Question

Graph each system of linear inequalities. State whether the graph is bounded or unbounded, and label the corner points. $$\left\\{\begin{array}{r}x \geq 0 \\\y \geq 0 \\\2 x+y \leq 6 \\\x+2 y \leq 6\end{array}\right.$$

Short answer

The graph is bounded and the corner points are (0,0), (3,0), (0,3), and (2,2).

Step-by-step solution

- Graph the boundary lines

First, write each inequality as an equation: 1. \(x = 0\)2. \(y = 0\)3. \(2x + y = 6\)4. \(x + 2y = 6\). Graph these lines on the coordinate plane.

- Identify the feasible region

The system of inequalities includes the lines (boundaries). Use test points to determine the side of the line where the inequalities are true. For example, substitute a point not on the line into each inequality.

- Shade the feasible region

Shade the region that satisfies all inequalities: x ≥ 0 lies to the right of the y-axis, y ≥ 0 lies above the x-axis, 2x + y ≤ 6 lies below the line, and x + 2y ≤ 6 lies below that line. The intersection of these shaded areas is the feasible region.

- Determine corner points

Find the intersection points of the boundary lines: 1. Intersection of \(x = 0\) and \(y = 0\) is \((0,0)\)2. Intersection of \(2x + y = 6\) and \(y = 0\) is \((3,0)\)3. Intersection of \(x = 0\) and \(x + 2y = 6\) is \((0,3)\)4. Intersection of \(2x + y = 6\) and \(x + 2y = 6\) solve for \((2, 2)\)

- Check if the region is bounded or unbounded

A region is bounded if it forms a closed area. In this case, the region is enclosed within all the constraints, so it is bounded.

- Label the corner points

The corner points of the feasible region are (0,0), (3,0), (0,3), and (2,2). Label these points on the graph.

Key concepts

Bounded and Unbounded Regions

When graphing systems of linear inequalities, it is important to determine if the resulting region is bounded or unbounded. This tells us if the solution region is confined within a specific area or if it extends infinitely in some direction.
To determine this, we first graph the boundary lines defined by the equations. After graphing, we analyze where the lines intersect and form a region. If this intersection creates a closed figure, such as a triangle or polygon, then the region is bounded.
For example, in the given exercise, the intersection of the inequalities forms a quadrilateral, meaning the region is enclosed within the boundary lines. This is an example of a bounded region.
On the other hand, if the lines do not form a closed shape and the shaded region extends indefinitely, then the region is unbounded. Bounded regions are important in optimization problems because the solutions are confined to a specific area, making it easier to identify maximum or minimum values.

Feasible Region

The feasible region is the set of points that satisfies all the given linear inequalities simultaneously. In simple terms, it is where the shaded areas overlap when you graph each inequality on the same coordinate plane.
To find this region, follow these steps:
1. Graph each inequality separately. Convert each inequality to an equation to plot its boundary line. For example, the inequality \(2x + y \leq 6\) becomes \(2x + y = 6\).
2. Determine which side of the boundary line satisfies the inequality. You can do this by selecting a test point (usually (0,0) if it's not on the line) and checking whether it satisfies the inequality. If it does, shade that side of the boundary line.
3. Repeat the process for all inequalities. The overlapping (shaded) region is the feasible region. In our exercise, the overlapping shaded region is confined within the boundaries created by the inequalities, forming a quadrilateral shape.
The feasible region is crucial in solving linear programming problems, as it represents all the potential solutions to the system of inequalities.

Corner Points

Corner points, also known as vertices, are where the boundary lines of the feasible region intersect. These points are significant because they are often used to find optimal solutions in optimization problems.
To identify corner points, follow these steps:
1. Solve the equations derived from the boundary lines to find their intersection points. For example, the boundary lines \(2x + y = 6\) and \(x + 2y = 6\) intersect at \( (2, 2)\).
2. Consider all possible intersections from the given inequalities. In our problem, the inequalities \(x \geq 0\) and \(y \geq 0\) intersect at \( (0, 0)\), and \(x + 2y = 6\) intersects the x-axis at \( (3, 0)\).
Identifying these corner points gives us the vertices \( (0, 0)\), \( (3, 0)\), \( (0, 3)\), and \( (2, 2)\).
These points help in evaluating the objective function in linear programming problems to find maximum or minimum values within the feasible region.