Problem 51 - Algebra and Trigonometry

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} {(x-1)^{2}+(y+1)^{2}<25} \\ {(x-1)^{2}+(y+1)^{2} \geq 16} \end{array}\right. $$

Step-by-Step Solution

Answer

The solution set for the system of inequalities is the set of points that lie between the circles \((x-1)^{2}+(y+1)^{2}=25\) and \((x-1)^{2}+(y+1)^{2} =16\), excluding the boundary of the circle with radius 5 but including the boundary of the circle with radius 4. The system does not have 'No solution' in this scenario.

Step 1: Understand the equations

The inequalities \((x-1)^{2}+(y+1)^{2}<25\) and \((x-1)^{2}+(y+1)^{2} \geq 16\) each represent a circle with center at (1,-1). The number on the right side of each inequality gives the square of the radius of each circle.

Step 2: Graph the solution

First, draw a circle with center at point (1,-1) with radius \(\sqrt{25}\), which is 5, based on the inequality \((x-1)^{2}+(y+1)^{2}<25\). Make sure to use a dashed line for the circle because the inequality sign < doesn’t include the boundary line of the circle. This circle represents the points (x, y) that are strictly within a distance of 5 from the center (1,-1). Next, draw another circle with the same center, but with radius \(\sqrt{16}\), which is 4. In this case, draw a solid line for the circle because the inequality \((x-1)^{2}+(y+1)^{2} \geq 16\) includes the boundary line of the circle. This circle represents all points that are at least a distance of 4 from the center. The solution set is the set of points that satisfy both inequalities, which makes it the area between the two circles, including the boundary of the circle with radius 4, but not the boundary of the circle with radius 5.

Step 3: No Solution Condition

The system wouldn’t have a solution if the circles represented by the inequalities didn’t overlap, but in this case, since the two circles share the same center and have different radii, there will be an area common to both circles, thus the system has a solution.

Key Concepts

Inequalities in Algebra

Inequalities are foundational components in algebra that express a relationship where two values are not equal and one has a greater or lesser value than the other. They come in various forms including '<', '>', '\(\leq\)', and '\(\geq\)'. In the context of a system of inequalities, multiple inequalities are combined, and the solution consists of all points that satisfy all inequalities simultaneously.

When graphing a system of inequalities, we're mapping a region defined by these conditions onto a coordinate plane. Important aspects include deciding whether to use a dashed or solid line to indicate whether a border is included in the solution set and shading the correct area that represents all possible solutions. For instance, the inequality '\((x-1)^2 + (y+1)^2 < 25\)' indicates a region within the boundary and this is why the boundary is not included, hence the dashed line.

Circle Graphs

Circle graphs, or graphs of circular inequalities, display the set of points that satisfy an inequality involving a circular equation. The standard form of a circle's equation is '\((x-h)^2 + (y-k)^2 = r^2\)', where '\((h,k)\)' is the center of the circle and '\(r\)' is the radius.

When dealing with inequalities, the region represented can be inside the circle ('<'), outside the circle ('>'), or include the boundary ('\(\leq\)' or '\(\geq\)'). To graph these inequalities, one would draw the circle to represent the limits of the solution region and then shade the appropriate area. The region inside the circle is shaded when the inequality sign is less than '<', whereas for greater than '>', one shades the region outside the circle.

Solving Algebraic Systems

Solving a system of algebraic equations or inequalities involves finding the set of all possible values that satisfy all equations or inequalities in the system. When graphing the system, each equation or inequality is represented as a line, curve, or region. The solution for a system of inequalities is the overlap or intersection of these regions.

To accurately represent the solution to a system on a graph, one must pay attention to the inequality signs. The exercises involving the inequalities '\((x-1)^2 + (y+1)^2 < 25\)' and '\((x-1)^2 + (y+1)^2 \geq 16\)' require graphing two circles and identifying the area of intersection that satisfies both conditions without error. It's essential to use different line styles for strict ('<') and inclusive ('\(\geq\)') inequalities.

Radius and Center of a Circle

Understanding the significance of the radius and center of a circle is crucial when graphing circular inequalities. The center '\((h,k)\)' of the circle represents the fixed point from which all points on the perimeter are equidistant. The radius '\(r\)' is this distance. In a system of inequalities referring to circles, like the one given, the center remains the same while the radius changes, creating circles with different sizes but the same origin.

To graph inequalities involving circles correctly, first identify the center of the circle, and then use the radius to determine the extent of the circle. A varied radius can result in concentric circles, representing different regions in the coordinate plane. Each region corresponds to a different inequality, creating a visual map of where the solutions to a system of inequalities can lie. In the provided system, radii of 5 and 4 are used to create two concentric circles, visualizing the solution as a ring-shaped region between them.