Problem 20 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the inequality. $$x^{2}+y^{2}>4$$

Step-by-Step Solution

Answer

The graph of the inequality \(x^{2}+y^{2}>4\) is the entire plane excluding a circle with a radius of 2 centered at (0, 0). This is shown by all the shaded region outside the dashed circle.

Step 1: Identify the radius of the circle

The equation is in the form \(x^{2}+y^{2}=r^{2}\) where r is the radius of the circle. Given, \(x^{2}+y^{2}=4\), we can infer that the radius is \(r=2\). Plot the circle with a center at (0, 0) and a radius of 2.

Step 2: Sketch the circle

Using the radius and center from step 1, sketch a circle with center at (0, 0) and radius 2. Be careful to represent it as a dashed line to show that points on the line are not included in the solution because our inequality is \(x^{2}+y^{2}>4\), not \(x^{2}+y^{2}\geq4\). This differentiates the solution to the problem as it does not include the points on the boundary.

Step 3: Identify the area satisfying the inequality

The inequality \(x^{2}+y^{2}>4\) represents the set of all points (x, y) in the plane that are strictly greater than 2 units away from the center. This means that the area outside of the circle represents the solution of the inequality. Shade all of the area outside of the circle to represent the solution set.

Key Concepts

Circles

Circles are one of the fundamental shapes in geometry, perfectly symmetric and defined by all points equidistant from a center point. The basic equation for a circle in the coordinate plane is given by the formula \(x^2 + y^2 = r^2\). Here, \(x\) and \(y\) represent the coordinates of a point on the circle, and \(r\) is the radius of the circle. This equation tells us that the distance from the center of the circle at the origin (0,0) to any point \((x,y)\) on the circle is equal to \(r\).

• Center: The center of the circle is the point about which the circle is perfectly symmetric.
• Radius: The radius is the distance from the center of the circle to any point on its perimeter.

Understanding these basic parts of a circle is essential when working with equations and inequalities involving circles.

Radius and Center

The radius and center of a circle are critical to graphing inequalities involving circles. They determine not only the size and position of the circle but also play a crucial role when deciphering inequalities.
Since the given inequality is \(x^2 + y^2 > 4\), it helps us identify quickly that:

• Center: The center of the circle in standard form \(x^2 + y^2 = r^2\) is at the origin, (0,0).
• Radius: The radius \(r\) is equal to 2, since \(r^2 = 4\).

It's important to note how the center and radius inform us about possible solutions to an inequality. In this problem, because we have a strict inequality \(x^2 + y^2 > 4\), solutions will lie outside the circle, meaning the radius tells us how far out the solutions start. Solutions begin beyond 2 units away from the center point at (0, 0).
Remember, graphing inequalities usually involves shading the region that satisfies the inequality. Here, it means shading the outside of the circle.

Inequality Notation

Inequality notation communicates different sets of solutions and what specific points are included or excluded. When dealing with inequalities like \(x^2 + y^2 > 4\), we use special graphs to illustrate solutions.
For \(x^2 + y^2 = 4\), all points on the circle are at the exact distance of 2 units from the center (0,0). However, the inequality indicates a different set: \(x^2 + y^2 > 4\), which includes:

• Strict Inequality (>): This means the boundary circle \(x^2 + y^2 = 4\) itself is not part of the solution set; points on the circle do not satisfy the inequality as the equality point makes \(x^2 + y^2 = 4\), not greater.
• Graph Representation: We use a dashed line to draw the circle to show that the boundary is not included.
• Shaded Region: The area outside the dashed circle represents the solution to the inequality. This shows points with a distance greater than 2 units from the center.

Understanding these symbols and how to graph them appropriately ensures that the solutions to inequalities are accurately communicated visually.