Problem 25 - Precalculus: Mathematical 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) and (-3, 0). Solution set is bounded.

Step 1: Identify the Inequalities

The given system of inequalities is \( y < 9 - x^2 \) and \( y \geq x + 3 \). The first inequality is a quadratic inequality representing a downward opening parabola, and the second inequality is a linear inequality representing a line with slope \(1\) and y-intercept \(3\).

Step 2: Graph the Parabola

Firstly, graph the parabola defined by \(y = 9 - x^2\). This is a downward-opening parabola with vertex at \((0, 9)\). Indicate the area below this parabola, as we have \(y < 9 - x^2\). Use a dashed line for the boundary, as it is a strict inequality.

Step 3: Graph the Line

Next, graph the line defined by \(y = x + 3\). This line crosses the y-axis at \((0, 3)\) and has a slope of 1. Indicate the area above this line using a solid line for the boundary as \(y \geq x + 3\).

Step 4: Find Intersection Points

Find the points of intersection by solving \(9 - x^2 = x + 3\). Rearrange this to form a quadratic equation: \(x^2 + x - 6 = 0\), which can be factored into \((x - 2)(x + 3) = 0\). This gives solutions \(x = 2\) and \(x = -3\). Substitute these back into the line equation \(y = x + 3\) for their respective y-values: when \(x = 2\), \(y = 5\) and when \(x = -3\), \(y = 0\). Thus, the coordinates are \((2, 5)\) and \((-3, 0)\).

Step 5: Plot the Possible Solutions

Shade the region that satisfies both inequalities: below the parabola and above the line. This region will be bounded by the curve of the parabola and the intersection points with the line.

Step 6: Determine if Solution Set is Bounded

The solution set is bounded as it lies within a finite region under the parabola and above the line. It does not extend to infinity in any direction.

Key Concepts

Graphing Inequalities

Graphing inequalities involves plotting the regions that satisfy the given inequalities on a coordinate plane. In this exercise, we are dealing with two inequalities: a quadratic \(y < 9 - x^2\) and a linear one \(y \geq x + 3\).
Understanding how to graph these is crucial.
- **Quadratic Inequality**: This is graphed as a parabolic curve. We start by plotting the boundary represented by \(y = 9 - x^2\), a downward-opening parabola. Since the inequality uses a less than symbol (\(<\)), the boundary is indicated with a dashed line.
- **Linear Inequality**: This is graphed as a straight line. The boundary for \(y = x + 3\) is drawn with a solid line, depicting \(>\) or \(\geq\). Shade the region that satisfies the inequality requirements above or below these boundaries.
By doing this, you'll visually represent all possible solutions that satisfy both conditions simultaneously.

Parabola and Linear Equations

Understanding the nature of a parabola and a line is key to solving inequalities.
- **Parabola**: A parabola like \(y = 9 - x^2\) opens downward and its vertex, or peak point, is at \( (0, 9)\). It is symmetrical around the y-axis and as you move away from the vertex, the "arms" of the parabola move downwards.
- **Line**: The line \(y = x + 3\) is straightforward. It has a slope of 1, indicating a 45-degree angle with the x-axis, and crosses the y-axis at point \( (0, 3)\).
Combining these two helps in identifying solution spaces that align with both the parabola's downward curve and the linear graph's inclined path.

Intersection Points

Intersection points are critical as they signify where two curves meet, detailing a change in the solution region.
To find these, solve the system of equations formed by setting the two equations equal: \(9 - x^2 = x + 3\). Rearranging yields \(x^2 + x - 6 = 0\), a factorable quadratic. Solve it to get \(x = 2\) and \(x = -3\).
Substituting these x-values back into the line equation provides the full coordinates: \( (2, 5) \) and \( (-3, 0) \).
These points help define the confines of the solution region in the graph, marking where the two equations intersect and where solutions are possible.

Bounded Solution Sets

A bounded solution set is contained within a finite area on the graph.
In this system of inequalities, the solution set is bounded.
- The parabola provides a curved limitation, not allowing solutions to extend beyond it.
- The line and its solid boundary further enclose the area from below.
- The intersection points \( (2, 5) \) and \((-3, 0) \) serve as limits on the graph, ensuring no solutions venture into infinity.This defined and closed space makes analyzing or interpreting the solution set easier since it's limited to a discrete region, helping in identifying feasible solutions.