Problem 2 - 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). $$ 4 x-y \leq 12 $$

Step-by-Step Solution

Answer

The region corresponding to the inequality \(4x-y \leq 12\) is unbounded and has only one boundary line with the slope-intercept equation \(y=4x-12\). There are no corner points in the region.

Step 1: Rearrange the equation to slope-intercept form

To rearrange the equation, we first isolate y: \(y=4x-12\) Now, we can see that the slope of our boundary line, m, is 4 and the y-intercept, b, is -12. #Step 2: Determine which side of the line contains the region that satisfies the inequality# To figure out which side of the line contains the region that satisfies the inequality, we can choose a test point (not on the line) and check whether the inequality holds true for that point. If it does, then the region that contains this test point satisfies the inequality.

Step 2: Choose a test point and plug it into the inequality

A common test point to use is the origin (0,0) if it does not lie on the boundary line. In this case, the origin is not on the line, so we can use it as our test point. Plug the origin into the inequality: \(4(0) - (0) \leq 12\) \(0 \leq 12\) Since the inequality holds true for the test point (0,0), the region that contains this point is the region that satisfies the inequality. #Step 3: Sketch the region on the coordinate plane# Now we need to sketch the boundary line and shade the region that satisfies the inequality, based on our test point.

Step 3: Sketch the line and shade the region

First, plot the y-intercept (-12) on the y-axis. Then, use the slope of 4 to plot additional points on the line. Since the slope is positive, go up 4 units and to the right 1 unit for each additional point. Now, draw the line connecting the plotted points. Since the inequality is less than or equal to (≤), it means that the line also belongs to the solution, and we will use a solid line to indicate this. Finally, shade the region that contains the origin to show the region that satisfies the inequality. #Step 4: Analyze the sketch to determine if the region is bounded or unbounded, and to find the coordinates of any corner points# By examining our sketch, we can make our conclusions about the region and corner points.

Step 4: Determine if the region is bounded or unbounded

Since the shaded region continues indefinitely to the right, the region is unbounded.

Step 5: Find the coordinates of all corner points

As there is only one boundary line, there are no corner points in our region. In conclusion, we have sketched the region corresponding to the inequality \(4x-y \leq 12\), determined that it is unbounded, and found that there are no corner points.