Work each problem. Write the inequality that represents the region inside a circle with center (-5, -2) and radius 4.

Verified step by step guidance

1. Recall the general equation of a circle with center \((h, k)\) and radius \(r\) is given by: \[ (x - h)^2 + (y - k)^2 = r^2 \]
2. Identify the center \((h, k)\) and radius \(r\) from the problem: Center: \((-5, -2)\) Radius: \(4\)
3. Substitute the values of \(h\), \(k\), and \(r\) into the circle equation: \[ (x - (-5))^2 + (y - (-2))^2 = 4^2 \] which simplifies to \[ (x + 5)^2 + (y + 2)^2 = 16 \]
4. Since the problem asks for the inequality representing the region inside the circle, replace the equal sign with a less than or equal to sign: \[ (x + 5)^2 + (y + 2)^2 \leq 16 \]
5. This inequality describes all points \((x, y)\) whose distance from the center \((-5, -2)\) is less than or equal to 4, which is the region inside and on the circle.

Key Concepts

Equation of a Circle

The equation of a circle with center (h, k) and radius r is given by (x - h)² + (y - k)² = r². This formula represents all points (x, y) that are exactly r units away from the center.

Inequalities Representing Regions

Inequalities involving circle equations describe regions inside or outside the circle. For points inside or on the circle, the inequality is (x - h)² + (y - k)² ≤ r², meaning the distance from the center is less than or equal to the radius.

Coordinate Geometry and Distance

Understanding how to calculate the distance between points in the coordinate plane is essential. The distance formula derives the circle equation and helps interpret inequalities as regions relative to the circle's boundary.