 Hello friends, let's work out the following problem. It says solve the following system of inequalities graphically. The first inequality given to us is 2x plus i greater than equal to 6 and the second inequality given to us is 3x plus 4 y less than equal to 12. Let us now move on to the solution. First inequality is 2x plus y greater than equal to 6 and its corresponding equation of line is 2x plus y is equal to 6. Now to draw the line we need to have two points. So if x is 0 that implies y is equal to 6 and if y is 0 this implies x is equal to 3. So to draw the line 2x plus y is equal to 6 we need to plot the ordered pairs 0, 6 and 3, 0. Let's now draw the line 2x plus y is equal to 6 or that we need to plot the ordered pairs 0, 6 and 3, 0. That is if x is 0, y is 6 and if y is 0 then x is 3. Let's now join these two points to get the line 2x plus y is equal to 6. Now we have to identify the region for the inequality 2x plus y greater than equal to 6. For that we take any point not lying on the line 2x plus y is equal to 6 and we will check whether that point satisfies this inequality or not. If that point satisfies this inequality we will shade the region which contains that point and if that point doesn't satisfy this inequality we will shade the region which doesn't contain that point. And we see that the point 0, 0 does not lie on the line 2x plus y is equal to 6. So if x is 0, y is 0 the inequality becomes 2 into 0 plus 0 is greater than equal to 6 that is 0 greater than equal to 6 which is not true. That means the point, 0, 0 does not satisfy the inequality 2x plus y is greater than equal to 6. So we will shade the region which does not contain the point, 0, 0 for the inequality 2x plus y greater than equal to 6. Now this is the region which does not contain the point 0, 0 for the inequality 2x plus y greater than equal to 6. greater than equal to 6. So, we shade this region. This is the solution region for the inequality 2x plus y greater than equal to 6 and this solution region also includes the line 2x plus y is equal to 6 because we are given the inequality as 2x plus y greater than equal to. That means the line is also included in the solution region. So, we need to darken this line. Now, the second inequality given to us is 3x plus 4y greater than equal to 12 and its corresponding equation of line is 3x plus 4y is equal to 12. Now, to draw this line we need to have at least two points. So, if x is 0, so this implies y is equal to 3 and if y is 0 this implies x is equal to 4. So, we need to plot the ordered pairs 0, 3 and 4, 0 to draw the line 3x plus 4y is equal to 12. So, let us now plot the ordered pairs 0, 3 and 4, 0. That is if x is 0 then y is 3 and if y is 0 then x is 4 this point. Now, join these two points to get the line 3x plus 4y is equal to 12. Now, we have to identify the region for the inequality 3x plus 4y less than equal to 12. For that we take any point not lying on the line 3x plus 4y is equal to 12 and we have seen that the point 0, 0 does not lie on the line 3x plus 4y is equal to 12 and if x is 0, y is 0 then inequality becomes 3 into 0 plus 4 into 0 less than equal to 12 that is 0 is less than equal to 12 which is true. That means the point 0, 0 satisfies the inequality 3x plus 4y less than equal to 12. So, we need to shape the region which contains the point 0, 0 for the inequality 3x plus 4y less than equal to 12. Now, this is the region which contains the point 0, 0 for the inequality 3x plus 4y less than equal to 12 right. So, we shape this region this is the solution region for the inequality 3x plus 4y less than equal to 12 and this solution region also includes the line 3x plus 4y is equal to 12 because the inequality contains the sign less than equal to. So, we need to darken this line to show that this line is also included in the solution region. Now, the region in the dark grey is common to both the solution regions and this is the required solution region and this completes the question. Bye for now. Take care. Have a good day.