Question

Graph each system of linear inequalities. State whether the graph is bounded or unbounded, and label the corner points. $$\left\\{\begin{array}{r}x \geq 0 \\\y \geq 0 \\\x+y \geq 2 \\\x+y \leq 8 \\\2 x+y \leq 10\end{array}\right.$$

Short answer

The graph is bounded with corner points at (0, 2), (2, 0), (2, 6), and (5, 0).

Step-by-step solution

- Identify each inequality

List each inequality given in the exercise:1. \( x \geq 0 \)2. \( y \geq 0 \)3. \( x + y \geq 2 \)4. \( x + y \leq 8 \)5. \( 2x + y \leq 10 \)

- Plot the boundary lines

First, plot the lines corresponding to the equalities for each inequality on the Cartesian plane:1. \( x = 0 \)2. \( y = 0 \)3. \( x + y = 2 \)4. \( x + y = 8 \)5. \( 2x + y = 10 \)

- Determine the shaded regions

For each inequality, determine which side of the boundary line to shade:1. \( x \geq 0 \) (Shade to the right of the y-axis)2. \( y \geq 0 \) (Shade above the x-axis)3. \( x + y \geq 2 \) (Shade above the line \( x + y = 2 \))4. \( x + y \leq 8 \) (Shade below the line \( x + y = 8 \))5. \( 2x + y \leq 10 \) (Shade below the line \( 2x + y = 10 \))

- Find the corner points

Calculate the intersection points of the boundary lines to determine the corner points of the feasible region:1. Intersection of \( x = 0 \) and \( x + y = 2 \): (0, 2)2. Intersection of \( y = 0 \) and \( x + y = 2 \): (2, 0)3. Intersection of \( x + y = 8 \) and \( 2x + y = 10 \): (2, 6)4. Intersection of \( 2x + y = 10 \) and \( y = 0 \): (5, 0)

- Determine bounded or unbounded

Observe the feasible region formed by the shaded areas. Since the region is enclosed on all sides within the first quadrant, the graph is bounded.

- Label the corner points

Mark the corner points on the graph: (0, 2), (2, 0), (2, 6), and (5, 0)

Key concepts

Graphing Linear Inequalities

To start with graphing linear inequalities, the first step is to identify each inequality from the given system. The inequalities split the Cartesian plane into different regions. To graph these inequalities:

• Plot the boundary lines: For each inequality, convert it into an equation by replacing the inequality sign with an equals sign. This gives you the boundary lines.
• Determine the shaded regions: Basing it on each inequality, determine which side of the boundary line the region should be shaded.

For example, for the inequality system provided in the solution:Remember, shading is essential because it specifies which part of the plane includes the solutions to the inequality.

Feasible Region

The feasible region is the area of the graph where all the inequalities overlap. This region contains all the possible solutions that satisfy all the given inequalities at once. To find the feasible region:

• Shade the regions that satisfy each inequality.
• The area where all shaded regions overlap is the feasible region.

This region is found in the first quadrant as defined by the inequalities in the exercise. It’s important to properly identify this region for accurate solutions.

Bounded and Unbounded Regions

The feasible region can be either bounded or unbounded. A bounded region is closed and contained within some limits, while an unbounded region extends infinitely in at least one direction. To determine if a region is bounded:

• Check if the feasible region forms a closed shape.
• If the shaded area is completely enclosed, it is bounded.

For the provided exercise, the graph is bounded due to the feasible region being enclosed within the first quadrant and limited by the given lines. Recognizing whether the region is bounded or unbounded helps in understanding the limits of possible solutions.

Corner Points

Corner points, or vertices, are where the boundary lines intersect within the feasible region. These points are essential because they can potentially be the max or min solutions for optimizing linear functions. To find the corner points:

• Calculate the intersection points of the boundary lines.
• List these intersection points as they represent the vertices of the feasible region.

For the example in the solution, the corner points are (0, 2), (2, 0), (2, 6), and (5, 0). These points help in graphical solution analyses and optimization problems.