In Exercises 1–26, graph each inequality. x^2+y^2>25

Verified step by step guidance

1. <Step 1: Understand the inequality.> The given inequality is $x^2 + y^2 > 25$. This represents a region outside a circle centered at the origin (0,0) with a radius of 5, since $x^2 + y^2 = 25$ is the equation of a circle with radius 5.
2. <Step 2: Identify the boundary.> The boundary of the region is the circle $x^2 + y^2 = 25$. This circle is not included in the solution set because the inequality is 'greater than' (>) and not 'greater than or equal to' (≥).
3. <Step 3: Graph the boundary.> Draw the circle $x^2 + y^2 = 25$ on the coordinate plane. Use a dashed line to indicate that points on the circle are not included in the solution set.
4. <Step 4: Determine the region to shade.> Since the inequality is $x^2 + y^2 > 25$, shade the region outside the circle. This represents all points (x, y) whose distance from the origin is greater than 5.
5. <Step 5: Verify with a test point.> Choose a test point not on the boundary, such as (0,0), and substitute it into the inequality: $0^2 + 0^2 = 0$, which is not greater than 25. This confirms that the region outside the circle is the correct solution.

Key Concepts

Inequalities

Inequalities express a relationship where one quantity is not equal to another, often using symbols like '>', '<', '≥', or '≤'. In this case, the inequality x² + y² > 25 indicates that we are looking for all points (x, y) in the coordinate plane where the sum of the squares of x and y exceeds 25.

Graphing Quadratic Equations

Graphing quadratic equations involves plotting the curve represented by the equation. The equation x² + y² = 25 describes a circle with a radius of 5 centered at the origin. Understanding this helps in visualizing the area that satisfies the inequality, which is the region outside this circle.

Coordinate Plane

The coordinate plane is a two-dimensional surface defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Each point on this plane is represented by an ordered pair (x, y). Familiarity with the coordinate plane is essential for accurately graphing inequalities and understanding the regions they define.