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: Fix the obvious constraints first.
The shaded set sits in the first quadrant, so we must have $x \ge 0$ and $y \ge 0$. That already rules out nothing yet, so we check the slanted lines.
Step 2: Test the origin against each line.
For a line $ax+by=c$, plug in $(0,0)$ and keep the sign that holds for the shaded side. For $3x+4y=18$: $0 \le 18$ is true, so $3x+4y \le 18$.
Step 3: Do the same for the rest.
For $2x+3y=3$ the origin gives $0 \ge 3$ which is false, so the region needs $2x+3y \ge 3$. For $x-6y=3$ the origin gives $0 \le 3$, true, so $x-6y \le 3$. For $-7x+14y=14$ the origin gives $0 \le 14$, true, so $-7x+14y \le 14$.
Step 4: Match the set.
These four signs with $x \ge 0,\ y \ge 0$ fit option 3 exactly.
\[ \boxed{3x+4y \le 18,\ x-6y \le 3,\ 2x+3y \ge 3,\ -7x+14y \le 14} \]
