91Ó°ÊÓ

When we come across an inequality like \(x^{2}+y^{2}>36\), it's hinting at something special in the world of graphing—it points us to a concept known as 'circle inequalities'. Just as the equation \(x^{2}+y^{2}= r^{2}\) represents a circle on a coordinate plane with a radius \(r\) and a center at the origin, an inequality involving a circle tells us about the interior or exterior of that circle.

The inequality \(x^{2}+y^{2}>36\) speaks of all the points that are outside of the circle with radius 6. It's like drawing a boundary that is not to be crossed. But instead of a line, this boundary is a circle, and instead of staying inside, we're coloring outside of our circular lines—every point that's not on or within this imaginary round border is a solution to the inequality.

Graphing inequalities on a coordinate plane is just as much an art as it is a science. Think of it as coloring; only certain regions of the graph satisfy the inequality. So when you're asked to graph an inequality like \(x^{2}+y^{2}>36\), you begin by graphing the boundary, which is the related equation, in this case, the circle defined by \(x^{2}+y^{2}=36\).

Now the 'greater than' sign (>) indicates we are interested in the points outside this circle. To illustrate this, the circle itself is drawn with a dashed line to signify that it is not included as part of the solution (if it were 'greater than or equal to', the circle would be drawn with a solid line). We then apply shading to the graph: the area outside the dashed circle. Any point that falls within this shaded region is a solution to the inequality.

An algebraic inequality, like \(x^{2}+y^{2}>36\), expresses a relationship between two expressions that aren't necessarily equal but have a specific order— one is greater or lesser than the other. This inequality involves a comparison between the sum of the squares of two variables and a constant. The inequality symbol 'greater than' (>), 'less than' (<), 'greater than or equal to' (≥), or 'less than or equal to' (≤) dictate which parts of the graph are solutions.

In the case of the circle inequality we're considering, there is no equal sign involved, which implies that the edge of the circle itself isn't a part of the solution—only the area outside of it makes the cut. This is an important distinction and a common source of mistakes. Remembering to draw a dashed line for such inequalities can be a helpful reminder.

To graph any inequality, we need a reliable stage, and that's precisely what the Cartesian coordinate system provides us. Formed by two perpendicular lines named axes—the horizontal x-axis and vertical y-axis—this system allows us to locate any point in a two-dimensional space by using a pair of numbers called 'coordinates'.

The point of intersection of the axes is called the origin, and it's from here that all our plotting starts. When graphing inequalities like the circle inequality \(x^{2}+y^{2}>36\), we center our circle at the origin, because the x and y in the inequality have no added or subtracted constants, which would otherwise move the center away from (0,0). The radius is determined by the constant on the other side of the inequality, and it gives us the distance from the origin to any point on the circle's border.