 Hi, and welcome to the session. Let's discuss the following question. The question says, solve the following inequality graphically in two-dimensional plane. Before solving this question, we should know that a line divides the Cartesian plane into two-half planes. And the graph of inequality will be one of the half planes. And we shall show the solution by shading in the corresponding half plane. We shall follow the following rules to identify the half plane represented by an inequality. According to the first rule, we have to take any point a v not on the line. And then we have to check whether it satisfies the inequality or not. If that point satisfies the inequality, then the inequality represents that particular half plane containing the point. But if the point does not satisfy the inequality, then the other half plane represents the solution. According to the second rule, if we have a sign of equality with inequality, then line is included in the solution region, and so we draw a dotted line. And according to the third rule, if we have a pure inequality, then the points on the corresponding line are not to be included in the solution region, and so we draw a dotted line. With the help of these three rules, we will solve this question. So always remember these rules. Now, we can put the solution. Given inequality is y plus 8 is greater than equal to 2x. Now, this implies y minus 2x is greater than equal to minus 8, and this implies 2x minus y is less than equal to 8. So this means y plus 8 greater than equal to 2x is equal into 2x minus y less than equal to 8. Now, converting this inequality into equation, we get 2x minus y is equal to 8. Now we have to plot the graph of this equation. For plotting the graph, we need at least two solutions of this equation. So let's first find the two solutions of this equation. When x is equal to 0, then y is equal to minus 8, and when y is equal to 0, then x is equal to 4. So the two solutions of this equation are 0 minus 8 and 4, 0. Now we will plot these two points on the graph. So let's make a graph now. The two points which we have to plot are 0 minus 8 and 4, 0. Now, the first point is 0 minus 8. This means when x is i is 0, then ordinate is minus 8. So this is the required point, 0 minus 8. Second point is 4, 0. This means when ordinate is 0, then if c is 4. So this is the required point, 4, 0. According to the second rule, we know that if sign of equality is there with the inequality, then the line is included in the solution region, and so we draw a dark line. Now in this question, we have y plus 8 is greater than equal to 2x. The sign of equality is there, so this means we will join these two points by a dark line, and this line will be included in the solution region. So let's now join these two points by a straight line. Now this line is representing the equation 2x minus y is equal to 8, which is equivalent to y plus 8 is equal to 2x. This line is dividing this plane into two half planes, that is 1 and 2. Now we have to identify the half plane represented by the given inequality. According to the first rule, we have to select a point which does not lie on the line, and then we have to check whether it satisfies the inequality or not. If the point satisfies the inequality, then the inequality represents that particular half plane containing the point. But if the point does not satisfy the inequality, then the other half plane, that means the half plane which does not contain the point, represents the solution. As the point 0, 0 does not lie on this line, so we can take the point as 0, 0. Now substitute this point in the given inequality by substituting x i 0 and y i 0. We get 0 less than equal to 8, which is true of plane containing 0, 0 represents the inequality x minus y is less than equal to 8, which is equivalent to y plus 8 greater than equal to 2x. Now look at the graph. Now the first half plane contains the point 0, 0. So this means the solution region of the given inequality consists of first half plane. And since we have equal to sign in the given inequality, so this means this line is also included in the solution region. Let's now shade this region. This is the recoup. So the shaded half plane first, including the line, is the required graphical solution of y plus 8 greater than equal to 2x. This is our required answer. So this completes the session. Bye and take care.