required solution region for all three regions. So this completes the question. Hope you enjoyed this session. Goodbye and take care.