Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

\(\left\\{\begin{array}{l}x^{2}+y^{2}>1 \\ x^{2}+y^{2}<4\end{array}\right.\)

Step-by-Step Solution

Answer

Points in the annular region between circles of radius 1 and 2.

Step 1: Interpret the Inequalities

The given inequalities describe circular regions on the coordinate plane. The inequality \(x^2 + y^2 > 1\) represents the region outside a circle centered at the origin with radius 1. The inequality \(x^2 + y^2 < 4\) represents the region inside a circle centered at the origin with radius 2.

Step 2: Identify the Shared Region

To find the solution to the system of inequalities, determine the region on the coordinate plane that satisfies both conditions. This means we are looking for the region that is outside the smaller circle (radius 1) and inside the larger circle (radius 2).

Step 3: Describe the Solution Set

The solution set is the annular (ring-shaped) region on the coordinate plane. It consists of all the points that are more than 1 unit but less than 2 units away from the origin.

Key Concepts

Understanding Inequalities

Inequalities are expressions that indicate a relationship between two values where one is smaller, larger, or simply not equal. Unlike equations which signify exact equality, inequalities allow for a range of possible values.
Imagine inequalities as a set of rules:

• The symbol "<" means less than.
• The symbol "<=" means less than or equal to.
• The symbol ">" means greater than.
• The symbol ">=" means greater than or equal to.

In our exercise, we have two inequalities:

• The first inequality, \(x^2 + y^2 > 1\), suggests values of \((x, y)\) that lie outside a circle centered at the origin with a radius 1.
• The second inequality, \(x^2 + y^2 < 4\), indicates conditions where\((x, y)\) are inside a circle with a larger radius of 2.

Motion on the Coordinate Plane

The coordinate plane serves as the foundational backdrop for graphing inequalities and understanding their solutions visually. It's essentially a two-dimensional grid, defined by the intersecting horizontal \((x)\) and vertical \((y)\)axes.
Understanding how points are plotted aids in visualizing these inequalities:

• Each point on the plane is represented by a coordinate pair \((x, y)\).
• The origin, \((0, 0)\), is where the two axes cross.

To determine where inequalities manifest on the plane, we check whether specific \((x, y)\) values satisfy the inequality conditions. By testing various points, you can plot lines and curves which delineate where the conditions of the inequalities hold true. In our task, we're looking for points that meet both inequality conditions, forming a special region on this coordinate setup.

Decoding with Circles

To tackle our problem involving inequalities, it's pivotal to recognize that circles come into play. A circle on the coordinate plane is defined by all points \((x, y)\) that maintain a constant distance from a central point, known as the radius.
For us, both inequalities are linked to circles centered at the origin, but with differing radii:

• The first inequality\(x^2 + y^2 > 1\) translates to all points outside a circle with a radius of 1.
• The second inequality \(x^2 + y^2 < 4\) speaks to points inside a circle with a radius of 2.

Visualizing these can assist in discerning the interconnected regions that satisfy both conditions, creating an annular pattern.

Defining the Solution Set

A solution set in systems of inequalities encompasses all points that satisfy every inequality in the system.
In essence, it is the collection of coordinates \((x, y)\) that keep both conditions in play.
Revisiting the problem, once we intertwine the inequalities, there's a clear picture:

• The points that fall outside the circle with radius 1 but inside the circle with radius 2.

This combination forms an annular region - akin to a doughnut shape - on the coordinate plane. It's an elegant solution showcasing how inequalities collaboratively unlock unique geometric formations.