91Ó°ÊÓ

Understanding circle equations is essential when working with shapes in algebra and geometry. A circle's equation in the cartesian coordinate system can be expressed in the standard form \(x - h)^2 + (y - k)^2 = r^2\), where \(h, k\) represents the center of the circle, and \(r\) is the radius.

For the given inequality \(x^{2} + y^{2} \leq 1\), it represents a circle centered at the origin \(0,0\) with radius 1, because it aligns with the standard form's structure where \(h = 0\), \(k = 0\), and \(r^2 = 1\). Thus, identifying the boundary of the circle is a matter of plotting a set of points that all lie exactly 1 unit away from the center at all points along the circumference.

Inequality solutions encompass finding the set of values that satisfy the given inequality. Graphing inequalities helps visualize the solution set. The key features to look for include the inequality symbol itself, such as '<', '>', '\leq', or '\geq', which indicate whether the boundary is included in the solution (solid line) or not (dashed line), and in which direction—above or below the line, or inside or outside a curve—to shade the region.

In our exercise \(x^{2} + y^{2} \leq 1\), the '\leq' symbol implies that all points inside and on the circle are part of the solution. Hence, we not only draw a solid boundary but also shade the entire area within the circle to depict all the points \( (x, y) \) that make the inequality true.

The cartesian coordinates system is a two-dimensional plane defined by a pair of perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point in this system is defined by an ordered pair of numbers \( (x, y) \), which represent its distance from the origin along these axes.

To plot inequalities like \(x^{2} + y^{2} \leq 1\), we use this coordinate system to identify the points that satisfy the inequality. By understanding how to navigate the coordinate system, one can easily decode the location and size of the solution set, such as the area of a circle in the context of our exercise. Graphing on cartesian coordinates, thus, provides a visual and intuitive understanding of the solutions to equations and inequalities.