Question

Shade the feasible region for the inequalities \[ x + y \geq 2, \quad 2x + 3y \leq 6, \quad x \geq 0, \quad y \geq 0 \] in a rough figure.

• A.

  

• B.

  

• C.

  

• D. None Of The Above

Hint:

For solving inequalities graphically, always plot the boundary lines first, then shade the regions that satisfy the inequalities. The overlapping shaded region gives the feasible region.

Answer 1

The Correct Option is B

We are given the system of inequalities:
1. \( x + y \geq 2 \)
2. \( 2x + 3y \leq 6 \)
3. \( x \geq 0 \)
4. \( y \geq 0 \) We need to sketch the feasible region by first plotting the boundary lines and then shading the region satisfying all the inequalities. Step 1: Plot the lines 1. \( x + y = 2 \) represents a line where the x-intercept is 2 (when \( y = 0 \)) and the y-intercept is 2 (when \( x = 0 \)). The region satisfying \( x + y \geq 2 \) lies above this line.
2. \( 2x + 3y = 6 \) represents a line with the x-intercept 3 (when \( y = 0 \)) and the y-intercept 2 (when \( x = 0 \)). The region satisfying \( 2x + 3y \leq 6 \) lies below this line. Step 2: Consider the restrictions \( x \geq 0 \) and \( y \geq 0 \)
The feasible region is constrained to the first quadrant (where both \( x \geq 0 \) and \( y \geq 0 \)). Step 3: Shade the feasible region To satisfy all inequalities, shade the area where all conditions overlap. This region is represented in Option (B). Thus, the correct answer is Option (B).