91Ó°ÊÓ

Graphing inequalities involves mapping out the areas of the plane that satisfy certain conditions. Here, we look at inequalities that define regions around circles. In our case, the inequalities are:

• \( x^2 + y^2 \le 36 \)
• \( x^2 + y^2 \ge 9 \)

These inequalities represent circles because the equation \( x^2 + y^2 = R^2 \) is the standard form for a circle centered at the origin with radius \( R \). Slightly different, inequalities can either indicate regions "inside" or "outside" a circle:

• The inequality \( \le \) depicts the region inside and including the boundary circle.
• The inequality \( \ge \) depicts the region outside and including the boundary circle.

To solve, you'll need to draw these circles on the graph, marking where the conditions of the inequalities are met. Shade the overlapping region that satisfies both inequalities. Then, identify critical intersection points where the circles meet the axes.

Circle equations give us vital information about a circle's position and size on a graph.

• A circle's standard equation is \( x^2 + y^2 = R^2 \), where \( R \) is the radius.
• It is centered at the origin \((0,0)\) when written like this without any additional terms.

For the inequality \( x^2 + y^2 \le 36 \), we have a circle with a radius \( R = 6 \).

• This means all points (x, y) are within or on this circle.
• Envision a boundary at 6 units from the origin in every direction.

Similarly, the inequality \( x^2 + y^2 \ge 9 \) corresponds to a circle with a radius \( R = 3 \).

• This helps us visualize a circle that contains all points starting at 3 units from the origin.
• Together with the first inequality, the solution set will appear as an annular (ring-shaped) region between these two circles.

Identifying the solution set in a system of inequalities involves determining the feasible region where the conditions from both inequalities are true.

• The problem consists of tube inequalities: one indicating all points from the origin up to 6 units and another from 3 units out.

In practical terms, the solution set is a ring-shaped area where these boundaries intersect:

• This region includes all points contained within the outer circle \( x^2 + y^2 = 36 \) and outside the inner circle \( x^2 + y^2 = 9 \).
• It involves the boundary of both circles as both inequalities are "equal to".
• Use the intersection coordinates, such as where each circle intersects x or y axis, to mark key vertices: \((6,0), (-6,0), (0,6), (0,-6)\) for the larger circle and \((3,0), (-3,0), (0,3), (0,-3)\) for the smaller circle.

Ultimately, these vertices and shaded regions graphically represent the solution set, helping to easily visualize satisfaction of the conditions in the inequalities.