Graph the solution of the system of linear inequalities.
y >= - x + 2
y <= 4x + 6
To graph the solution of the system of linear inequalities, we will first graph the boundary lines of the inequalities and then shade the region that satisfies both inequalities.
The first inequality, y >= -x + 2, can be graphed by creating a straight line with a slope of -1 and a y-intercept of 2. Since y is greater than or equal to, we will draw a solid line to represent this inequality.
The second inequality, y <= 4x + 6, can be graphed by creating a straight line with a slope of 4 and a y-intercept of 6. Since y is less than or equal to, we will draw a solid line to represent this inequality.
Now, we will shade the region that satisfies both inequalities. To determine this shaded region, we will choose a point within the overlapping regions. Let's use the origin (0,0) as this point.
When we substitute (0,0) into the first inequality, we get 0 >= -0 + 2, which is true. Therefore, the region above the line y = -x + 2 satisfies the first inequality.
When we substitute (0,0) into the second inequality, we get 0 <= 4(0) + 6, which is true. Therefore, the region below the line y = 4x + 6 satisfies the second inequality.
The overlapping region between y >= -x + 2 and y <= 4x + 6 is shaded.
Here is the graph of the solution: