Problem 50 - Algebra and Trigonometry

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} {x^{2}+y^{2}>1} \\ {x^{2}+y^{2}<9} \end{array}\right. $$

Step-by-Step Solution

Answer

The solution set of the system of inequalities is the annulus or ring-shaped area between the circle of radius 1 and the circle of radius 3, both centered at the origin.

Step 1: Understanding the Inequalities

The inequality \(x^{2}+y^{2}>1\) represents all the points outside the circle of radius 1 centered at the origin in the xy-plane. Similarly, the inequality \(x^{2}+y^{2}<9\) represents all the points inside the circle of radius 3 (since \(\sqrt{9}=3\)) centered at the origin.

Step 2: A Graphical Representation

Draw two circles with radii 1 and 3 respectively, both centered at the origin. The area satisfied by \(x^{2}+y^{2}>1\) would be outside the smaller circle but inside the larger one because that’s the region common to both inequalities.

Step 3: Determining the Solution Set

The common region to both inequalities is the annulus between the two circles. This is the solution set of the system of inequalities.

Key Concepts

Graphing Inequalities

When it comes to understanding systems of inequalities, graphing them on a Cartesian plane is a crucial skill. In our exercise, you are confronted with two inequalities that represent conditions on the coordinates \(x, y\). Specifically, these inequalities are linked to circles with different radii. To graph such inequalities, you'll first sketch the boundary of the inequality. For example, the equation \(x^2 + y^2 = 1\) corresponds to a circle boundary, and the inequality \(x^2 + y^2 > 1\) includes all points outside this circle.

Similarly, \(x^2 + y^2 < 9\) involves all points inside the circle with a radius of 3. To graph an inequality, use a dashed or solid line for the boundary (solid if the boundary is included in the solution, dashed if not) and shade the region that satisfies the inequality. In our case, the area between the two circles (an annulus) will get shaded, indicating that any point falling within this shaded region satisfies both inequalities.

Circle Equations

Circle equations in the form \(x^2 + y^2 = r^2\) play a pivotal role in systems of inequalities, especially when the solutions are based on geometric shapes. The equation represents a circle centered at the origin (0, 0) on the Cartesian plane with a radius \(|r|\). The left-hand part of the equation, \(x^2 + y^2\), indicates the sum of squares of the distances from the x and y axes, which is constant for all points on the circumference of a circle. In the given exercise, we have two such circle equations with radii 1 and 3, representing two circles with different sizes.

Inequality Solution Set

The inequality solution set refers to the collection of all possible ordered pairs \(x, y\) that satisfy a given inequality or system of inequalities. In a graphical approach, this is the region where all the solutions literally 'live'. In our exercise, the solution set is not a single line or a simple zone but an annulus – it's the space between two circles on the Cartesian plane. This region represents all the \(x, y\) coordinates that meet the conditions imposed by the inequalities \(x^2 + y^2 > 1\) and \(x^2 + y^2 < 9\). It's important to clearly understand and visualize this set to grasp the full scope of possible solutions allowed by the system.

Cartesian Plane

The Cartesian plane is a two-dimensional surface formed by two perpendicular axes: the horizontal axis (x-axis) and the vertical axis (y-axis). Each point on this plane is determined by an ordered pair of numbers known as coordinates. The first number in the pair corresponds to the position along the x-axis, while the second number corresponds to the position along the y-axis. Graphing inequalities, circle equations, or any geometric figure all rely on this fundamental coordinate system. Understanding how to use the Cartesian plane effectively is therefore essential to solving many mathematical problems, including systems of inequalities that include geometric constraints like circles.