Problem 56 - Algebra and Trigonometry

Question

Graph the system of inequalities. Tell whether the graph is bounded or unbounded, and label the corner points. $$ \left\\{\begin{array}{r} x \geq 0 \\ y \geq 0 \\ x+y \leq 6 \\ 2 x+y \leq 10 \end{array}\right. $$

Step-by-Step Solution

Answer

The graph is bounded with corner points \((0, 0)\), \((0, 6)\), and \((4, 2)\).

Step 1: Graph the Boundary Lines

Start by graphing the boundary lines of each inequality. These lines are found by turning the inequalities into equations. Graph the lines: 1. For the inequality \(x \geq 0\), draw the vertical line at \(x = 0\). 2. For the inequality \(y \geq 0\), draw the horizontal line at \(y = 0\). 3. For the inequality \(x + y \leq 6\), draw the line \(x + y = 6\). 4. For the inequality \(2x + y \leq 10\), draw the line \(2x + y = 10\).

Step 2: Identify the Feasible Region

Shade the regions that satisfy each inequality. \(x \geq 0\) is to the right of the line \(x = 0\). \(y \geq 0\) is above the line \(y = 0\). The region for \(x + y \leq 6\) is below the line \(x + y = 6\). The region for \(2x + y \leq 10\) is below the line \(2x + y = 10\). The overlapping shaded region is the feasible region for the system of inequalities.

Step 3: Find the Intersection Points

Determine the corner points by finding the intersections of the boundary lines: 1. The intersection of \(x = 0\) and \(y = 0\) is \((0, 0)\). 2. The intersection of \(x = 0\) and \(2x + y = 10\) is \((0, 6)\). 3. The intersection of \(x + y = 6\) and \(2x + y = 10\) can be found by solving the system of equations: \(x + y = 6\) and \(2x + y = 10\). Subtract the first from the second to get \(x = 4\). Plugging \(x = 4\) into \(x + y = 6\) gives \(y = 2\). Hence, the point is \((4, 2)\).

Step 4: Determine the Bounded or Unbounded Nature

Since the feasible region is enclosed and forms a closed area, it is bounded.

Step 5: Label Corner Points

Write down the corner points of the feasible region which are: \((0, 0)\), \((0, 6)\), and \((4, 2)\). These are the vertices of the bounded feasible region.

Key Concepts

Graphing Inequalities

Graphing inequalities helps us visually understand the solutions to a system of inequalities. It involves plotting boundary lines on a graph and shading the regions that satisfy the inequality conditions. Here’s how to do it:
Start by turning each inequality into an equation. For example, turn \(x \geq 0\) into the line \(x = 0\). Do the same for the other inequalities:
1. For \(y \geq 0\), draw the line \(y = 0\).
2. For \(x + y \leq 6\), draw the line \(x + y = 6\).
3. For \(2x + y \leq 10\), draw the line \(2x + y = 10\).
Once the lines are on the graph, shade the areas that satisfy each inequality. \(x \geq 0\) is to the right of the \(x = 0\) line. \(y \geq 0\) is above the \(y = 0\) line. The region for \(x + y \leq 6\) is below the line \(x + y = 6\), and for \(2x + y \leq 10\) is below the line \(2x + y = 10\). The overlapping shaded area where all conditions meet is our solution. This is known as the feasible region.

Feasible Region

The feasible region is the area on the graph where all the inequalities overlap. It represents all the possible solutions that satisfy the system of inequalities. To find it, graph each inequality and identify where the shaded areas intersect.
The process involves:

• Graphing the boundary lines for each inequality.
• Shading the regions that satisfy each inequality.
• Noting the overlapping shaded area that meets all conditions.

For our given system, the feasible region is bound by the lines \(x = 0\), \(y = 0\), \(x + y = 6\), and \(2x + y = 10\). This region forms a closed shape, indicating the solutions' area.

Bounded and Unbounded Regions

A feasible region can either be bounded or unbounded. A bounded region is enclosed by the boundary lines, forming a finite area. An unbounded region, on the other hand, is open and extends infinitely in at least one direction.
For our system of inequalities, the feasible region is bounded. It's enclosed by all boundary lines and forms a closed polygon with the corner points or vertices at \((0, 0)\), \((0, 6)\), and \((4, 2)\).
This means:

• There’s a finite area where all solutions lie.
• The solutions are limited to this enclosed space.

Knowing whether a region is bounded or unbounded helps understand the nature and limits of the solutions for systems of inequalities.