Problem 46 - College Algebra

Question

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

Step-by-Step Solution

Answer

The solution to the system of inequalities is the region that lies between the circle with radius 1 and the circle with radius 3.

Step 1: Determine the regions defined by each inequality

The inequality \(x^{2}+y^{2}>1\) represents the region outside the circle with center at the origin and radius 1. Likewise, the inequality \(x^{2}+y^{2}<9\) represents the region inside the circle with center at the origin and radius 3.

Step 2: Graph the regions

Plot the two circles on a 2-dimensional coordinate system. The first circle should have a radius of 1 and the second should have a radius of 3. Color the region inside the larger circle and outside the smaller circle.

Step 3: Identify the solution region

The solution to the system of inequalities is given by the region of the plane that satisfies both inequalities simultaneously. This region is the one that lies between both circles, which is the region of the plane that both inequalities have in common.

Key Concepts

Graphing Inequalities

Graphing inequalities lets you visualize where solutions exist on a plane. It involves shading specific regions on the graph that satisfy the conditions of the inequality. For an inequality like \(x^{2}+y^{2}>1\), you are looking at an area outside a circle with a radius of 1 in a coordinate system. The inequality suggests any point (x, y) where the distance from the origin, calculated using the formula \(\sqrt{x^2 + y^2}\), is greater than 1.
For the inequality \(x^{2}+y^{2}<9\), imagine a circle, but this time with a radius of 3. Any point (x, y) within this circle has a distance from the origin that is less than 3.
The graph of these inequalities involves drawing these circles and understanding which parts of the plane are included or excluded by these conditions. After drawing these circles, you identify and shade the correct regions: outside for the first inequality, inside for the second.

• The boundary line (or curve) of \(x^{2}+y^{2}=1\) is not included (shown with a dashed line).
• The circle \(x^{2}+y^{2}=9\) borders the part where points stop being solutions (shown with a solid line if it’s inclusive but generally dashed in these conditions to signify exclusion).

In graphing inequalities, always verify the solution area clearly with points from the region during the shading process.

Solution Set

A solution set in a system of inequalities consists of all the points that simultaneously satisfy all inequalities. This concept is crucial because it tells us exactly where the solutions to the inequalities exist.
In the exercise, you're dealing with two inequalities. The solution set lies in the region found between two circles:

• Start by identifying each circle's respective region.
• For \(x^{2}+y^{2}>1\), solutions are found outside the smaller circle.
• For \(x^{2}+y^{2}<9\), solutions are within the larger circle.

Thus, the solution set is not the full ring area but exclusively the area not touching or crossing either boundary line. This creates an annular (donut-shaped) region between the two circles. Every point in this area satisfies both inequalities, representing "the solution set." For clarity, choosing a test point within this region always helps to confirm it satisfies both conditions effectively.

Coordinate System

The coordinate system is a mathematical plane defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). It enables the plotting and visual representation of numbers and algebraic expressions such as inequalities.
In the example exercised, a coordinate system helps us draw and visualize two circles. These circles represent all points where the distance from the origin equals certain values, defined by their respective inequalities. Understanding this system is essential to manage the solutions clearly displayed here.
The origin, denoted by (0,0), is the center point where both axes intersect. For inequalities in the form of \(x^{2}+y^{2}\) representing circles, the origin acts as the circle's center. Knowing this helps determine where boundaries like "greater than" or "less than" are correctly interpreted on the plane.
Using the coordinate system, we define:

• The horizontal (x) axis measures distance left or right from the origin.
• The vertical (y) axis measures distance up or down from the origin.

Competency in interpreting points on this system ensures you can effectively graph inequalities and identify solution regions within this space. Remember that coordinate graphs often require a logical understanding of these axes to navigate solving mathematical problems effectively.