Problem 42 - College Algebra with Modeling and Visualization

Question

Shade the region of feasible solutions for the following constraints. $$ \begin{array}{l} x+2 y \leq 8 \\ 2 x+y \geq 2 \\ x \geq 0, y \geq 0 \end{array} $$

Step-by-Step Solution

Answer

Graph the boundary lines, shade the correct sides, and identify the overlapping region in the first quadrant.

Step 1: Identify the Constraints

We have three inequalities that form the constraints: 1) \(x + 2y \leq 8\), 2) \(2x + y \geq 2\), and 3) both \(x \geq 0\) and \(y \geq 0\). The goal is to find the region where all these constraints are satisfied simultaneously.

Step 2: Graph the Boundary Lines

Transform each inequality into an equation by replacing inequalities with equal signs. This gives the lines: 1) \(x + 2y = 8\), 2) \(2x + y = 2\). Plot these lines on the Cartesian plane to find their intersection points and boundaries of the feasible region. The lines \(x = 0\) and \(y = 0\) are already boundaries for the non-negative constraints.

Step 3: Determine the Half-Planes

Determine which side of each boundary line satisfies the inequality. For \(x + 2y \leq 8\), shade the region below the line. For \(2x + y \geq 2\), shade the region above the line. Use a test point like \((0,0)\) to verify which side of each line to shade.

Step 4: Identify the Feasible Region

The feasible region is the area where all shaded regions overlap, considering the non-negative requirements \(x \geq 0\) and \(y \geq 0\). This gives the region in the first quadrant where all constraints are satisfied.

Key Concepts

System of Inequalities

A system of inequalities consists of multiple inequalities which need to be satisfied simultaneously. In our exercise, the system of inequalities is given by three specific conditions:

• \( x + 2y \leq 8 \)
• \( 2x + y \geq 2 \)
• \( x \geq 0 \) and \( y \geq 0 \)

Each inequality represents a constraint that defines a part of the plane where solutions are valid. The solution to the system is the set of all point coordinates \( (x, y) \) that satisfy all these inequalities at once. Essentially, you are looking for a common area on a graph that fits all these rules.

This concept is essential in linear programming, where finding the set of permissible solutions helps in optimizing functions like cost, profit, or resource allocation.

Feasible Region

In linear programming, the feasible region is a crucial concept that depicts all possible solutions for a given system of inequalities. For the current problem, to identify the feasible region, graphically represent the system on a coordinate plane.

The feasible region is the overlap where all inequalities are satisfied. Here, your boundaries are defined by:

• The line \( x + 2y = 8 \), showing where exactly \( x + 2y \) is equal to 8.
• The line \( 2x + y = 2 \), indicating where exactly \( 2x + y \) is equal to 2.
• The axes \( x = 0 \) and \( y = 0 \), representing the non-negative constraints.

The feasible region is in the first quadrant of the Cartesian plane and is the intersection of all these half-planes. This area is the possible region for the solutions, representing all valid \((x, y)\) combinations that meet the constraints provided.

Graphical Solution

Graphical solutions provide a visual method to understand and solve systems of inequalities, making it easier to identify feasible regions. To find the graphical solution, start by converting inequalities into equations to form boundaries on the graph.

For the constraints \( x + 2y \leq 8 \) and \( 2x + y \geq 2 \):

• Replace \(\leq\) and \(\geq\) with \(=\) to get the boundary lines: \( x + 2y = 8 \) and \( 2x + y = 2 \).
• Plot these lines on the graph using intercepts. For example, for \( x + 2y = 8 \), when \( x = 0 \), \( y = 4 \) and when \( y = 0 \), \( x = 8 \).
• Determine which side of the line to shade. Use a test point like \((0,0)\) to find which half-plane satisfies each inequality.

The intersection of these shaded regions represents the feasible region, showing all possible \((x, y)\) solutions that fulfill the system of inequalities. This method helps visualize constraints and potential solutions, bridging numerical and visual understanding.