Problem 42 - Algebra and Trigonometry

Question

41–44 Use a graphing calculator to graph the solution of the system of inequalities. Find the coordinates of all vertices, correct to one decimal place. $$\left\\{\begin{aligned} x+y & \geq 12 \\ 2 x+y & \leq 24 \\ x-y & \geq-6 \end{aligned}\right.$$

Step-by-Step Solution

Answer

Vertices are (3, 9), (6, 12), and (12, 0).

Step 1: Identify Inequalities

First, let's identify the inequalities from the given system: \( x + y \geq 12 \), \( 2x + y \leq 24 \), and \( x - y \geq -6 \). These inequalities will define our constraint area on the graph.

Step 2: Graph the Boundary Lines

Each inequality can be transformed into an equation by replacing the inequality sign with an equal sign. The boundary lines are: \( x + y = 12 \), \( 2x + y = 24 \), and \( x - y = -6 \). Graph these lines on the calculator.

Step 3: Identify the Feasible Region

The feasible region is where all given inequalities overlap on the graph. Using the graphing calculator, shade the area where \( x + y \geq 12 \), \( 2x + y \leq 24 \), and \( x - y \geq -6 \) all intersect. This area represents the solution set.

Step 4: Find Intersection Points

Find the intersection points of the boundary lines graphically or by solving equations pairwise. They are potential vertices of the feasible region. Solve the equations: \( x + y = 12 \), \( 2x + y = 24 \), and \( x - y = -6 \) two at a time to find these points.

Step 5: Calculate Intersection Points

Calculate coordinates of intersection points:- Solving \( x + y = 12 \) and \( 2x + y = 24 \) gives (12, 0).- Solving \( 2x + y = 24 \) and \( x - y = -6 \) gives (6, 12).- Solving \( x + y = 12 \) and \( x - y = -6 \) gives (3, 9).

Step 6: Verify Within Constraints

Confirm each intersection point lies within all inequalities by substituting back into the original inequalities. - (12, 0), (6, 12), and (3, 9) must satisfy all inequalities.

Step 7: Round the Coordinates

All calculated coordinates are already correct to one decimal place as they are integers. Ensure to round any further calculations correctly if needed.

Key Concepts

Feasible Region

When dealing with a system of inequalities, the feasible region is one of the most important concepts to understand. It represents all the possible solutions to the inequalities, which is visually represented on a graph as the area where the inequalities overlap.
To find the feasible region, you first need to graph each inequality. Instead of looking at them as inequalities at first, view them as equations. For example:

• Transform the inequality \(x + y \geq 12\) into the line \(x + y = 12\).
• Do the same for the other inequalities: \(2x + y \leq 24\) becomes \(2x + y = 24\), and \(x - y \geq -6\) turns into \(x - y = -6\).

Shade the regions corresponding to each inequality, where the feasible region is the overlapping shaded area from all inequalities. This helps you see where all conditions are met, indicating the potential solution area. If an inequality is a 'greater than or equal to (\(\geq\))', shade above the line, whereas 'less than or equal to (\(\leq\))' is below.
This bounded area is vital, as any point within it satisfies all the inequalities simultaneously.

Intersection Points

Intersection points occur where the boundary lines of the inequalities meet on the graph. These points are critical because they often form the vertices of the feasible region. To find these points, solve the equations pairwise, i.e., take two at a time and solve them simultaneously.
For example:

• Solve \(x + y = 12\) and \(2x + y = 24\) to find one intersection point: \((12, 0)\).
• Solve \(2x + y = 24\) and \(x - y = -6\) to obtain another point: \((6, 12)\).
• Finally, solve \(x + y = 12\) and \(x - y = -6\) to get \((3, 9)\).

These intersection points are effectively where two conditions meet. Check each of these coordinates back against all original inequalities to confirm that they lie within the feasible region. If they satisfy all inequalities, they contribute as potential boundary points for the solution set.

Graphing Calculator

A graphing calculator is an excellent tool when working with systems of inequalities. It not only helps in plotting the lines accurately but also shades the feasible region where all inequalities intersect. This graphical representation aids in visually comprehending the solutions quickly.
To use a graphing calculator effectively:

• Input each inequality separately as its corresponding equation.
• Adjust the viewing window to ensure that all relevant points and shaded regions are visible.
• Look for points of intersection formed by the lines, which will appear in the overlapping shaded area.
• Use the calculator's intersection or trace feature to find exact coordinates of these points, usually displayed with high precision.

The calculator simplifies the process of identifying the feasible region and intersection points, making it easier and more precise to find the solution set. It is invaluable in cross-verifying manual calculations and ensuring graphical accuracy.