In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. x^2+y^2>1, x^2+y^2<16

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. In this context, the inequalities x² + y² > 9 and x² + y² < 25 define regions in the coordinate plane. The first inequality represents the area outside a circle with radius 3, while the second represents the area inside a circle with radius 5.

The equations x² + y² = r² represent circles centered at the origin (0,0) with radius r. When graphing inequalities involving these equations, the area inside or outside the circle is shaded based on the inequality sign. For example, x² + y² < 25 shades the area inside the circle of radius 5, while x² + y² > 9 shades the area outside the circle of radius 3.

The solution set of a system of inequalities is the region where the conditions of all inequalities are satisfied simultaneously. In this case, the solution set is the area between the two circles defined by the inequalities, where points satisfy both x² + y² > 9 and x² + y² < 25. This region represents all points that are more than 3 units away from the origin but less than 5 units away.