Problem 51 - College Algebra

Question

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} (x-1)^{2}+(y+1)^{2}<25 \\ (x-1)^{2}+(y+1)^{2} \geq 16 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of inequalities represents the region between two circles centered at (1,-1), one with a radius of 5 (not included) and the other with a radius of 4 (included).

Step 1: Identify the Circles

The circles' equations are \((x-1)^{2}+(y+1)^{2}<25\) and \((x-1)^{2}+(y+1)^{2} \geq 16\). The center of both is (1,-1) and their radii are \( \sqrt{25} = 5 \) and \( \sqrt{16} = 4 \) respectively.

Step 2: Draw the Circles and the Region They Define

Now, draw both of these circles on the coordinate plane. The circle with a radius of 5 will be the larger one and the circle with a radius of 4 will be inside it. The area between these two circles is the solution to the system of inequalities. The inequality \( (x-1)^{2}+(y+1)^{2}<25 \) shows that the solution doesn't include the outer circle's circumference, and the inequality \( (x-1)^{2}+(y+1)^{2} \geq 16 \) shows the solution includes the inner circle's circumference. Shade this region to represent the solution set.

Key Concepts

Graphing Inequalities

Graphing inequalities on a coordinate plane involves more than just plotting simple lines. It requires understanding which regions these inequalities represent. To begin, transform each inequality into an equation to make it easier to visualize.
For example, consider \[(x-1)^2 + (y+1)^2 < 25\]This represents a circle with a radius of 5 and center at (1, -1). The inequality means points inside the circle are part of the solution, excluding the boundary. Now, let's consider solving the second inequality:\[(x-1)^2 + (y+1)^2 \geq 16\]This describes a boundary where the circle’s radius is 4, and the inequality includes all points outside or on this circle. Taken together, these represent a 'ring' of valid solutions between the two circles. The key lies in precise shading: exclude the outer edge, include the inner circle's perimeter.

Circles

Circles are essential in graphing, especially with systems involving inequalities.The equation \((x-h)^2 + (y-k)^2 = r^2\)represents a circle with:

• Center at \((h, k)\)
• Radius \\(r\)

In inequalities, such as:\((x-1)^2 + (y+1)^2 < 25\)You're dealing with every point whose distance from (1, -1) is less than 5 on the graph. The circle represents a boundary. For strict inequalities like "less than," the boundary isn't included in your solution set, thus it's often expressed as a dashed line. In contrast, a boundary for "greater than or equal to" like:\((x-1)^2 + (y+1)^2 \geq 16\)Does include the circle's perimeter.Understanding these nuances helps accurately determine the solution region.

Solution Set

The solution set is where all conditions from a system of inequalities are met. Finding it isn't just about plotting; it’s about interpreting the shaded regions, which translate into viable combinations of x and y.
Think of the area that satisfies all inequalities as the common ground where they overlap. In the case of circles, the solution set is a region bound by two circles. Here, it's the ring or "annulus" that lies between a larger circle of radius 5 and a smaller one with radius 4. The solution excludes the larger circle’s edge due to the "less than" constraint but includes the smaller circle’s circumference because "greater than or equal to" allows it.
The correct shading of this annular region provides a visual map of all point-solutions that meet both inequalities' demands.

Coordinate Plane

The coordinate plane is a fundamental tool for visually representing equations and inequalities. It comprises two dimensions:

• The x-axis (horizontal)
• The y-axis (vertical)

It's divided into four quadrants that allow plotting of positive and negative values for both x and y.
When dealing with inequalities, the plane helps see where solutions lie. Every point in the plane corresponds to a potential solution. As you graph the inequalities, you often shade certain regions. These shaded areas reveal where sets of inequalities simultaneously hold true. For our circle equations, the plane becomes a canvas depicting concentric circles to visualize valid points. The careful use of a coordinate plane lets you see, rather than just calculate, complex solutions.