Problem 77 - College Algebra with Modeling and Visualization

Question

Shade the solutions set to the system. $$ \begin{aligned} (x-1)^{2}+(y+1)^{2} &<4 \\ (x+1)^{2}+y^{2} &>1 \end{aligned} $$

Step-by-Step Solution

Answer

Shade the area inside the first circle that is also outside the second circle.

Step 1: Recognize the shapes

The inequalities represent circles. The first inequality \((x-1)^2 + (y+1)^2 < 4\) is a circle centered at \((1, -1)\) with radius 2, and the second inequality \((x+1)^2 + y^2 > 1\) is a circle centered at \((-1, 0)\) with radius 1.

Step 2: Identify regions

For the inequality \((x-1)^2 + (y+1)^2 < 4\), we consider the interior of the circle centered at \((1, -1)\) with a radius of 2. For the inequality \((x+1)^2 + y^2 > 1\), consider the exterior of the circle centered at \((-1, 0)\) with a radius of 1.

Step 3: Find the intersection

The solution set is where the shaded region from the first circle (inside \((x-1)^2 + (y+1)^2 < 4\)) overlaps with the unshaded area outside the second circle \((x+1)^2 + y^2 > 1\). So, we need to find the regions where these conditions meet.

Step 4: Graph the circles and shade the region

Draw both circles on a coordinate plane. The first circle is centered at \((1,-1)\) with radius 2, and the second circle is centered at \((-1,0)\) with radius 1. Shade the area inside the first circle and leave out the area inside the second circle to get the solution set.

Key Concepts

Understanding Systems of Inequalities

Systems of inequalities involve two or more inequalities that are solved together. Each inequality represents a region on a coordinate plane. When combined, these inequalities create a common area where all conditions are satisfied. To solve such systems, follow these steps:

• Graph each inequality separately. This involves drawing the region that satisfies each inequality on the coordinate plane.
• Identify the region where all inequalities intersect. This is known as the solution set.

In our exercise, we dealt with a system of two inequalities. The first inequality defines a region inside a circle, and the second describes a region outside another circle. Therefore, the solution set is the overlapping area where conditions from both inequalities are met. This intersection gives us the viable solutions to the system. The solution set can be quite complex depending on the shapes and positions of the regions described by each inequality.

Graphing Circles

Graphing circles on a coordinate plane is straightforward using the circle equation. 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. To understand how circles are graphed from inequalities, remember:

• A circle's equation depicts its boundary. In our exercise, inequalities define parts inside or outside these boundaries.
• For \((x-1)^2 + (y+1)^2 < 4\), the circle is centered at \((1, -1)\) with radius 2, signifying points within two units of this center.
• For \((x+1)^2 + y^2 > 1\), the circle is centered at \((-1, 0)\) with radius 1, indicating points outside one unit from this center.

Graphing these circles aids in visualizing solution regions. You would typically draw solid or dashed boundaries depending on whether the inequality symbol includes equality (\( \leq, \geq \)) or is strictly less than/greater than (\(<, >\)). A solid line indicates inclusion of the boundary, whereas a dashed line denotes exclusion.

Exploring Solution Sets in a System of Inequalities

The solution set to a system of inequalities in two variables is a collection of coordinates that satisfy all inequalities in the system simultaneously. Finding this solution set often involves looking at the regions on a graph where these inequalities overlap. Here's how to approach finding the solution set:

• Identify each inequality's corresponding region by graphing.
• The solution set is the overlapping region where all these individual inequality conditions hold true. In our exercise, this is where the points lie inside one circle and outside another simultaneously.
• Shade the appropriate region which reflects all conditions met by the inequalities. The shades and unshaded areas help in identifying overlaps visually.

In our example, the solution set was the area inside the first circle minus any part inside the second circle. This can sometimes create unusual shapes depending on the way regions intersect. Clearly visualizing each overlaid inequality helps find the correct solution area more effectively.