Question: Write a system of inequalities for the graph. Select the correct choice below and, fill in the…

Write a system of inequalities for the graph.

Select the correct choice below and, fill in the answer box to complete your choice.

A. \(\begin{cases} y > \boxed{} \\ y \geq \boxed{} \end{cases}\)

B. \(\begin{cases} y < \boxed{} \\ y \geq \boxed{} \end{cases}\)

C. \(\begin{cases} y > \boxed{} \\ y < \boxed{} \end{cases}\)

D. \(\begin{cases} y < \boxed{} \\ y \leq \boxed{} \end{cases}\)

E. \(\begin{cases} y \geq \boxed{} \\ y \leq \boxed{} \end{cases}\)

F. \(\begin{cases} y > \boxed{} \\ y \leq \boxed{} \end{cases}\)

[Graph of a shaded region on a coordinate plane]

Solution

To find the system of inequalities for the given shaded region on the graph, let’s first determine the equations of the boundary lines. 1. Identify the Boundary Lines The region is bound by two non-parallel lines. Let’s find their equations using two points they pass through. 2. Equation of the First Line - Points: \((0, 4)\) and \((4, 0)\) Find the slope \( m \): \[ m = \frac{0 - 4}{4 - 0} = -1 \] Use point-slope form: \[ y - y_1 = m(x - x_1) \] Using point \((0, 4)\): \[ y - 4 = -1(x - 0) \\ y = -x + 4 \] 3. Equation of the Second Line - Points: \((-2, -2)\) and \((2, 2)\) Find the slope \( m \): \[ m = \frac{2 - (-2)}{2 - (-2)} = \frac{4}{4} = 1 \] Use point-slope form: \[ y - y_1 = m(x - x_1) \] Using point \((-2, -2)\): \[ y + 2 = 1(x + 2) \\ y = x \] 4. Writing the System of Inequalities For shading above \(y = -x + 4\): \[ y \leq -x + 4 \] For shading below \(y = x\): \[ y \geq x \] Therefore, the system of inequalities is: \[ \begin{cases} y \leq -x + 4 & \\ y \geq x & \end{cases} \] The correct choice from the options would be selection that includes these inequalities.