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A coordinate plane is a two-dimensional surface where we can plot points, lines, and curves using pairs of numbers. The plane is divided by two axes: the horizontal axis known as the x-axis and the vertical axis called the y-axis.
The point where these axes meet is called the origin, represented by the coordinates (0,0). To plot points on the coordinate plane, we write them as ordered pairs (x, y), where x is the distance from the origin along the x-axis, and y is the distance along the y-axis. Understanding this layout is essential for graphing equations, such as circles, because it helps us visualize where they exist on this plane.When graphing inequalities like \(x^2 + y^2 \geq 1\), the coordinate plane allows us to determine whether points satisfy the inequality based on their location relative to a defined boundary, in this case, a circle.

The circle equation in a two-dimensional coordinate plane is expressed as \(x^2 + y^2 = r^2\). Here, \(r\) denotes the radius of the circle, and the circle is centered at the origin (0,0). This equation represents all points (x, y) that are exactly \(r\) units away from the origin.
If you change \(r\), the size of the circle changes correspondingly.For the inequality \(x^2 + y^2 \geq 1\), the equation \(x^2 + y^2 = 1\) represents a circle with a radius of 1 centered at the origin. The symbol \(\geq\) shows that you need to consider points on the circle as well as those outside of it.This circle forms the boundary line for our inequality, and understanding the circle equation allows us to graph it accurately on the coordinate plane, ensuring that we include the right set of points.

Shading regions in graphing refers to marking a part of a plane to signify it holds solutions to an inequality.
In our exercise, the inequality \(x^2 + y^2 \geq 1\) means we need to shade the area outside and on the circle, as these are the points where the sum of the squares of their coordinates is 1 or more.To accurately shade this region:

• First, draw the circle corresponding to the equation \(x^2 + y^2 = 1\) using a solid line, indicating that the boundary is included in the solution.
• Then, choose a test point not on the line, such as (0,0), to see if it satisfies the inequality. Since \(0^2 + 0^2 = 0\) is less than 1, it doesn't satisfy the inequality, so areas inside the circle shouldn't be shaded.
• Proceed to shade the region outside the circle. This visual representation helps in understanding the set of all points that are solutions to the inequality.

By following these steps, students can correctly shade and graph inequalities involving circles.