Linear Inequalities

The inequalities represent a system of constraints defining the shaded region in the graph.
1. represents a line that cuts the region into two parts. We are interested in the region above this line.
2. represents a vertical line at and restricts the region to the left of this line.
3. represents a horizontal line at and restricts the region to below this line.
4. and ensure that the region is in the first quadrant. Thus, the solution set of the inequalities that defines the shaded region is option (C).

The correct answer is (C) : 5x + 4y ≥ 20, x ≤ 6, y ≤ 3, x ≥ 0, y ≥ 0.

The correct answer is: (B): .

We are tasked with identifying the system of inequalities that describes the shaded region in the given figure.

Step 1: Understand the inequalities

The first inequality is . This represents a half-plane where the sum of and is less than or equal to 7. This forms a line with slope -1 and a y-intercept of 7. The region of interest is below this line (since ).

Step 2: Analyze the second inequality

The second inequality is , which can be rearranged to . This represents a half-plane above the line . The line has a slope of and a y-intercept of -2. The region of interest is above this line.

Step 3: Consider the third and fourth inequalities

The third inequality, , restricts the solution to the right of the y-axis (positive x-axis). Similarly, the fourth inequality, , restricts the solution to the upper half-plane (positive y-axis).

Step 4: Combine the inequalities

By combining all the inequalities, we find the region that satisfies all these conditions is the shaded region in the figure. Thus, the correct answer is (B): .