Problem 49 - College Algebra

Question

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} x^{2}+y^{2}>1 \\ x^{2}+y^{2}<16 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution is the set of all points in the 2-dimensional plane that lie between the circles of radii 1 and 4 centered at the origin, not including the boundary circles themselves.

Step 1: Understanding the Inequalities

The system of inequalities represents two circles with origin as the center. The first one \(x^2 + y^2 > 1\) signifies the area outside the circle of radius 1, i.e. all the points that lie further than one unit distance from the origin, but not on the circle of radius 1. The second one \(x^2 + y^2 < 16\) signifies the area inside the circle of radius 4 (square root of 16), i.e. all the points within the four units distance from the origin but not on the circle of radius 4.

Step 2: Graphing the Inequalities

On a coordinate plane, draw two circles. One centered at the origin with radius 1 and the other also centered at the origin but with a radius 4. Label them accordingly. As the boundary of the circles are not included in the solution, the circles should be drawn with dotted lines.

Step 3: Identifying the Solution Region

The solution to this system of inequalities is the region where both these conditions are satisfied. This area can be visually represented as the area lying between the circles of radius 1 and 4, excluding the circles themselves. Highlight this area on the graph.

Key Concepts

Understanding the Coordinate Plane

The coordinate plane is a two-dimensional surface where each point is determined by a pair of numbers, known as coordinates. Think of it as a vast sea with a grid, where each point is a unique location that can be found with its own 'address' given by the x (horizontal) and y (vertical) coordinates.

The horizontal axis is called the X-axis and the vertical axis is termed the Y-axis. Intersection of these two axes creates the origin, marked as point (0,0), which is the starting point for measuring distances along the axes. It's akin to the point where you'd start when giving someone directions on a map.
For graphing inequalities, we use shading to denote areas that contain solutions to an inequality. Understanding the coordinate plane is crucial because it's our canvas for presenting these solutions in a way that's visually intuitive and easily grasped.

Graphing Inequalities on a Coordinate Plane

Graphing inequalities is a bit like painting by numbers but with math rules. Unlike equations that have a single answer, inequalities showcase ranges of possible solutions. To graph a system of inequalities, we plot the region that satisfies all the inequalities involved.

Here's how it works: first, treat each inequality like an equation and draft its boundary. If the inequality is strict (like < or >), use a dotted line to draw the boundary, which implies that the points on the line are not part of the solution. Similarly, a solid line would mean the border includes the solutions (<= or >=). Then we select a test point not on the line, often the origin if it's not on the boundary, and see if it meets the inequality. If it does, shade in the side of the line where the test point lies; that's your solution area. In a system, this gets a bit more colorful - our solution set will be the area where the shading overlaps, as all conditions must be met simultaneously. It's like finding where different spotlights on a stage overlap, making that section brighter.

Circle Equations in the Coordinate Plane

Circle equations give us the 'rules' for drawing a circle in our grid-like sea, the coordinate plane. The standard form of a circle equation is \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center and \( r \) is the radius.

Think of \( h \) and \( k \) as the 'home address' for the center of the circle, and the radius \( r \) tells us how far we can go from home, forming a perfect loop. Plugging different points into the equation helps us trace out the circle's edge. When inequalities enter the picture, the circle's equation helps us identify the 'no-go' zones or 'free roam' areas for solutions. For instance, when we have \( x^2 + y^2 < r^2 \) it paints the inside of the circle as our play area, while \( x^2 + y^2 > r^2 \) marks off the territory outside as our playing field, excluding the boundary line itself. Understanding circle equations is key in figuring out not just where your solution can't go, but where it's free to wander - the confines and freedoms of your mathematical playground.