Problem 40 - Precalculus Mathematics for Calculus

Question

Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ y & \leq 4 \\ 2 x+y & \leq 8 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution set forms a trapezoid with vertices at \((0,0), (0,4), (2,4), (4,0)\). It is bounded.

Step 1: Plot the Line for the Inequality

Start by considering the line from the equality of the inequality: \(2x + y = 8\). Rearrange it to solve for \(y\), giving \(y = -2x + 8\). Plot this line by identifying two points. For instance, when \(x = 0\), \(y = 8\) and when \(y = 0\), \(x = 4\). Connect these points to form the line. Indicate that the inequality \(2x + y \leq 8\) includes the area below this line.

Step 2: Plot y = 4

Next, plot the line for the inequality \(y \leq 4\). This line is horizontal because \(y\) does not depend on \(x\), and it passes through the point \((0, 4)\). Since the inequality is \(y \leq 4\), shade the region below this line.

Step 3: Consider Non-negativity Constraints

The inequalities \(x \geq 0\) and \(y \geq 0\) restrict the solution to the first quadrant. This means that only the region above the x-axis and to the right of the y-axis is valid.

Step 4: Identify the Intersection Points

Determine the intersection points where the lines cross within the feasible region. The equations \(2x + y = 8\) and \(y = 4\) intersect at \((2, 4)\). Check other intersections: \(2x + y = 8\) intersects the x-axis at \((4, 0)\), and \(y = 4\) intersects the y-axis at \((0, 4)\). Since all lines start or intersect at \((0,0)\), which is also on the x and y axes, include \((0, 0)\) as a vertex.

Step 5: Determine the Bounded Region

Check the region that is shaded based on all inequalities. The vertices formed by \((0,0)\), \((0,4)\), \((2,4)\), and \((4,0)\) create a trapezoid which is closed. This indicates that the solution set is bounded.

Key Concepts

Graphing Inequalities

Graphing inequalities involves plotting lines or curves on a coordinate plane to represent the solution of an inequality. However, instead of just a line, you need to shade a region that satisfies the inequality condition.

• Identify equations from inequalities, such as converting the inequality to an equation. For example, change \(2x + y \leq 8\) to \(2x + y = 8\).
• Draw the line corresponding to the equation. For instance, when \(x = 0\), \(y = 8\) and vice versa when \(y = 0\), \(x = 4\).
• Determine which region to shade by testing a point not on the line. If it satisfies the inequality, shade that side.

The plot for inequalities is critical because it visually depicts all possible solutions within a defined region on the graph.

Bounded Solution Set

A bounded solution set refers to a region on the graph where all solutions to the system of inequalities are confined within a closed and finite area. This means there's no way for these solutions to extend indefinitely in any direction on the plane.

• Examine if the shaded region forms a closed shape. For example, in the system given, vertices like \((0,0), (0,4), (2,4),\) and \((4,0)\) form a trapezoid.
• Determine whether this polygon has borders that restrict the solutions, ensuring they don't extend to infinity.

If the vertices connect to form a distinct and enclosed shape, the solution set is bounded, implying it has a specific boundary.

Intersection Points

Intersection points are where two lines or curves cross on the graph, presenting critical solutions for the system of inequalities. Identifying these points can provide important vertices for the solution area.

• Solve simultaneously by equating two line equations. For example, solve \(2x+y=8\) and \(y=4\) to find \((2, 4)\).
• Check others like where \(2x+y=8\) intersects at \((4,0)\) and where \(y=4\) meets the y-axis at \((0,4)\).
• Check boundaries against the axes, such as \((0,0)\) which is both on the x and y axes.

These intersection points, once confirmed in the feasible region, are often vertices that bound the solution set.

Feasible Region

The feasible region is the common overlapping area of all shaded regions from each inequality on the graph. It is the portion that satisfies all constraints placed by the system of inequalities.

• Start by plotting each inequality and shading the related area. Consider the constraints \(x \geq 0\), \(y \geq 0\), \(y \leq 4\), and \(2x+y \leq 8\). Each adds a layer of shading.
• Identify where these shaded areas overlap. This common zone is your feasible region.
• Include edges according to the inequality type (\(\leq\) or \(\geq\) includes the line).

The feasible region is essential because it is the exact area that contains all solution points which satisfy every inequality condition in the system.