Problem 4 - Finite Mathematics and Applied Calculus

Question

Sketch the region that corresponds to the given inequalities, say whether the region is bounded or unbounded, and find the coordinates of all corner points (if any). $$ -x+2 y \geq 4 $$

Step-by-Step Solution

Answer

The region corresponding to the given inequality is graphed by shading the area above the line \(y = \frac{1}{2}x + 2\). The region is unbounded and there are no corner points.

Step 1: Graph the inequality

First, we need to graph the inequality. To do this, we can rewrite the inequality as an equation: $$ -x + 2y = 4 $$ Now, we can solve for \(y\): $$ y = \frac{1}{2}x + 2 $$ Next, we'll graph the line \(y = \frac{1}{2}x + 2\). The inequality states that the region of interest is above the line, so we'll shade the region above the line.

Step 2: Determine if the region is bounded or unbounded

Now, let's determine if the region is bounded or unbounded. Since there's only one inequality, there aren't any other lines that constrain the region. Hence, the region is unbounded.

Step 3: Identify corner points (if any)

Since the region is unbounded, there are no corner points for the region. So, the final answer is: - The region is graphed by shading the area above the line \(y = \frac{1}{2}x + 2\). - The region is unbounded. - There are no corner points.