In Exercises 1–26, graph each inequality. x²+y²>25

Verified step by step guidance

1. Recognize that the inequality $x^2+y^2>\; 25$ represents all points $(x,\; y)$ whose distance from the origin is greater than 5, since $25\; =\; 5^2$.
2. Start by graphing the boundary circle defined by the equation $x^2+y^2=\; 25$. This circle has center at $(0,0)$ and radius 5.
3. Since the inequality is strict ($>$), the points on the circle itself are not included, so represent the boundary with a dashed circle to indicate that points on the circle are excluded.
4. Determine which side of the circle to shade by testing a point not on the circle, such as the origin $(0,0)$. Substitute into the inequality: $0^2\; +\; 0^2\; =\; 0$, which is not greater than 25, so the origin is not part of the solution.
5. Shade the region outside the circle because the inequality $x^2+y^2>\; 25$ means all points whose distance from the origin is greater than 5.

Key Concepts

Graphing Inequalities in Two Variables

Graphing inequalities involves shading regions of the coordinate plane that satisfy the inequality. For inequalities with two variables, the boundary is found by replacing the inequality with an equation, and then the solution region is determined by testing points to see which side satisfies the inequality.

Equation of a Circle

The equation x² + y² = r² represents a circle centered at the origin with radius r. Understanding this helps identify the boundary curve for inequalities involving x² + y², such as x² + y² > 25, where the radius is 5.

Strict vs. Non-Strict Inequalities and Boundary Lines

A strict inequality (>, <) means the boundary line or curve is not included in the solution set, so it is drawn as a dashed line or curve. For x² + y² > 25, points on the circle are excluded, and the solution is the region outside the circle.