Problem 13 - College Algebra

Question

In Exercises 1–26, graph each inequality. $$x^{2}+y^{2} \leq 1$$

Step-by-Step Solution

Answer

The inequality \(x^{2}+y^{2} \leq 1\) represents a graph of a filled circle centered at the origin (0,0) with radius 1.

Step 1: Recognize the Circle

Recognize that \(x^{2}+y^{2} \leq 1\) represents a circle with radius 1. The inequality sign \( \leq \) suggests that both the boundary of the circle and its inside satisfies the inequality.

Step 2: Draw the Circle Graph

Draw a circle centered at (0, 0) with a radius equal to 1. This can be done by using a compass, or by creating a dot at the center point (0,0), then making another dot at a point 1 unit away and connecting them to form a circle.

Step 3: Shade In the Area

As the inequality is \( \leq \), it indicates that not only points on the boundary, but also inside the circle satisfies the inequality. Therefore, shade in the interior of the circle on the graph.

Key Concepts

Graphing Inequalities

Graphing inequalities is an essential concept in mathematics that helps visualize the solution set of an inequality. When an equation involves inequalities, it signifies that the solutions encompass a range of points rather than exact locations. For instance, the inequality \(x^2 + y^2 \leq 1\) describes a set of points within a specific boundary.

• The symbol \( \leq \) suggests that the solution includes every point where \(x^2 + y^2\) is less than or equal to 1.
• The graph of the inequality includes not only the edge defined by the equation \(x^2 + y^2 = 1\) but also all the points inside that boundary.

To accurately represent this on a graph, you’ll need to shade in all the regions that satisfy the inequality, indicating the vast pool of potential solutions.

Circle Equations

Circle equations are a fascinating aspect of geometry defining a set of points. The basic circle equation is in the form \(x^2 + y^2 = r^2\), where \(r\) represents the radius of the circle, and the center is at the origin \((0,0)\).

• In the equation \(x^2 + y^2 \leq 1\), the term \(1\) is actually \(r^2\), so the circle has a radius of 1.
• The circle's center is at \((0, 0)\) since there are no terms of \(x\) or \(y\) inside the circle equation to shift the center.

By understanding these concepts, you can easily draw the circle on a graph, knowing exactly where its center and edge will reside.

Shading Solutions

Shading solutions on a graph is a key technique to demonstrate which areas satisfy the given inequality. This involves identifying all the points that fall within the permissible range described by the inequality.

• In the inequality \(x^2 + y^2 \leq 1\), shading the area inside the circle shows all possible solutions.
• The boundary of the circle \(x^2 + y^2 = 1\) is also part of the solution set since the inequality includes \( \leq \).

By visually showing these solutions, shading in the graph helps clarify which points within the Cartesian plane meet the condition set by the inequality.

Geometry of Inequalities

Understanding the geometry of inequalities is crucial to comprehend how they describe spaces in a plane. Inequalities involving geometric shapes, like circles, define regions that satisfy the condition set forth by the inequality.

• The inequality \(x^2 + y^2 \leq 1\) geometrically represents all points inside and on a circle centered at the origin.
• The boundary specified by the equality \(x^2 + y^2 = 1\) serves as a limit that confines the solution region of the inequality.

Recognizing how inequalities sketch boundaries within geometry helps visually solve complex problems by observing the relationships and interactions among shapes and spaces.