Sure, we need to find the points that satisfy both of the inequalities x + y > 2 and 4x + y >= -1.
We have the following four points to test: (0, 0), (1, 2), (2, 2), and (-1, -1).
Let's start by plugging these points into our inequalities.
1. For point (0, 0), we substitute these values to the inequalities:
Inequality 1: x + y > 2 becomes 0 + 0 > 2 which is false, as 0 is not greater than 2.
Hence, point (0, 0) does not satisfy all inequalities, so we do not need to check the second inequality for this point.
2. For point (1, 2), we substitute these values to the inequalities:
Inequality 1: x + y > 2 becomes 1 + 2 > 2, which is true as 3 is greater than 2.
Then we check the second inequality: 4x + y >= -1 becomes 4*1 + 2 >= -1, which is also true as 6 is greater than -1.
Hence, point (1, 2) satisfies all inequalities, so this point is included in our solution set.
3. For point (2, 2), we substitute these values to the inequalities:
Inequality 1: x + y > 2 becomes 2 + 2 > 2, which is true as 4 is greater than 2.
Then we check the second inequality: 4x + y >= -1 becomes 4*2 + 2 >= -1, which is also true as 10 is greater than -1.
Hence, point (2, 2) satisfies all inequalities, so this point is also included in our solution set.
4. For point (-1, -1), we substitute these values to the inequalities:
Inequality 1: x + y > 2 becomes -1 + -1 > 2 which is false, as -2 is not greater than 2.
Hence, point (-1, -1) does not satisfy all inequalities, so don't need to check the second inequality for this point.
Thus, among the given points, (1, 2) and (2, 2) are the points that lie in the solution set of the system of inequalities.