Problem 24 - Precalculus: Functions and Graphs

Question

Sketch the graph of the system of Inequalities. $$\left\\{\begin{array}{l}x^{2}+y^{2}>1 \\\x^{2}+y^{2}<4\end{array}\right.$$

Step-by-Step Solution

Answer

The solution is the annular region between the circles of radius 1 and radius 2.

Step 1: Identify the Inequalities

The first inequality is \(x^2 + y^2 > 1\), which represents all the points outside the circle centered at the origin \((0,0)\) with radius 1. The second inequality is \(x^2 + y^2 < 4\), representing all the points inside the circle centered at the origin with radius 2.

Step 2: Interpret the System of Inequalities

For the system of inequalities, we need to find the region where both conditions are satisfied simultaneously. It means we are looking for the region that is outside the smaller circle \((x^2 + y^2 = 1)\) and inside the larger circle \((x^2 + y^2 = 4)\).

Step 3: Visualize the Solution

Sketch the two circles. The smaller circle with radius 1 is dashed to show it includes points that are not part of the solution. The larger circle with radius 2 is also dashed. The region that satisfies both inequalities is the ring-shaped area between these two circles.

Step 4: Shade the Solution Region

Shade the annular region between the two dashed circles. This shaded area represents the solution to the system \(x^2 + y^2 > 1\) and \(x^2 + y^2 < 4\), indicating all points lying between the two circles but not on the circles themselves.

Key Concepts

System of Inequalities

A system of inequalities consists of multiple inequalities that must be considered together. In this case, we have two inequalities involving circles:

• \(x^2 + y^2 > 1\) and
• \(x^2 + y^2 < 4\).

To solve the system, we need to find a set of points that satisfy both of these inequalities at the same time. Each inequality represents a condition on the possible points in the coordinate space. Ultimately, we are looking for values of \(x\) and \(y\) that satisfy both conditions simultaneously, forming what is known as the 'solution region.' Think of each inequality as setting a boundary, and the solution is where these boundaries intersect in a way that they meet both conditions.

Solution Region

The solution region in the context of a system of inequalities is the area where all inequalities in the system are true. For this problem, we identify the area where both \(x^2 + y^2 > 1\) and \(x^2 + y^2 < 4\) conditions hold simultaneously.
The first inequality \(x^2 + y^2 > 1\) tells us we need to look at all the points outside the circumference of a circle with radius 1, centered at the origin. However, the second inequality \(x^2 + y^2 < 4\) restricts us to points inside the circumference of a larger circle with radius 2.
When you examine both of these together, the solution region lies between these two circles. This means we have a ring-shaped area which includes all the points that are outside the smaller circle but inside the larger one.

Shaded Area

The shaded area is crucial for visualizing and conveying the solution to a system of inequalities. In graph problems, shading indicates where the solution lies.
For the given inequalities, \(x^2 + y^2 > 1\) and \(x^2 + y^2 < 4\), the solution is visually represented by a ring shape formed by two dashed circles. The dashed lines show that points on the circles themselves do not satisfy the inequalities because \(> \) and \(< \) are strict inequalities.
You shade the area between the dashed circles to show the solution region graphically. This shaded ring doesn't touch the actual circles, and it visually encodes the points that satisfy both inequalities. Drawing such diagrams can significantly help to understand which parts of the coordinate plane are included and excluded from the solution.

Circle Equations

Circle equations, such as those used here, show all the points equidistant from a center point. A standard circle equation is \(x^2 + y^2 = r^2\), where \(r\) is the radius, and \((0, 0)\) is the center if no other shifting terms are present.
In our exercise, the inequalities are based on such equations:

• \(x^2 + y^2 > 1\): Points outside a circle of radius 1
• \(x^2 + y^2 < 4\): Points inside a circle of radius 2

To express points outside or inside the circle rather than along it, the symbol \(eq\) changes to \(>\) or \(<\). These circle equations are essential for setting up the boundaries on the coordinate plane, around which we then determine our shaded (solution) region.