Problem 65 - 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: \((0, 8)\), \((1.8, 4.8)\), \((3,0)\).

Step 1: Graph the First Inequality

Start by graphing the line of the inequality \( y \geq x-3 \). To do this, graph the line \( y = x - 3 \) by plotting two points: one when \( x = 0 \), giving \( y = -3 \), and one when \( y = 0 \), giving \( x = 3 \). Shade the region above the line because we have \( y \geq x-3 \).

Step 2: Graph the Second Inequality

Next, graph the line of the inequality \( y \geq -2x + 6 \). Start by plotting two points: one when \( x = 0 \), giving \( y = 6 \), and a second when \( y = 0 \), giving \( x = 3 \). Shade the region above this line since \( y \geq -2x + 6 \).

Step 3: Graph the Third Inequality

Graph the horizontal line \( y = 8 \) for the inequality \( y \leq 8 \). Shade the region below this line because \( y \leq 8 \).

Step 4: Identify the Solution Region

The solution region is where all three shaded areas overlap. This region represents all the points that satisfy all three inequalities at the same time.

Step 5: Identify and Calculate Vertices of the Solution Region

Identify the intersection points of the lines: \((0, 8)\), \((1.8, 4.8)\), and \((3,0)\). These points are where the boundary lines intersect, and they serve as the vertices of the solution region. Calculate each vertex by solving the systems of equations given by the intersection of the lines and check which intersection points lie within the shaded regions.

Key Concepts

Graphing Calculator

A graphing calculator is an incredibly useful tool when dealing with systems of inequalities. It helps visually represent the solution by accurately plotting lines and shading regions.
To use a graphing calculator:

• Enter each inequality. In our exercise, these are: \( y \geq x - 3 \), \( y \geq -2x + 6 \), and \( y \leq 8 \).
• Graph each inequality one at a time. Observe where the lines intersect and where the shaded areas overlap.
• Use functionality to zoom in or out if necessary, allowing a clearer view of the overlapping regions.

With a graphing calculator, you can easily see where all inequalities hold true and visualize the solution region.

Intersection Points

When graphing a system of inequalities, identifying intersection points is crucial. Intersection points are where two or more lines cross each other.
To find these points manually, solve the equations formed by setting the expressions for two lines equal. For example:

• For \( y = x - 3 \) and \( y = -2x + 6 \), solve \( x - 3 = -2x + 6 \) to find the intersection point.
• Repeat for other pairs within the system of inequalities.

These solutions give coordinates, such as \((1.8, 4.8)\), where the lines intersect. This helps determine the vertices of the solution region.

Shaded Region

In a graph representing systems of inequalities, a shaded region shows where the solutions lie.
For each inequality:

• Graph the boundary line. Use dashed lines for \( < \)/\( > \) and solid lines for \( \leq \)/\( \geq \).
• Determine which side of the line to shade. This is based on whether the inequality denotes a greater-than or a less-than condition.

The area where all individual shaded regions overlap is the actual solution region. This region satisfies all the inequalities collectively.

Vertices of Solution Region

Vertices of the solution region are key points, as they represent the boundaries of the feasible area.
After shading the overlapping region:

• Locate points of intersection that lie at the boundary of this region.
• For our system, vertices are \((0, 8)\), \((1.8, 4.8)\), and \((3, 0)\). These points mark the corners of the polygonal solution space.

Knowing the vertices is essential, especially in linear programming and optimization problems, as they often include the optimal solutions.