Graph each inequality. (x−2)²+(y+1)²<9

Verified step by step guidance

1. Identify the inequality as representing a circle in the coordinate plane. The general form of a circle's equation is \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) is the center and \(r\) is the radius.
2. Compare the given inequality \( (x - 2)^2 + (y + 1)^2 < 9 \) to the general form. Here, the center is at \( (2, -1) \) and the radius \( r \) is \( \sqrt{9} = 3 \).
3. Since the inequality is '<' (less than), the graph represents all points inside the circle but not on the boundary. This means the circle itself is not included, so the boundary should be drawn as a dashed curve.
4. Plot the center point at \( (2, -1) \) on the coordinate plane. Then, using the radius 3, mark points 3 units away from the center in all directions (up, down, left, right) to help sketch the circle.
5. Shade the interior region of the circle to represent all points \( (x, y) \) that satisfy the inequality \( (x - 2)^2 + (y + 1)^2 < 9 \). This shaded area shows the solution set.

Key Concepts

Graphing Circles

The inequality (x−2)² + (y+1)² < 9 represents all points inside a circle centered at (2, -1) with radius 3. Understanding how to graph circles involves identifying the center and radius from the equation and plotting the circle accordingly.

Inequalities and Regions

An inequality like (x−2)² + (y+1)² < 9 describes a region inside the circle, not just the boundary. Points satisfying the inequality lie strictly inside the circle, so the graph includes all interior points but excludes the circle itself.

Coordinate Plane and Plotting Points

Plotting the graph requires familiarity with the coordinate plane, including locating points using (x, y) coordinates. This skill helps in accurately drawing the circle's center and radius, and shading the correct region defined by the inequality.