Problem 15 - College Algebra Essentials

Question

Graph each inequality. $$x^{2}+y^{2}>25$$

Step-by-Step Solution

Answer

The solution to the inequality is the entire region outside of the circle with origin (0,0) and radius 5 units, excluding the boundary. This is represented on the graph by a dashed circle and shading or arrows indicating the area outside of the circle.

Step 1: Identify the form of inequality

The inequality given, \(x^{2}+y^{2}>25\), is the form of a circle's equation \(x^2 + y^2 = r^2\), where the radius \(r= \sqrt{25} = 5 \). The inequality symbol '>' indicates that the solution will be the region outside the circle, not including the boundary of the circle.

Step 2: Sketch the circle

Plot the circle with center at the origin (0,0) and radius of 5 units. As the inequality does not include the 'equal to' (=) sign, the circle is to be drawn with a dashed line to show that points on the circle are not included in the solution.

Step 3: Identify the solution region

The solution to the inequality is given by the region outside the circle. This means that any point (x,y) that you choose which lies outside the circle will satisfy the inequality \(x^{2}+y^{2}>25\). This region outside the dashed circle represents the solution to the inequality.

Key Concepts

Understanding Circles

Circles play a vital role in graphing inequalities, especially when the equation involves terms like \(x^2 + y^2\). A circle is defined as the set of all points in a plane that are equidistant from a fixed point known as the center. The distance from the center to any point on the circle is called the radius. In the standard form of a circle equation, \(x^2 + y^2 = r^2\), the center is at the origin (0,0), and \(r\) stands for the radius.

• The equation \(x^2 + y^2 = 25\) describes a circle centered at the origin with a radius of 5 units.
• The inequality \(x^2 + y^2 > 25\) modifies this equation to describe a region.

With inequalities, you'll often see circles represented with dashed lines. This indicates that points on the line itself are not included in the solution set if the inequality is strict \(>\) or \(<\).

Exploring Inequalities

Inequalities allow us to explore a range or region instead of specific solutions, which is often indicated by using symbols such as \(>\), \(<\), \(\geq\), or \(\leq\). When graphing in terms of circles, the inequality \(x^2 + y^2 > 25\) indicates that we are looking for a set of points outside of the circle.

• The expression "greater than" \((>)\) tells us that all points (x,y) outside of the boundary meet the inequality.
• If it were "less than" \((<)\), our focus would shift to the interior region.

Examining inequalities can highlight areas that meet or do not meet a particular condition, which in graphical terms helps in quickly visualizing complex relationships.

Navigating the Coordinate Plane

The coordinate plane is a two-dimensional surface where points are defined using a pair of numerical coordinates \((x, y)\). Understanding this plane is essential for graphing equations and inequalities. In the graphing context:

• The x-axis runs horizontally, and the y-axis runs vertically, intersecting at the origin (0,0).
• Coordinate points (x,y) form the basic building blocks for plotting any equation or inequality.

To graph the inequality \(x^2+y^2>25\), we start by plotting a circle at the origin with a radius of 5 using a dashed circle. Then, the entire region outside this dashed circle on the coordinate plane is shaded to represent the solution. This visual representation helps in understanding which points satisfy the inequality.