Problem 31 - Precalculus Mathematics for Calculus

Question

Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{array}{l} y<9-x^{2} \\ y \geq x+3 \end{array}\right.$$

Step-by-Step Solution

Answer

Vertices: (2, 5), (-3, 0). The solution set is bounded.

Step 1: Graph the first inequality

To graph the inequality \( y < 9 - x^2 \), first graph the boundary line \( y = 9 - x^2 \). This is a downward-opening parabola with the vertex at (0, 9). Shade the area below the parabola since the inequality is \( y < 9 - x^2 \).

Step 2: Graph the second inequality

To graph the inequality \( y \geq x + 3 \), first graph the boundary line \( y = x + 3 \). This is a straight line with a slope of 1 and a y-intercept at (0, 3). Shade the area above the line since the inequality is \( y \geq x + 3 \).

Step 3: Determine the intersection of the graphs

The solution set is the region where the shaded areas from both inequalities overlap. Identify this region by locating where the parabola \( y < 9 - x^2 \) intersects with the line \( y = x + 3 \).

Step 4: Find the intersection points

To find the intersection points, solve the equation \( 9 - x^2 = x + 3 \). Rearrange to get \( x^2 + x - 6 = 0 \), then factor to \((x - 2)(x + 3) = 0\). Thus, the solutions are \( x = 2 \) and \( x = -3 \). For these x-values, substitute into \( y = x + 3 \) to find the y-values: \((2, 5)\) and \((-3, 0)\).

Step 5: Determine the boundedness of the solution region

The region is bounded because both curves intersect and close off the region. It has clear vertices at the intersection points previously found: (2, 5), (-3, 0), and also includes points derived from intersections with the axes, as analyzed from shading.

Key Concepts

Graphing Inequalities

When graphing systems of inequalities, the task is to represent not just one but multiple inequalities on the same plane. Starting with a single inequality like \( y < 9 - x^2 \), we initially graph the boundary line or curve. In this example, the boundary is the parabola \( y = 9 - x^2 \). To sketch it,

• Identify the vertex and the direction of the parabola. Here, the vertex is at \((0, 9)\), and it opens downwards.
• Shade the area below the parabola, since the inequality is 'less than'.

Next, take \( y \geq x + 3 \), and similarly graph the boundary, a straight line passing through the y-intercept \((0, 3)\) with a slope of 1. For this,

• Draw a line through \( (0, 3) \) that rises one unit up and one across right for each unit on the x-axis.
• Shade above the line, representing the region that satisfies the 'greater than or equal to' condition.

The overall graph should clearly show where the shaded areas of these inequalities overlap, forming the potential solution set.

Bounded Solution Sets

A bounded solution set refers to a region that looks like it is enclosed or closed off in all directions when plotted on a graph. In the context of this exercise, we determine boundedness by observing how the solution set forms from the interaction of both inequalities.

• For our example, shaded regions corresponding to the inequalities \( y < 9 - x^2 \) and \( y \geq x + 3 \) overlap and intersect.
• This intersection is a tell-tale sign of the region being bounded.

This concept is critical because bounded sets form finite regions, meaning that the area is limited, ensuring all points within the region meet the conditions of both inequalities.
The vertices of the region outlined by the intersections, such as points where the curves meet, illustrate that the solution does not extend infinitely in any direction, confirming boundedness.

Intersection Points

Intersection points are where the graphs of two or more equations meet or cross each other. For inequalities, these points are crucial for defining the vertices of the solution set.

• To find these points, set the equations of the boundaries equal: \( 9 - x^2 = x + 3 \).
• Solving the resultant equation \( x^2 + x - 6 = 0 \) by factoring yields \( (x - 2)(x + 3) = 0 \).
• This gives \( x = 2 \) and \( x = -3 \).

Substitute these x-values back into one of the boundary equations to find y-values, confirming the intersection points at \((2, 5)\) and \((-3, 0)\). These points are essential as they help define the corners, or vertices, of the bounded solution region. Recognizing these points allows you to complete the graph accurately and understand the setup's geometrical structure.