Problem 44 - 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. $$\left\\{\begin{array}{l}{y \geq x^{3}} \\ {2 x+y \geq 0} \\ {y \leq 2 x+6}\end{array}\right.$$

Step-by-Step Solution

Answer

The vertices are approximately at (0,0), (-1.5,3), and (1.6,4.1).

Step 1: Understand the Inequalities

The system of inequalities is:\( y \geq x^3 \), \( 2x + y \geq 0 \), \( y \leq 2x + 6 \). We want to shade the region where all these inequalities intersect on the graph.

Step 2: Graph the First Inequality

Plot the graph of \( y = x^3 \). The region of interest is the area above the curve since \( y \geq x^3 \).

Step 3: Graph the Second Inequality

Plot the line \( y = -2x \). The inequality \( 2x + y \geq 0 \) indicates shading the region above this line.

Step 4: Graph the Third Inequality

Plot the line \( y = 2x + 6 \). The inequality \( y \leq 2x + 6 \) requires shading the region below this line.

Step 5: Find Regions of Overlap

Identify the region where all shaded areas from Step 2, 3, and 4 overlap on the graph. This is the solution region for the system of inequalities.

Step 6: Identify the Vertices

Calculate the coordinates where the boundaries intersect to find vertices in the overlap region. Solve for intersections: \( y = x^3 \) and \( y = -2x \), \( y = x^3 \) and \( y = 2x + 6 \), and \( y = -2x \) and \( y = 2x + 6 \).

Step 7: Calculate Intersections

Solve the following equations to find intersection points:- \( x^3 = -2x \): This gives points at \( (0,0) \), \( (\sqrt{2}, -2\sqrt{2}) \), and \( (-\sqrt{2}, 2\sqrt{2}) \).- \( x^3 = 2x + 6 \): Requires numeric methods or graphing calculator for accurate decimals, intersection approximately around \( (1.6, 4.1) \).- \( -2x = 2x + 6 \): Solve to get the point \( (-1.5, 3) \).

Step 8: List Vertices

The vertices of the overlapping solution region, rounded to one decimal place, are approximately: \( (0,0) \), \( (-1.5, 3) \), and \( (1.6, 4.1) \).

Key Concepts

Graphing Calculator

A graphing calculator is an incredibly useful tool when working with systems of inequalities.

• It allows you to visually represent equations and inequalities.
• You can observe where different regions overlap.

To graph an inequality, input each equation into the calculator. The tool will help you see the areas that satisfy one or more inequalities. For example, consider the system from the exercise:

• The calculator will visualize the curves and lines.
• It shades the regions where inequalities are true.

By observing these shaded areas, you can identify the intersection region. This makes it much easier to find solutions compared to doing it manually.

Intersection Points

Intersection points are where two or more lines or curves meet on a graph. Finding these points helps determine the vertices of the solution region for a system of inequalities.To find these points:

• Use your calculator to trace the curves.
• The calculator often automatically marks intersection points.

For manual calculations, set the equations equal to each other and solve for the variables. In this exercise:- For \(y = x^3\) and \(y = -2x\), solve \(x^3 = -2x\) to find the intersections at points such as \((0,0)\).- For \(y = x^3\) and \(y = 2x + 6\), a numerical method is required to find intersection, which might give an approximate solution like \((1.6, 4.1)\).Finding these points accurately is crucial since they define the boundaries and vertices of the region.

Inequalities

Inequalities in mathematics are expressions involving the symbols \( \geq \), \( > \), \( \leq \), and \( < \).

• They describe a range of values rather than a single value.
• Graphically, they represent areas or regions.

In the exercise, three inequalities define a system:- \( y \geq x^3 \) suggests the region above or on the curve.- \( 2x + y \geq 0 \) implies shading the area above the line \(y = -2x\).- \( y \leq 2x + 6 \) requires that shading be below or on the line.By graphing each inequality, you find the area that satisfies all the conditions simultaneously. Recognizing this region is key to solving the system and finding the vertices.

Vertices Calculation

Finding vertices involves calculating the points where the boundaries of the inequality regions intersect. These are crucial for outlining the shape of the solution region. To determine vertices:

• Identify and solve for intersections between boundaries using algebra and your calculator.
• Ensure the points lie within the shaded overlap region.

The given system includes points like:- \((-1.5, 3)\) from \(-2x = 2x + 6\).- \((0,0)\) from \(y = x^3\) and \(y = -2x\).Make sure calculations offer decimal accuracy, like the approximate vertex \((1.6, 4.1)\). Knowing vertices accurately provides a complete solution to the system of inequalities.