Question

The solution set for the system of linear inequations $x+y \ge 1 ; 7x+9y \le 63 ; y \le 5 ; x \le 6, x \ge 0$ and $y \ge 0$ is represented graphically in the figure. What is the correct option?

• A.

  

• B.

  

• C.

  

• D.

  

Hint:

To quickly find the correct region, use the test point $(0,0)$. For $x+y \ge 1$, $0 \ge 1$ is false, so shade away from origin. For $7x+9y \le 63$, $0 \le 63$ is true, so shade towards origin.

Answer 1

The Correct Option is A

Step 1: Read the inequalities.
The region must satisfy $x + y \ge 1$, $7x + 9y \le 63$, $y \le 5$, $x \le 6$, with $x \ge 0$ and $y \ge 0$.

Step 2: Understand each boundary.
Each inequality is a straight line. The allowed side of each line is decided by its sign.

Step 3: Handle the "greater than" line.
$x + y \ge 1$ keeps the side away from the origin, since the origin gives $0 \ge 1$, which is false.

Step 4: Handle the "less than" lines.
$7x + 9y \le 63$, $y \le 5$, and $x \le 6$ all keep the side towards the origin, since the origin satisfies them.

Step 5: Combine all conditions.
The feasible region is bounded below-left by $x + y = 1$ and bounded above-right by the three "less than" lines, all in the first quadrant.

Step 6: Conclusion.
The graph that shows exactly this shaded region, cut off near the origin by $x + y = 1$, is the correct one. \[ \boxed{\text{Option 1}} \]