Short Answer 
The process involves identifying two inequalities: the first is ( y geq x – 2 ) with a solid line representing the area above it, and the second is ( y < -frac{1}{2}x + 2 ) with a dashed line indicating the area below. The complete system of inequalities can be expressed as ( y geq x - 2 ) and ( x + 2y < 4 ), which can be visualized on a graph to show the shaded regions.
Step 1: Determine the First Inequality
Start by identifying the first inequality which is given as ( y geq x – 2 ). To find the equation of the line, use the points (0, -2) and (2, 0) to calculate the slope. The slope formula is ( m = frac{y_2 – y_1}{x_2 – x_1} ). By substituting the given points, you find that the slope ( m = 1 ) and the y-intercept is -2. Thus, the equation of the line is ( y = x – 2 ) and you will shade the area to the left of the solid line.
Step 2: Determine the Second Inequality
Next, for the second inequality, we use the points (0, 2) and (4, 0) to find its equation. Again apply the slope formula ( m = frac{y_2 – y_1}{x_2 – x_1} ), yielding a slope of ( m = -frac{1}{2} ) and a y-intercept of 2. The equation of this dashed line is ( y = -frac{1}{2}x + 2 ). Since we are looking at values below this line, the inequality is ( y < -frac{1}{2}x + 2 ).
Step 3: Combine and Rewrite the System of Inequalities
Finally, combine the results from the first two steps to write the complete system of inequalities. Start by rewriting the second inequality ( y < -frac{1}{2}x + 2 ) as ( 2y < -x + 4 ) leading to ( x + 2y < 4 ). Therefore, your complete set of inequalities is ( y geq x - 2 ) and ( x + 2y < 4 ). Make sure to visualize this on a graph to clearly identify the shaded regions.