Problem 10 - 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). $$ \frac{x}{3}+\frac{2 y}{3} \geq 2 $$

Step-by-Step Solution

Answer

First, we convert the inequality, \(\frac{x}{3}+\frac{2 y}{3} \geq 2\), into an equation and find the x- and y-intercepts for plotting the line. The x-intercept is (6,0) and the y-intercept is (0,3). We then plot the line and identify the region satisfying the inequality by choosing a test point, like the origin (0,0). Since the test point doesn't satisfy the inequality, the region is on the opposite side of the line from the origin. After shading the region, we can see that it is unbounded and has no corner points.

Step 1: Convert the inequality into equation

Firstly, replace the inequality symbol with an equal sign to obtain the equation of the line: \[ \frac{x}{3}+\frac{2 y}{3} = 2 \]

Step 2: Find x and y intercepts

Determine the x- and y-intercepts to help draw the line. The x-intercept occurs when y=0, and the y-intercept occurs when x=0. For x-intercept, set y=0: \[ \frac{x}{3}+\frac{2(0)}{3} = 2 \Rightarrow x = 6 \] For y-intercept, set x=0: \[ \frac{0}{3}+\frac{2 y}{3} = 2 \Rightarrow y = 3 \]

Step 3: Plot the line

Using the x-intercept (6,0) and y-intercept (0,3), plot the line on the coordinate plane.

Step 4: Identify the region satisfying the inequality

The inequality states that the region is greater than or equal to 2: \[ \frac{x}{3}+\frac{2 y}{3} \geq 2 \] We need to identify which side of the line contains the points that satisfy this inequality. A common method is to pick a test point not on the line and see if it meets the inequality. A good test point to use is the origin (0,0) because it often makes calculations easier. Substitute the test point (0,0) into the inequality: \[ \frac{0}{3}+\frac{2(0)}{3} \geq 2 \] This simplifies to 0 ≥ 2, which is false. Therefore, the region that satisfies the inequality is on the opposite side of the line from the origin. Shade this region in the coordinate plane.

Step 5: Determine if the region is bounded or unbounded

Since the shaded region extends infinitely, the region is unbounded.

Step 6: Find the corner points (if any)

Our region is unbounded, and there are no corner points or vertices as the shaded region continues indefinitely in one direction. In conclusion, the region corresponding to the given inequality is unbounded and has no corner points.