The shaded figure given below is the solution set for the linear inequations. Choose the correct option.
Show Hint
When graphing inequalities, plugging in the origin $(0,0)$ is the fastest way to check shading direction! If $0 \le \text{positive number}$, shade towards the origin. If $0 \ge \text{positive number}$, shade away from it!
Step 1: Understanding the Question:
We are presented with a bounded feasible region defined by a system of linear inequalities and must identify the correct set of inequalities that mathematically describes this shaded area.
Step 2: Key Formula or Approach:
For each boundary line $ax + by = c$, we can determine the correct inequality sign ($\le$ or $\ge$) by selecting a test point clearly inside the shaded region (often the origin $(0,0)$ if it's not on the line, or another convenient point) and checking which inequality holds true.
Step 3: Detailed Explanation:
Based on the provided standard solution for this exam figure:
1. The region lies strictly in the first quadrant, establishing the non-negativity constraints: $x \ge 0$, $y \ge 0$.
2. The boundary line $3x + 4y = 18$ bounds the region from above (towards the origin). Testing $(0,0)$ gives $0 \le 18$ (True), so the inequality is $3x + 4y \le 18$.
3. The boundary line $2x + 3y = 3$ bounds the region from below (away from the origin). Testing $(0,0)$ gives $0 \ge 3$ (False), so the region must satisfy $2x + 3y \ge 3$.
4. For the line $x - 6y = 3$, the origin $(0,0)$ is on the valid side since $0 - 0 \le 3$ is true, giving $x - 6y \le 3$.
5. For the line $-7x + 14y = 14$ (which simplifies to $-x + 2y = 2$), testing the origin $(0,0)$ gives $0 \le 14$, which is true for the shaded area below it, resulting in $-7x + 14y \le 14$.
Compiling all these verified conditions precisely matches the set in option (C).
Step 4: Final Answer:
The correct system of linear inequalities is $3x + 4y \le 18$; $x - 6y \le 3$; $2x + 3y \ge 3$; $-7x + 14y \le 14$; $x \ge 0$; $y \ge 0$, which corresponds to option (C).
