Step 1: Understanding the Concept:
The problem asks to identify the system of linear inequalities that corresponds to a given shaded region on a graph.
The boundaries of the region are defined by equations. The shaded area represents the inequalities.
We can determine the correct inequality signs by picking a test point clearly inside the shaded region and plugging it into the options. Step 2: Key Formula or Approach:
1. Identify the boundary lines from the graph.
2. Choose a test point $(x_0, y_0)$ inside the shaded region.
3. Substitute the coordinates of the test point into the given inequalities. The set of inequalities that are all true for this point is the correct answer. Step 3: Detailed Explanation:
From the figure, the boundary lines are given as:
Line 1: $x - y = 0$ (which is $y = x$)
Line 2: $x + y = 0$ (which is $y = -x$)
The shaded region lies to the right of the y-axis, between these two lines.
Let's select a simple test point clearly inside the shaded region, for example, a point on the positive x-axis.
Let's choose the test point $(1, 0)$.
Now, test this point $(x=1, y=0)$ against each option:
(A) $x - y \ge 0 \Rightarrow 1 - 0 \ge 0 \Rightarrow 1 \ge 0$ (True)
$x + y \ge 0 \Rightarrow 1 + 0 \ge 0 \Rightarrow 1 \ge 0$ (True)
Both conditions are satisfied. This is the likely answer.
(B) $x - y \le 0 \Rightarrow 1 - 0 \le 0 \Rightarrow 1 \le 0$ (False)
No need to check the second inequality.
(C) $x - y \ge 0 \Rightarrow 1 - 0 \ge 0 \Rightarrow 1 \ge 0$ (True)
$x + y \le 0 \Rightarrow 1 + 0 \le 0 \Rightarrow 1 \le 0$ (False)
(D) $x - y \le 0 \Rightarrow 1 - 0 \le 0 \Rightarrow 1 \le 0$ (False)
Only option (A) holds true for our test point.
Alternatively, we can analyze geometrically:
The region is below the line $y = x$, which translates to $y \le x$ or $x - y \ge 0$.
The region is above the line $y = -x$, which translates to $y \ge -x$ or $x + y \ge 0$.
Combining these confirms the result. Step 4: Final Answer:
The solution set is $x - y \ge 0, x + y \ge 0$.
