Question

The shaded region shown in the figure is given by the inequations: [height=5cm]{Screenshot 2026-05-01 102102.png}

• A. \( 14x + 5y \ge 70, \, y \le 14 \) and \( x - y \ge 5 \)
• B. \( 14x + 5y \le 70, \, y \le 14 \) and \( x - y \ge 5 \)
• C. \( 14x + 5y \ge 70, \, y \ge 14 \) and \( x - y \ge 5 \)
• D. \( 14x + 5y \ge 70, \, y \ge 14 \) and \( x - y \le 5 \)
• E. \( 14x + 5y \ge 70, \, y \le 14 \) and \( x - y \le 5 \)

Hint:

To quickly find the correct inequality, pick a point clearly inside the shaded region (e.g., \( (6, 10) \)) and see which set of inequalities it satisfies. This saves time compared to deriving every line equation from scratch.

Answer 1

Answer: (E)

Concept: A shaded region in a 2D plane is defined by a system of linear inequalities. Each boundary line corresponds to an equation, and the shading direction is determined by testing a point (like the origin or a point inside the region).

Step 1: Identify the equations of the boundary lines.
The triangle has vertices at \( (0, 14) \), \( (5, 0) \), and \( (19, 14) \). 1. Line joining (0, 14) and (5, 0): Slope \( m = \frac{0 - 14}{5 - 0} = -\frac{14}{5} \). Equation: \( y - 0 = -\frac{14}{5}(x - 5) \Rightarrow 5y = -14x + 70 \Rightarrow 14x + 5y = 70 \). For the shaded region (to the right of this line), test \( (6, 1) \): \( 14(6) + 5(1) = 84 + 5 = 89 \ge 70 \). Inequality: \( 14x + 5y \ge 70 \).

Step 2: Identify the horizontal boundary.
2. Line joining (0, 14) and (19, 14): This is a horizontal line where \( y = 14 \). The shaded region is below this line. Inequality: \( y \le 14 \).

Step 3: Identify the third boundary line.
3. Line joining (5, 0) and (19, 14): Slope \( m = \frac{14 - 0}{19 - 5} = \frac{14}{14} = 1 \). Equation: \( y - 0 = 1(x - 5) \Rightarrow y = x - 5 \Rightarrow x - y = 5 \). For the shaded region (to the left/above this line), test \( (6, 14) \): \( 6 - 14 = -8 \le 5 \). Inequality: \( x - y \le 5 \).