Question

Graph each system of linear inequalities. State whether the graph is bounded or unbounded, and label the corner points. $$\left\\{\begin{array}{r}x \geq 0 \\\y \geq 0 \\\x+y \geq 2 \\\x+y \leq 8 \\\x+2 y \geq 1\end{array}\right.$$

Short answer

The feasible region is bounded with corner points: (2, 0), (0, 2), (8, 0), (1,0), (2,0.4), (6, 1).

Step-by-step solution

Identify and Graph Each Inequality

Start by graphing each inequality on the same coordinate plane: 1. For the inequality \(x \geq 0\), shade the region to the right of the y-axis.2. For \(y \geq 0\), shade the region above the x-axis.3. For \(x + y \geq 2\), shade the region above the line \(x + y = 2\). This line intersects the axes at points \((2, 0)\) and \((0, 2)\).4. For \(x + y \leq 8\), shade the region below the line \(x + y = 8\). This line intersects the axes at points \((8, 0)\) and \((0, 8)\).5. For \(x + 2y \geq 1\), shade the region above the line \(x + 2y = 1\). This line intersects the axes at points \((1, 0)\) and \((0, 0.5)\).

Determine the Feasible Region

The feasible region is where all the shaded areas overlap. Specifically, it is the intersection of the half-planes defined by the inequalities. Begin looking for the intersection points of all the considered lines, as these points will help form the region's borders.

Find the Corner Points

Find the corner points by solving the system of equations obtained by the intersection of the lines: 1. \((2, 0)\) intersection of \(x + y = 2\) and \(x = 0\).2. \((0, 2)\) intersection of \(x + y = 2\) and \(y = 0\).3. \((8, 0)\) intersection of \(x + y = 8\) and \(x = 0\).4. \((1, 0)\) intersection of \(x + 2y = 1\) within the boundary.5. \((0.2, 0.4)\) found through simultanous solving \(x + 2y = 1\) and \(x + y = 2\).6. \((6, 1)\) solving \(x + y = 8\) and \(y = 0\).

Determine if the Region is Bounded or Unbounded

Check if the feasible region is enclosed. Since the inequalities \(x \geq 0\) and \(y \geq 0\) constrain the region to the first quadrant and other boundaries are formed by lines like \(x + y \geq 2\), \(x + y \leq 8\) and \(x + 2y \geq 1\), which result in no directions extending to infinity, the region is bounded.

Label the Corner Points

List and label the corner points of the feasible region: \((2, 0)\), \((0,2)\), \((8,0)\), \((1.0)\), \((0.2,0.4)\) and \((6, 1)\). These points are found at the intersections of the boundary lines.

Key concepts

graphing inequalities

When graphing inequalities, the first step is to visualize each inequality on the same coordinate plane. Each inequality represents a half-plane. Here's how you can graph them:

• For the inequality \(x \geq 0\), shade the region to the right of the y-axis.
• For \(y \geq 0\), shade the region above the x-axis.
• For \(x + y \geq 2\), shade the region above the line \(x + y = 2\). This line intersects the x-axis at \((2, 0)\) and the y-axis at \((0, 2)\).
• For \(x + y \leq 8\), shade the region below the line \(x + y = 8\). This line intersects the x-axis at \((8, 0)\) and the y-axis at \((0, 8)\).
• For \(x + 2y \geq 1\), shade the region above the line \(x + 2y = 1\). This line intersects the x-axis at \((1, 0)\) and the y-axis at \((0, 0.5)\).

To ensure clarity, always use different lines or colors for each inequality and consistently shade their respective areas. This overlapping of shaded regions will help you identify the feasible region.

feasible region

The feasible region is the area on the graph where all the shaded areas from each inequality overlap. This region represents all possible solutions that satisfy all the given inequalities simultaneously.

To identify the feasible region:
• Graph each inequality as described previously.
• Look for the common overlapping area of all shaded regions.

In this example, after graphing the given inequalities, the feasible region is bounded by the lines \(x + y = 2\), \(x + y = 8\), and \(x + 2y = 1\), all within the first quadrant. The region where all these conditions meet is the feasible region.

Inside the feasible region, any point will represent a possible solution of the set of inequalities. It's important to carefully graph and shade the inequalities to correctly identify the feasible region.

bounded and unbounded regions

A region is termed 'bounded' if it is enclosed and all its boundaries do not extend to infinity. Conversely, an 'unbounded' region extends indefinitely in at least one direction.

To determine whether a feasible region is bounded or unbounded, observe the intersection lines and the constraints on the inequalities within the graph:
• Check if the feasible region lies entirely within certain boundary lines or extends beyond them.

In this exercise, the feasible region is constrained within the first quadrant due to the inequalities \(x \geq 0\) and \(y \geq 0\). Other boundaries such as \(x + y \geq 2\), \(x + y \leq 8\) and \(x + 2y \geq 1\) further restrain the region, preventing it from extending to infinity.

Consequently, the feasible region here is bounded because all of its edges are defined by the intersection of equalities, leading to a closed shape within the coordinate plane.

intersection points

Intersection points play a crucial role as they define the corners of the feasible region. These points are found at the points where the boundary lines of the inequalities intersect.

To identify intersection points:
• Solve equations obtained by the intersection of the boundary lines.

From the exercise, important intersection points are:
• \((2, 0)\) from the intersection of \(x + y = 2\) and the x-axis.
• \((0, 2)\) from the intersection of \(x + y = 2\) and the y-axis.
• \((8, 0)\) from the intersection of \(x + y = 8\) and the x-axis.
• \((1, 0)\) from the intersection of \(x + 2y = 1\) within the boundary.
• \((0.2, 0.4)\) by solving \(x + 2y = 1\) and \(x + y = 2\).
• \((6, 1)\) by solving \(x + y = 8\) and \(y = 0\).

The feasible region of the system of inequalities is thus enclosed within these corner points. It's vital to accurately calculate and plot these points to correctly represent the feasible region.