Here are the essential concepts you must grasp in order to answer the question correctly.
Circle Equation
The standard equation of a circle in a Cartesian plane is given by (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. For a circle centered at (-5, -2) with a radius of 4, the equation becomes (x + 5)² + (y + 2)² = 16. This equation describes all points that are exactly 4 units away from the center.
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Inequalities
Inequalities are mathematical expressions that show the relationship between two values when they are not equal. In the context of a circle, to represent the region inside the circle, we use the inequality (x + 5)² + (y + 2)² < 16. This indicates that we are interested in all points (x, y) whose distance from the center is less than the radius.
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Graphing Regions
Graphing inequalities involves shading a region on the coordinate plane that satisfies the inequality. For the inequality (x + 5)² + (y + 2)² < 16, the shaded area will be the interior of the circle, excluding the boundary. Understanding how to graph these regions is essential for visualizing solutions to inequalities.
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Graphs & the Rectangular Coordinate System