Question

The shaded region in the figure given is the solution of which of the inequations?

• A. x + y ≥ 7, 2x - 3y + 6 ≥0, x ≥ 0, y ≥ 0
• B. x + y ≤ 7, 2x - 3y + 6 ≥0, x ≥ 0, y ≥ 0
• C. x + y ≤ 7, 2x - 3y + 6 ≤0, x ≥ 0, y ≥ 0
• D. x + y ≥ 7, 2x - 3y + 6 ≤0, x ≥ 0, y ≥ 0

Hint:

When dealing with systems of inequalities, it is helpful to first graph the boundary lines of each inequality and then determine which side of each line the solution lies on. This method makes it easier to visually identify the region that satisfies all inequalities.

Answer 1

The Correct Option is B

The correct answer is: (B): \( x + y \leq 7, \, 2x - 3y + 6 \geq 0, \, x \geq 0, \, y \geq 0 \).

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 \( x + y \leq 7 \). This represents a half-plane where the sum of \( x \) and \( y \) 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 \( x + y \leq 7 \)).

Step 2: Analyze the second inequality

The second inequality is \( 2x - 3y + 6 \geq 0 \), which can be rearranged to \( 2x - 3y \geq -6 \). This represents a half-plane above the line \( 2x - 3y = -6 \). The line has a slope of \( \frac{2}{3} \) 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, \( x \geq 0 \), restricts the solution to the right of the y-axis (positive x-axis). Similarly, the fourth inequality, \( y \geq 0 \), 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): \( x + y \leq 7, \, 2x - 3y + 6 \geq 0, \, x \geq 0, \, y \geq 0 \).