Problem 12 - Precalculus: Mathematical for Calculus

Question

Graph the inequality. $$x^{2}+y^{2} \geq 9$$

Step-by-Step Solution

Answer

Graph a circle with radius 3 centered at the origin and shade outside the circle.

Step 1: Identify the Basic Shape

The inequality is related to a circle because it resembles the equation of a circle, which is \(x^2 + y^2 = r^2\). In this case, the circle has a radius \(r\) where \(r^2 = 9\), so \(r = 3\). This makes the circle centered at the origin \((0,0)\) with a radius of 3.

Step 2: Understand the Inequality Sign

The given inequality is \(x^2 + y^2 \geq 9\), which means we need to include the points on the circle \(x^2 + y^2 = 9\) as well as outside of it. This represents the area outside and including the boundary of this circle.

Step 3: Graph the Circle

Start by graphing the circle. Draw a circle with center \((0,0)\) and radius 3. Make sure to draw it with a solid line since the inequality is \(\geq\), including the line itself along with the shaded region.

Step 4: Shade the Correct Region

Shade the area outside the circle. Since the inequality is \(x^2 + y^2 \geq 9\), shade the region outside the circle, indicating that points in this region satisfy the inequality (i.e., the distance from the origin is greater or equal to 3).

Key Concepts

Graphing Inequalities

Graphing inequalities involves visually representing the solution set of an inequality. It is a critical skill in algebra. When we have an inequality like \(x^2 + y^2 \geq 9\), it involves understanding both the boundary and the area described by the equation.

- **Boundary**: The boundary is the circle described by the equation \(x^2 + y^2 = 9\). This is a circle centered at the origin with a radius of 3.
- **Inequality Sign**: The inequality \(\geq\) tells us that we need to include all the points on and outside the circle. Points on the circle satisfy \(x^2 + y^2 = 9\), and points outside satisfy \(x^2 + y^2 > 9\).
- **Graph**: To graph the inequality, we draw the circle with a solid line indicating the boundary is included. Next, we shade the area outside the circle since all those points fulfill the inequality condition. This shading visually shows the solution set, making it easier to understand which regions satisfy the inequality and which do not.

Circle Equations

Circle equations provide a way to describe the set of all points in a plane that are at a fixed distance (the radius) from a central point (the center). The basic form of a circle equation is \(x^2 + y^2 = r^2\).

- **Center and Radius**: For the equation \(x^2 + y^2 = 9\), the center of the circle is at the origin, \((0, 0)\), and the radius is 3, since \(r^2 = 9\) implies \(r = 3\).
- **Graphing**: When you graph a circle, you plot all points \((x, y)\) that satisfy this equation. For our specific problem, drawing a circle with a 3-unit radius centered at the origin covers exactly all points whose combined distances along x and y from the origin equal 3 squared.
- **Applications**: Understanding circle equations is essential in many applications, from designing gears in engineering to creating graphics in video games. In math, it helps us explore geometric properties and relationships between different shapes.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This lets us use algebraic techniques to solve geometric problems, such as graphing a circle or solving an inequality.

- **Coordinate Plane**: The coordinate plane is composed of two axes, x and y, which intersect at the origin (0,0). It provides a framework for plotting points, lines, and shapes like circles. By assigning coordinates to these elements, we can describe and analyze their properties mathematically.
- **Equation Interpretation**: In the example of \(x^2 + y^2 = 9\), coordinate geometry lets us interpret this equation as describing a circle in the coordinate plane. This understanding helps us derive solutions, such as finding the radius or determining which points satisfy an inequality.
- **Tools**: Tools like graphing software or graph paper make it easier to visualize and solve such problems. They offer a practical approach to understand how algebraic equations represent geometric figures, enhancing both comprehension and problem-solving capabilities.