Problem 49 - Algebra and Trigonometry

Question

\(47-50\) . Use a graphing calculator to graph the solution of the system of inequalities. Find the coordinates of all vertices, rounded to one decimal place. $$ \left\\{\begin{array}{c}{y \leq 6 x-x^{2}} \\ {x+y \geq 4}\end{array}\right. $$

Step-by-Step Solution

Answer

Vertices of the intersection area, rounded to one decimal place, are approximately (0.7, 3.3) and (6.3, -2.3).

Step 1: Understand the Inequalities

The system of inequalities consists of two inequalities: \( y \leq 6x - x^2 \) and \( x + y \geq 4 \). The graph will show the region where both inequalities overlap.

Step 2: Graph the Parabola

Graph the quadratic inequality \( y \leq 6x - x^2 \). This is a downward-opening parabola because the coefficient of \( x^2 \) is negative. Since the inequality is \( \leq \), shade the area below the parabola.

Step 3: Graph the Line

Graph the linear inequality \( x + y \geq 4 \). Rearrange to \( y \geq -x + 4 \) to identify the line's slope and y-intercept. Since the inequality is \( \geq \), shade above the line.

Step 4: Determine the Feasible Region

Locate the intersection area where the shaded regions from both inequalities overlap. This area represents the solutions to the system of inequalities.

Step 5: Find Intersection Points

To find the vertices of the intersection area, solve the equality system: \( y = 6x - x^2 \) and \( x + y = 4 \). Substitute \( y = 4 - x \) into the first equation: \( 4 - x = 6x - x^2 \). Simplify to get \( x^2 - 7x + 4 = 0 \).

Step 6: Solve the Quadratic Equation

Solve the quadratic equation \( x^2 - 7x + 4 = 0 \) using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -7, c = 4 \). Applying this, we get: \( x = \frac{7 \pm \sqrt{49 - 16}}{2} = \frac{7 \pm \sqrt{33}}{2} \). Calculate to find approximate values for \( x \).

Step 7: Calculate y-coordinates

Substitute each value of \( x \) back into \( y = 4 - x \) to find the corresponding \( y \) values for the intersection points.

Step 8: Identify Vertices

Using the roots from the quadratic equation, calculate and round the coordinates of the vertices to one decimal place.

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool in visualizing and solving systems of inequalities. It can graph complex equations, making it easier to identify solution areas.
Here's how it can help with inequalities:

• Input Equations: You can directly enter the inequalities into the calculator.
• Visual Representation: The graphing feature shows how different equations interact visually.
• Feasible Region: The overlapping shaded areas solution can be clearly seen.
• Precision: Vertices and points of intersections can be approximated precisely.

For the given system, use your graphing calculator to plot each inequality, observing where they overlap to understand the solution region.

Quadratic Inequalities

Quadratic inequalities are equations where the variable is of the second degree, such as in the expression \( y \leq 6x - x^2 \). For these inequalities:

• The quadratic term \( x^2 \) influences the curve's shape.
• If negative, like in our case, it forms a downward-opening parabola.

To solve:

• Graph the corresponding equation as if it were an equality.
• Determine which side to shade based on the inequality sign.

Understanding how to graph this parabola is crucial in identifying the part of the graph that contains solutions.

Linear Inequalities

Linear inequalities, like \( x + y \geq 4 \), involve linear equations expressed with inequality signs. These are straight lines in a graph.

• Transform the inequality to \( y \geq -x + 4 \) for easier graphing.
• Identify the slope (here \(-1\)) and y-intercept (here \(4\)).
• Plot the line: this is just like drawing the usual line, only that now the inequality sign will guide us where to shade.

In our case, we shade the region above the line to represent all spaces where the inequality holds true. This shaded region, in connection with the quadratic inequality, will help identify possible solution areas.

Feasible Region

A feasible region in the context of systems of inequalities is where all constraints are satisfied. When graphing, it's the intersection of shaded areas from each inequality.
For problems like ours:

• Graph each inequality separately to identify individual shaded regions.
• The feasible region is where these shaded areas overlap.
• This region contains all possible solutions that satisfy both inequalities at the same time.
• Vertices of this region can be found by solving the equality of the system for intersection points.

Understanding the concept of the feasible region helps clarify where potential solutions reside, an essential step in tackling systems of inequalities.