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Inequalities are mathematical expressions that show the relative sizes or order of two values. They tell us how numbers compare to each other, whether they are less than (<), greater than (>), less than or equal to (\(leq\)), or greater than or equal to (\(geq\)). Unlike equations, which have one solution, inequalities typically have a range of solutions that satisfy the conditions set by the inequality.

For example, in the system of inequalities from the provided exercise, each inequality defines a set of points in the coordinate plane that fulfills its condition. The first inequality, \(x^2 + y^2 \leq 36\), tells us that the solution includes all points within and on the circle with radius 6 centered at the origin. Similarly, the second inequality, \(x^2 + y^2 \geq 9\), designates an area that includes all points outside of and on a circle with radius 3, again centered at the origin. These graphical representations turn abstract inequalities into visual regions where solutions exist.

The solution set of a system of inequalities comprises all the ordered pairs that satisfy all of the inequalities in the system at once. In the context of our exercise, it means finding the set of points that lie in both of the shaded regions described by the two inequalities.

In practice, to mark the solution set, we simultaneously consider the regions dictated by each inequality. We then overlap these regions to find where they intersect. In our exercise, this intersection is the annular region (the 'ring' or 'donut shape') between two circles. This shared area is where the points satisfy both the condition of being within or on the larger circle (\(x^2 + y^2 \leq 36\)) and outside or on the smaller circle (\(x^2 + y^2 \geq 9\)). To convey understanding, it's helpful to visualize or sketch the individual regions first before overlapping them, making it easier to identify the solution set.

Sketching the graph of an inequality on the coordinate plane translates the inequality from a symbolic expression to a visual representation, allowing students to better understand and solve the problem. Here are simplified steps to sketch the systems of inequalities like those in the exercise:

• Sketch the boundaries of each inequality, typically using solid lines for '\(leq\)' or '\(geq\)' and dashed lines for '<' or '>'.
• Decide which side of the boundary line defines the region that satisfies the inequality. This can often be determined by testing a point not on the line (the origin (0,0) is a convenient choice if it's not on the line).
• Shade the region of the plane that satisfies the inequality.
• For a system of inequalities, find the overlapping shaded regions from all inequalities. This common area represents the solution set for the system.

It's crucial to sketch accurately because the graph provides a visual proof of the solution set, and any error in sketching can misrepresent the actual solution.