Problem 38 - Precalculus: Mathematics for Calculus

Question

Graph the solution set of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{aligned} x &>2 \\ y &<12 \\ 2 x-4 y &>8 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The vertices are \((2, -1)\) and \((28, 12)\); the solution set is unbounded.

Step 1: Graph the first inequality

The first inequality is \( x > 2 \). On a graph, draw a vertical line at \( x = 2 \). Since it is a strict inequality (not \( x \geq 2 \)), use a dashed line. Shade the region to the right of this line since you want \( x \) to be greater than 2.

Step 2: Graph the second inequality

The second inequality is \( y < 12 \). Draw a horizontal dashed line at \( y = 12 \). Shade the area below this line to show where \( y \) is less than 12.

Step 3: Graph the third inequality

The third inequality is \( 2x - 4y > 8 \). Rewrite it in slope-intercept form: \( y < \frac{1}{2}x - 2 \). Draw a dashed line for \( y = \frac{1}{2}x - 2 \) and shade below this line.

Step 4: Determine the feasible region

The feasible region is where all shaded areas overlap: to the right of \( x = 2 \), below \( y = 12 \), and below \( y = \frac{1}{2}x - 2 \). The region forms a triangular shape on the graph.

Step 5: Find intersection points (vertices)

Find where the lines intersect: 1. \( x = 2 \) and \( y = \frac{1}{2}(2) - 2 = -1 \) gives \((2, -1)\).2. \( y = 12 \) and \( y = \frac{1}{2}x - 2 \). Setting \( \frac{1}{2}x - 2 = 12 \) gives \( x = 28 \), resulting in the point \((28, 12)\). 3. None for \( x = 2 \) and \( y = 12 \) since \( x > 2 \) and \( y < 12 \).

Step 6: Check if the solution is bounded

The solution set is attached to the lines and rays at the vertices \( (2, -1) \) and \( (28, 12) \) but is open to the right. So, it's not bounded since it can extend indefinitely to the right and downwards beyond the intersection with \( y < \frac{1}{2}x - 2 \).

Key Concepts

Solution Set

In the context of graphing inequalities, the term "solution set" refers to all the possible solutions that satisfy a given system of inequalities. In simpler terms, it's the region on the graph where all the conditions of the inequalities overlap. For instance, when you consider the inequalities given in the exercise, the solution set is the portion of the graph where the conditions \( x > 2 \), \( y < 12 \), and \( 2x - 4y > 8 \) hold true simultaneously. When graphing, each inequality is represented by a line, typically dashed for strict inequalities (indicating that points on the line are not included). The solution set is visually obvious after shading the appropriate sides of these lines.

• The shaded areas help show where solutions exist.
• The solution set can be bounded or unbounded, depending on whether it extends infinitely or not.

Feasible Region

The "feasible region" is a graphical representation of all possible solutions to a system of inequalities. It's the area where all shaded regions from each inequality overlap. Within our exercise, the feasible region is the triangular area that forms when you combine the three inequalities: \( x > 2 \), \( y < 12 \), and \( 2x - 4y > 8 \).Here's a simplified view:

• The feasible region lies to the right of the vertical line at \( x = 2 \).
• It resides below the horizontal line at \( y = 12 \).
• It is beneath the slanted line represented by \( y = \frac{1}{2}x - 2 \).

The feasible region is crucial for understanding the range of possible solutions that satisfy all inequalities. It can be bounded by lines that close off the area or unbounded if the region opens indefinitely in a direction, as in our example.

System of Inequalities

A system of inequalities involves multiple inequalities that are considered together to understand what solutions can satisfy all of them simultaneously. Unlike a system of equations where you find specific intersecting points, a system of inequalities typically results in a broader solution—a region on the graph. For learning purposes:

• Each inequality forms a boundary line on the graph.
• Dashed lines are used for strict inequalities; solid lines for inclusive ones.
• The solution satisfies all inequalities within the system, forming an overlap or feasible region.

Understanding a system of inequalities is key for applications that require maximizing or minimizing within certain constraints (such as optimization problems). This concept helps identify which areas of a graph align with provided conditions, thus defining the feasible or viable solutions for a real-world problem.