Problem 47 - Precalculus Mathematics for Calculus

Question

Use a graphing calculator to graph the solution of the system of inequalities. Find the coordinates of all vertices, rounded to one decimal place. $$\left\\{\begin{array}{l} y \geq x-3 \\ y \geq-2 x+6 \\ y \leq 8 \end{array}\right.$$

Step-by-Step Solution

Answer

Vertices are approximately (3.7, 0.7), (11.0, 8.0), and (5.5, 8.0).

Step 1: Graph the Inequalities

First, on your graphing calculator, input each inequality. Start with the equation of the line for each. For example, for the inequality \(y \geq x-3\), graph the line \(y = x-3\). Do the same for \(y \geq -2x+6\) and \(y \leq 8\). Ensure the shaded region is represented correctly, above for \(\geq\) inequalities and below for \(\leq\) inequalities.

Step 2: Identify the Intersection Points

Locate the points where these lines intersect. These points define the region of feasible solutions to the system of inequalities. Use the intersection function of your calculator or plot points until the intersection clearly shows at the intersection of these lines.

Step 3: Find Intersection Coordinates

For each pair of lines, calculate their intersection point. Solve \(y = x-3\) and \(y = -2x+6\) simultaneously to find one vertex, \(y = x-3\) and \(y = 8\) for another, and \(y = -2x+6\) with \(y = 8\) for the last vertex.

Step 4: Solve for Each Vertex

Solve the equations for each intersection pair to find the coordinates of the vertices. \(y = x-3\) intersect with \(y = -2x+6\); set \(x-3 = -2x+6\), solve for \(x\), substitute back to get \(y\). Repeat similarly for other pairs. For \(y = x-3\) and \(y = 8\); \(x-3=8\), solve for \(x\). For \(y = -2x+6\) and \(y = 8\); \(-2x+6=8\), solve for \(x\).

Step 5: Round to One Decimal Place

Once the vertices are found, round the coordinates of each to one decimal place. Check calculator or perform the calculation by hand to assure rounding accuracy is maintained.

Key Concepts

Intersection Coordinates

When solving a system of inequalities graphically, finding intersection coordinates is essential for identifying the vertices of the solution region. To find these intersections, you start by graphing each inequality as an equation, which turns them into lines on a graph. The point where any two lines on your graph intersect is an intersection point.

These intersection points are crucial because, in the context of inequality systems, they help define the boundaries of the feasible region. Solving for these coordinates involves setting the equations equal to each other and solving for the variables.

• For the inequality system \( y = x-3, \) and \( y = -2x + 6 \), you set \(x-3 = -2x + 6 \) and solve for \(x\).
• The intersections are evaluated line by line, and typically you'll solve for both \(x\) and \(y\) coordinates.

This ensures that you correctly determine the vertices of the region that satisfies all inequalities.

Graphing Calculator

Using a graphing calculator to solve systems of inequalities can greatly simplify the process. By inputting each inequality as an equation, the calculator can graph the lines and provide a visual representation of the solution region. This tool is especially helpful for visual learners who benefit from seeing the relationships between lines and shaded regions directly.

• First, enter each inequality as an equation in your graphing calculator.
• Ensure the graph is correctly shaded. Above for inequalities like \( y \geq \), and below for \( y \leq \).
• Utilize the intersection feature on the calculator to find precise intersection points.

Many graphing calculators also provide the option to zoom and adjust the view, which can help in identifying small or crowded intersections. They offer both precision in the numeric output and visual clarity for interpreting the whole system.

Vertices of Solution Region

The vertices of the solution region are the corner points that define the polygon created by the intersection of graphed inequalities. These points are essential as they represent the boundaries of the feasible region that satisfies all the given inequalities.

To identify the vertices, solve the systems of equations formed by intersecting lines. In this exercise, the inequalities such as \( y = x-3 \) and \( y = 8 \), when solved simultaneously, give a point that is part of the solution region.

• Solve \( y = x-3 \) with \( y = 8 \) to find the \( x \) coordinate and substitute back to find \( y \).
• Calculate similarly for other pairs like \( y = -2x+6 \) with \( y = 8 \).

Each vertex identified in this manner marks a critical point on the boundary of the region. Remember to round each coordinate to one decimal place for precision. This ensures the solution is easy to interpret and accurate, resulting in well-defined vertices for any further analysis or graph interpretation.