Problem 59 - Precalculus: Mathematics for Calculus

Question

Graph the system of inequalities, label the vertices, and determine whether the region is bounded or unbounded. $$\left\\{\begin{array}{c} x+2 y \leq 14 \\ 3 x-y \geq 0 \\ x-y \leq 2 \end{array}\right.$$

Step-by-Step Solution

Answer

The region is bounded with vertices at the intersection points of the inequalities' lines.

Step 1: Graph Each Inequality

Start by converting each inequality into an equation and graph it. For the first inequality, convert \(x + 2y = 14\) into a line. For the second inequality, convert \(3x - y = 0\) into a line. For the third inequality, convert \(x - y = 2\) into a line.

Step 2: Determine the Feasible Region

Shade the appropriate side of each line according to the inequality sign. For \(x + 2y \leq 14\), shade below the line. For \(3x - y \geq 0\), shade above the line. For \(x - y \leq 2\), shade below the line. The area where all these shaded regions overlap is the feasible region.

Step 3: Label the Vertices

Find the intersection points of the lines to label the vertices of the feasible region. Solve each pair of equations to find their intersection points. These intersections are the vertices of the region.

Step 4: Analyze Boundedness

Observe the feasible region on the graph. If it forms a closed shape with all vertices connecting to provide a restricted area, then the region is bounded. Otherwise, if it extends infinitely in any direction, it is unbounded.

Key Concepts

Linear inequalities

Linear inequalities are like equations, but they define a range of solutions rather than a single line or point. Understanding them helps us solve problems involving conditions where multiple outcomes are possible rather than a specific one.
In a linear inequality, you might see symbols like \( \leq \), \( \geq \), \( < \), or \( > \). Each of these indicates a different range of solutions. For example:

• \( x + 2y \leq 14 \) means all points below or on the line.
• \( 3x - y \geq 0 \) means all points above or on the line.
• \( x - y \leq 2 \) indicates all points below or on this line as well.

Graphing these inequalities involves plotting the line as if it were an equation, then shading the area where the inequality is true.
This visual method works well for demonstrating where multiple conditions overlap.

Feasible region

The feasible region is where the magic of graphing inequalities happens. It represents the solution set that meets all the conditions of a system of inequalities.
To find the feasible region, we shade the parts of the graph that satisfy each inequality condition. The overlapping shaded regions form the feasible region. The vertices of these regions—points where the boundaries intersect—are crucial. By solving pairs of equations for each line intersection, we can pinpoint these vertices.
This area can tell us things like:

• If there's a viable solution that meets all criteria.
• Where optimal solutions might lie.

Analyzing feasible regions is beneficial for optimization and decision-making problems in numerous fields, such as economics and engineering.

System of inequalities

A system of inequalities involves multiple inequality conditions that must be satisfied simultaneously. Each inequality contributes to limiting or confining the potential solutions into a specific region or set of points.
Graphically, these are represented by different linear plots and shaded sections. The system is resolved step-by-step:

• Convert each inequality into a linear equation for easy graph plotting.
• Shade the area corresponding to each inequality.
• The intersection of these shaded areas is the feasible region, offering a visual representation of solutions.

Understanding systems of inequalities is essential when dealing with situations where a balance of conditions is required. This is common in areas like resource allocation, budgeting, and planning, where multiple constraints are frequent and need to be visualized and managed.