Problem 48 - Finite Mathematics and Applied Calculus

Question

Explain the advantage of shading the region of points that do not satisfy the given inequalities. Illustrate with an example.

Step-by-Step Solution

Answer

Shading the region of points that do not satisfy the given inequalities provides better visualization and simpler intersections, ultimately making it easier to identify the set of solutions satisfying a system of inequalities. For example, consider the system of inequalities \(y ≥ x + 1\) and \(y ≤ -x + 3\). By shading the undesired regions (below y = x + 1 and above y = -x + 3), we can quickly identify the unshaded overlapping area, which represents the possible solutions for this system of inequalities.

Step 1: Understanding Inequalities and Shading Regions

Inequality problems involve finding the region of possible solutions in a coordinate plane. To do this, we first need to understand which side of the inequality boundary should be shaded to represent the solutions. We can either shade the desired region or the undesired region. Shading the undesired region can sometimes be more convenient as it often provides better clarity and understanding of the solution.

Step 2: Advantages of Shading the Undesired Region

Shading the undesired region has the following advantages: 1. Better visualization: Shading the undesired region helps students to easily see which side of the inequality boundary is not the solution. This makes it clear where the solutions are and where they are not. 2. Simpler intersections: When working with multiple inequalities, shading the undesired regions can make it easier to visualize the intersections of the inequalities. By identifying the unshaded region, one can quickly determine the overlapping area.

Step 3: Example of Shading the Undesired Region

As an example, let's consider the following system of inequalities: \(y ≥ x + 1\) \(y ≤ -x + 3\) First, let's graph these inequalities. 1. For the inequality \(y ≥ x + 1\): Draw a solid boundary line along y = x + 1 (since it's a '≥' sign, we need a solid line). Test a point below the line, say (0, 0), to see if the inequality holds. Since 0 ≥ 1 is not true, we shade the region below the line. 2. For the inequality \(y ≤ -x + 3\): Draw a solid boundary line along y = -x + 3 (since it's a '≤' sign, we need a solid line). Test a point above the line, say (0, 4), to see if the inequality holds. Since 4 ≤ 3 is not true, we shade the region above the line. Now, the unshaded overlapping area represents possible solutions for this system of inequalities. In this example, by shading the undesired regions, we can quickly identify the unshaded overlapping region, which represents the set of solutions satisfying the given system of inequalities. This illustrates the advantage of shading the region of points that do not satisfy the given inequalities.