Problem 50 - 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. $$\left\\{\begin{array}{l} y \geq x^{3} \\ 2 x+y \geq 0 \\ y \leq 2 x+6 \end{array}\right.$$

Step-by-Step Solution

Answer

Vertices are approximately (-2.2, 2.2), (1.0, -2.0), and (1.8, 9.6).

Step 1: Graph Each Inequality Separately

Using the graphing calculator, graph the inequality \( y \geq x^3 \). This represents the region above or on the curve of the function \( y = x^3 \). Next, graph \( 2x+y \geq 0 \), which is equivalent to \( y \geq -2x \). This represents the region above or on the line \( y = -2x \). Lastly, graph \( y \leq 2x + 6 \), which is the region below or on the line \( y = 2x + 6 \).

Step 2: Identify the Intersection Region

Observe the graphs to find the region where all three inequalities overlap. This area will typically appear shaded or highlighted on your graphing calculator, indicating the solution set for the system of inequalities.

Step 3: Find Points of Intersection

To find the vertices of the intersection region, identify the points where the curves intersect. These will be found at points where two boundaries meet. Solve the equations simultaneously if necessary:1. \( y = x^3 \) and \( y = 2x + 6 \).2. \( y = x^3 \) and \( y = -2x \).3. \( y = 2x + 6 \) and \( y = -2x \).

Step 4: Calculate the Intersection Points

Solve the equations set in step 3 to find intersection points:1. For \( x^3 = 2x + 6 \), solve the equation for \( x \) and calculate corresponding \( y \).2. For \( x^3 = -2x \), solve the equation for \( x \) and calculate corresponding \( y \).3. For \( 2x + 6 = -2x \), solve for \( x \) and calculate \( y \). This will give possible vertices.

Step 5: Round and Record Coordinates

After solving, round each coordinate to one decimal place. The final vertices, representing the corners of the intersection region, will be the set of points found in step 4.

Key Concepts

systems of inequalities

When working with systems of inequalities, we are dealing with multiple inequality statements that need to be satisfied simultaneously. This means finding a region on the graph that meets the conditions of all inequalities involved. In our exercise, we had three inequalities:

• \( y \geq x^{3} \) - The region above or on the curve \( y = x^{3} \)
• \( 2x + y \geq 0 \) - Equivalent to \( y \geq -2x \), representing the region above or on the line \( y = -2x \)
• \( y \leq 2x + 6 \) - The region below or on the line \( y = 2x + 6 \)

To solve systems of inequalities, you graph all individual inequalities on the same coordinate plane and identify the overlapping region. This overlapping area indicates where all the conditions are satisfied and represents the solution to the system.

intersection points

Finding intersection points in a system of inequalities is crucial because these points often serve as the vertices of the solution region. Intersection points are where the lines or curves defined by the inequalities cross each other.For the given exercise, we had to find intersections where the boundaries of these inequalities meet. This can be done by solving pairs of equations simultaneously:

• Find where \( y = x^3 \) meets \( y = 2x + 6 \).
• Find where \( y = x^3 \) meets \( y = -2x \).
• Find where \( y = 2x + 6 \) meets \( y = -2x \).

By solving these equation pairs, either graphically on the calculator or algebraically, we locate the coordinates where these lines intersect. These intersection points are critical as they define the vertices' corners of the solution region, giving us the boundaries of where all inequalities hold true on the graph.

graphing calculator usage

A graphing calculator is an essential tool for visualizing and solving systems of inequalities. It simplifies the process by providing a clear visual representation of multiple inequalities on the same screen.When using a graphing calculator, follow these steps to effectively graph the inequalities:

• Enter each inequality form, such as \( y \geq x^3 \) or \( y \leq 2x + 6 \), into the calculator.
• The calculator will shade or highlight the solution regions for each inequality. The overlapping shaded areas indicate where all inequalities are true simultaneously.
• Use the calculator's intersection or trace function to identify the exact coordinates of intersection points. This is particularly useful for finding precise values without solving algebraically.

By leveraging a graphing calculator, complex systems become manageable, and understanding the graphical solution it presents aids in confirming algebraic solutions and enhances comprehension of the systems of inequalities.